An in-depth exploration of current electricity, resistivity, and circuit analysis using Kirchhoff's Rules, designed for the calculus-based AP Physics C curriculum.
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
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Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
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24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
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24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
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24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
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24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
Real Battery a b \(\mathcal{E}\) r V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
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24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
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24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
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Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
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24.3 Resistance and Resistivity
When an electric field is established in a material, the resulting current density \(J\) depends on the material's internal structure. For many materials, \(J\) is linearly proportional to \(E\): \[ \vec{J} = \sigma \vec{E} \] where \(\sigma\) is the conductivity. The reciprocal of conductivity is resistivity (\(\rho = 1/\sigma\)): \[ \vec{E} = \rho \vec{J} \] This is the microscopic form of Ohm’s Law. Materials that obey this linear relationship are called "ohmic," while those that do not (such as diodes) are "non-ohmic."
Deriving Macroscopic Resistance
For a uniform conductor of length \(L\) and area \(A\), the potential difference \(V = EL\) and current \(I = JA\). Substituting these into the microscopic law:
\( \rho = \frac{E}{J} = \frac{V/L}{I/A} \implies V = I \left( \frac{\rho L}{A} \right) \)
We define the Resistance (\(R\)) as the proportionality constant between \(V\) and \(I\):
\( R = \frac{\rho L}{A} \)
Resistance is measured in Ohms (\(\Omega\)), where \(1 \Omega = 1 \text{ V/A}\).
Temperature Dependence
Resistivity is a function of temperature. In metals, as temperature increases, lattice ions vibrate more vigorously, increasing the probability of electron collisions and thus increasing \(\rho\). For most metals: \[ \rho(T) = \rho_0 [1 + \alpha(T - T_0)] \] where \(\alpha\) is the temperature coefficient of resistivity. In semiconductors, the opposite occurs; thermal energy excites more electrons into the conduction band, significantly increasing conductivity and dropping resistivity.
Conductors
Example: Copper (\(1.72 \times 10^{-8} \Omega \cdot \text{m}\)). High carrier density \(n\), low scattering.
As charge \(dQ\) moves through a potential difference \(V\), its potential energy changes by \(dU = dQ \cdot V\). The rate at which this energy is transferred or dissipated is defined as Electric Power (\(P\)): \[ P = \frac{dU}{dt} = \frac{dQ}{dt} V = IV \] This relationship is universal for any circuit element. For a purely resistive load, we apply Ohm's Law (\(V=IR\)) to derive two additional expressions: \[ P = I^2 R \quad \text{and} \quad P = \frac{V^2}{R} \] In resistors, this electrical energy is converted entirely into internal thermal energy, a process known as Joule heating.
24.5 Electromotive Force (EMF)
To maintain a steady current, a "charge pump" or source of EMF (\(\mathcal{E}\)) is required to move charges against an electric field. Examples include batteries, solar cells, and generators.
Real EMF sources possess an internal resistance (\(r\)). Consequently, the terminal voltage (\(V_{ab}\)) measured across a battery is less than its ideal EMF when a current is flowing: \[ V_{ab} = \mathcal{E} - Ir \]
Power in Real Sources
The total power produced by an EMF source is \(\mathcal{E}I\). However, the power delivered to the external circuit is \(V_{ab}I\). The difference, \(I^2 r\), is the power dissipated as heat inside the battery itself. This internal dissipation is why batteries become warm during heavy use.
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24.6 Series and Parallel Combinations
Complex networks can often be simplified by identifying clusters of resistors that share either the same current (series) or the same potential difference (parallel).
Series
Current must pass through each resistor sequentially. Total potential difference is the sum of individual drops.
\( R_{eq} = \sum_{i=1}^n R_i \)
Equivalent resistance is always larger than any individual resistor.
Parallel
Resistors are connected across the same two junctions. Total current is the sum of the branch currents.
Equivalent resistance is always smaller than the smallest individual resistor.
Conservation Laws in Action
These reduction rules are not arbitrary; they are derived from two fundamental conservation laws:
Conservation of Charge: The total current entering a junction must equal the total current leaving it. This leads to the parallel summation rule.
Conservation of Energy: The work done on a charge around any closed path in a static field must be zero. This leads to the series summation rule.
Conceptual Insight
Think of resistance as an obstacle. In series, the obstacles are encountered one after another, increasing the total difficulty. In parallel, the circuit provides multiple paths for the flow, effectively "widening" the road and reducing the total difficulty.
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24.7 Kirchhoff’s Rules
When a circuit cannot be reduced to simple series and parallel components (e.g., bridge circuits or multiple EMF sources in different branches), we utilize Kirchhoff’s Rules.
1 The Junction Rule
At any junction (node), the sum of currents entering must equal the sum of currents leaving.
\( \sum I_{in} = \sum I_{out} \)
2 The Loop Rule
The algebraic sum of the changes in potential around any closed circuit loop must be zero.
\( \sum \Delta V = 0 \)
Procedural Network Analysis
To solve a multiloop circuit, follow this systematic approach:
Label Currents: Assign a name and direction to the current in every branch. If your direction is "wrong," the final answer will simply be negative.
Apply Junction Rule: Write junction equations for all but one of the junctions in the circuit.
Apply Loop Rule: Choose independent loops and sum the potential changes. Use consistent sign conventions:
Traversing a resistor with current: \(\Delta V = -IR\)
Traversing a resistor against current: \(\Delta V = +IR\)
Traversing EMF from \(-\) to \(+\) terminal: \(\Delta V = +\mathcal{E}\)
Solve System: Solve the resulting linear system of equations.
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24.8 Electrical Measuring Instruments
Measuring a circuit's state always introduces a small perturbation. Understanding the ideal characteristics of meters is vital for accurate data collection.
The Ammeter
Measures current. Must be placed in series with the branch.
IDEAL: \(R_A = 0\)
Purpose: To avoid adding resistance to the circuit and reducing the current it intends to measure.
The Voltmeter
Measures potential difference. Must be placed in parallel with the element.
IDEAL: \(R_V = \infty\)
Purpose: To avoid drawing current away from the branch and changing the potential drop.
Chapter Synthesis
Field Foundation
Current is not just a scalar; it is the manifestation of a vector field \(\vec{J}\) driven by an electric field \(\vec{E}\) that exists within the conductor.
Energy Transformation
The resistor acts as a transducer, converting electrical potential energy into thermal kinetic energy through macroscopic "friction" (scattering).
Final Challenge
"If a circuit has no resistance, can a potential difference be maintained across it?"
Reflect on the definition of Superconductors for next lesson.
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24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
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24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
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24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
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24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) r V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is measured between points \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
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24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
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24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
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24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
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24.7 The Nature of Ohm's Law
In 1827, Georg Simon Ohm discovered that for many materials, the current is directly proportional to the applied voltage. This leads to the fundamental statement used in circuit analysis.
The Statement of Ohm's Law
"The current flowing through a conductor between two points is directly proportional to the potential difference across those points, provided physical conditions—specifically temperature—remain constant."
\( V = IR \)
Where V is potential difference (Volts), I is current (Amperes), and R is resistance (Ohms, \(\Omega\)).
It is critical to remember that Ohm's Law is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics like Gauss's Law.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve. Materials such as semiconductors or gases often exhibit non-linear behavior.
V I Ohmic (Linear) Non-Ohmic (e.g., Diode)
Figure 24.4: I-V characteristics comparison. Ohmic materials follow a straight line through the origin.
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Differential Resistance
For non-linear components, we define differential resistance (or dynamic resistance) as the derivative of the voltage with respect to current at a specific operating point: \[ r = \frac{dV}{dI} \]
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire of length \(L\) and cross-sectional area \(A\): \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is found by integrating the differential resistance elements along the length:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms. By definition (\(P = dU/dt = V dq/dt\)): \[ P = IV \] In a resistor, this energy is dissipated as heat (Joule heating): \[ P = I^2 R = \frac{V^2}{R} \]
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24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) + - r Terminal Voltage V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is the potential difference between terminals \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
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24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
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24.13 Matrix Methods
For complex networks with many loops, the number of algebraic substitutions becomes unmanageable. We utilize Mesh Analysis or Nodal Analysis to set up a matrix equation: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \] Where \(\mathbf{R}\) is the resistance matrix, \(\mathbf{I}\) is the vector of unknown currents, and \(\mathbf{V}\) is the vector of EMF sources.
This systematic approach is the foundation of modern circuit simulation software. In AP Physics C, you are expected to be able to set up these equations, even if you solve the final arithmetic with a calculator.
A Note on Symmetry
Many advanced physics problems involve high-symmetry grids (e.g., an infinite lattice of resistors or a cube of resistors). In these cases, symmetry arguments can often bypass the need for full matrix inversion by identifying points of equipotential.
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24.14 Electrical Instruments
To analyze a circuit experimentally, we must measure its properties using ammeters and voltmeters. However, the act of measurement itself can alter the circuit's behavior—a phenomenon known as the loading effect.
A
The Ammeter
How to connect:
Must be connected in SERIES. This forces the current to flow through the meter, allowing it to "count" the charge passing per second.
The Physics Logic (Why):
Ideal \(R_a \to 0\). If it had significant resistance, it would increase the loop's total resistance and decrease the very current it is trying to measure.
V
The Voltmeter
How to connect:
Must be connected in PARALLEL. This allows the meter to "bridge" two points and measure the potential difference between them.
The Physics Logic (Why):
Ideal \(R_v \to \infty\). If it had low resistance, it would provide an alternate path for current, drawing it away from the load and changing the potential drop.
A In Series R V In Parallel current I
Figure 24.6: Correct meter placement. Ammeters must be part of the loop path; voltmeters must bridge the component.
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Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Ammeters must be in series (\(R \to 0\)); Voltmeters in parallel (\(R \to \infty\)).
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
\( P = IV = I^2 R \)
"The ability to measure and model charge flow is the bridge between theoretical physics and the electronic reality of our modern age."
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Review Questions
1. Instantaneous Current
The charge passing through a certain cross-section of wire is given by \(Q(t) = 3t^2 - 4t + 5\), where \(Q\) is in Coulombs and \(t\) is in seconds.
(a) Derive an expression for the instantaneous current \(I(t)\).
(b) At what time \(t > 0\) is the current zero?
2. The Drude Model
Explain using the microscopic Drude model why an increase in the temperature of a metallic conductor typically leads to an increase in its resistivity. Reference the mean free time \(\tau\) in your answer.
3. Non-Uniform Current Density
A cylindrical wire of radius \(R\) carries a current density that varies with distance \(r\) from the central axis according to \(J(r) = J_0(1 - r/R)\). Calculate the total current \(I\) flowing through the wire in terms of \(J_0\) and \(R\).
4. Empirical Nature
Why is Ohm's Law considered an "empirical relationship" rather than a fundamental principle like the Conservation of Charge? Provide an example of a common electronic component that is non-Ohmic.
5. Variable Geometry
A resistor is shaped like a truncated cone (frustum) of length \(L\). The radius at one end is \(r_1\) and the radius at the other end is \(r_2\). If the material has resistivity \(\rho\), derive an expression for the total resistance \(R\) of the component.
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6. Power Dissipation
Two resistors, \(R_1\) and \(R_2\) (where \(R_1 > R_2\)), are connected to an ideal battery of voltage \(V\). In which configuration (series or parallel) will the total power dissipation of the circuit be greater? Justify your answer using the formula \(P = V^2/R\).
7. The Real Battery
A battery has an EMF of \(12.0 \text{ V}\) and an internal resistance of \(0.5 \text{ \(\Omega\)}\). When it is connected to a load resistor \(R\), the terminal voltage drops to \(10.5 \text{ V}\).
(a) Calculate the current flowing in the circuit.
(b) Find the value of the load resistance \(R\).
8. Network Analysis
A circuit contains two loops and three branches. State the minimum number of independent equations required to solve for the currents in each branch using Kirchhoff's Rules. Which rule (Junction or Loop) corresponds to the Conservation of Energy?
9. Loading Effects
A student mistakenly connects an ammeter in parallel with a resistor in a high-voltage circuit.
(a) Why is this dangerous for the ammeter?
(b) Why would a voltmeter connected in series with a load prevent the load from functioning properly?
10. Advanced Symmetry
Consider twelve identical resistors \(R\), each forming one edge of a cube. If a potential difference is applied across opposite corners of the cube, explain qualitatively why points of symmetry must be at the same potential. (Note: You do not need to solve for \(R_{eq}\) here).
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greater total power dissipation
7. The Real Battery
(a) Current: Using the terminal voltage formula \(V = \mathcal{E} - Ir\):
(b) Load Resistance: Using Ohm's Law across the load \(V = IR\):
\(10.5 = (3.0)R \Rightarrow R = 3.5 \text{ \(\Omega\)}\)
8. Network Analysis
To solve for 3 unknown branch currents, we need 3 independent equations. This is typically achieved using 1 Junction Rule equation and 2 Loop Rule equations.
The Loop Rule (\(\sum \Delta V = 0\)) is a statement of the Conservation of Energy, as it implies the net work done on a charge moving around a closed loop is zero.
9. Loading Effects
(a) Ammeter in Parallel: Ammeters have near-zero resistance. Connecting one in parallel creates a short circuit. The extremely high current flow can blow the meter's fuse or destroy its internal circuitry.
(b) Voltmeter in Series: Voltmeters have near-infinite resistance. If placed in series, the total circuit resistance becomes massive, causing the current to drop to nearly zero and preventing the load from receiving functional power.
10. Advanced Symmetry
In a cube of identical resistors, current enters at one corner and splits. Because all paths from that corner look identical, the current must divide equally (\(I/3\)) into the three adjacent edges.
The three nodes at the ends of those first edges are symmetric relative to the source and sink. Therefore, there is no physical reason for their potentials to differ. These equipotential points can be "shorted" together in a theoretical model without changing the circuit's behavior, greatly simplifying the calculation of equivalent resistance.
End of Solutions — Circuit Mastery Unit
Page 2
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) r V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is measured between points \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
Page 11
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) r V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is measured between points \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
Page 11
Important Conceptual Result:
decreases
Page 2
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) r V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is the potential difference between the output terminals.
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
Page 11
Parallel Derivation
Starting with the Junction Rule (\(I_T = \sum I_i\)) and applying Ohm's Law (\(I=V/R\)):
Physical Intuition: Resistors in parallel act like adding extra lanes to a highway. More lanes decrease the total obstruction.
Equivalent Circuit Diagram
From the perspective of nodes \(a\) and \(b\), the entire internal network behaves like a single "Black Box" macroscopic resistor.
a b R_eq
Page 3
24.13 Compound Network Reduction
In practice, most circuits are combinations of series and parallel sub-blocks. To solve these, we use a recursive reduction approach:
01
Identify Blocks
Locate small groups that are clearly strictly series or parallel.
02
Reduce Blocks
Calculate the \(R_{eq}\) for that sub-block and replace it with a single resistor symbol.
03
Re-Iterate
Draw the simplified circuit and repeat until only a single total resistance remains.
24.14 Power in Networks
The total power delivered (\(P_{total}\)) equals the sum of power dissipated by every component. Note the relationship with resistance:
\( P = IV \)
\( P = I^2 R = \frac{V^2}{R} \)
Series Analysis
Common current: use \(P = I^2 R\). Higher resistance = Higher power.
Parallel Analysis
Common voltage: use \(P = V^2 / R\). Lower resistance = Higher power.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 4
For Advanced Learners
Introduce the Superposition Theorem or Thevenin's Theorem as alternative methods for solving complex networks before the end of the unit.
drift velocity
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). In a time interval \(dt\), the carriers move a distance \(dx = v_d dt\). The total number of carriers that pass through a cross-section \(A\) is \(n(A dx) = n A v_d dt\). The total charge \(dQ\) is: \[ dQ = q(n A v_d dt) \] Dividing by \(dt\), we find the relationship between current and drift velocity: \[ I = n q A v_d \quad \implies \quad \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
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24.4 What is Resistance?
If electrons in a conductor are being pushed by an electric field, why don't they move at the speed of light? The answer lies in Resistance. Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
The Scattering Mechanism
Imagine an electron trying to navigate through a crowded forest (the crystal lattice of a metal). Even though there is a steady wind (the electric field) pushing it in one direction, the electron constantly bumps into trees (ions).
Microscopic Scattering Model
Collision Point Mean Free Path (\(\lambda\))
Figure 24.2: The "pinball" model of resistance. Electrons accelerate in the electric field but lose their kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat.
Resistance is not just a nuisance; it is a fundamental property of how matter interacts with energy. Every collision transfers some of the electron's kinetic energy to the lattice ions, causing them to vibrate more—this is the physical origin of Joule Heating.
Page 3 of 12
24.5 The Drude Model: Deriving Ohm's Law
Paul Drude proposed a classical model in 1900 to derive the relationship between current and electric field. Let \(\tau\) be the mean free time—the average time between collisions.
Between collisions, an electron experiences an acceleration \(a = F/m = eE/m\). The drift velocity is the average velocity gained between collisions: \[ v_d = a \tau = \left( \frac{eE}{m} \right) \tau \] Substituting this into our expression for current density (\(J = n e v_d\)): \[ J = n e \left( \frac{e E \tau}{m} \right) = \left( \frac{n e^2 \tau}{m} \right) E \]
The Microscopic Ohm's Law
We define conductivity (\(\sigma\)) as the constant of proportionality between \(J\) and \(E\):
\( \vec{J} = \sigma \vec{E} \)
Where, from the Drude model:
\( \sigma = \frac{n e^2 \tau}{m} \)
This derivation is significant because it connects macroscopic conductivity to microscopic constants: the density of charge carriers (\(n\)), the charge of an electron (\(e\)), the mass of an electron (\(m\)), and the material's internal timing (\(\tau\)).
The reciprocal of conductivity is resistivity (\(\rho\)): \[ \rho = \frac{1}{\sigma} = \frac{m}{n e^2 \tau} \] Resistivity is a material-dependent property that measures how strongly a substance opposes the flow of electric current.
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24.6 Resistance and Geometry
While resistivity is a property of the material itself, Resistance (\(R\)) depends on both the material and its physical dimensions. For a uniform conductor with length \(L\) and cross-sectional area \(A\): \[ R = \rho \frac{L}{A} \] This relationship makes intuitive sense: a longer wire (\(L\)) provides more opportunities for collisions, increasing resistance. A wider wire (\(A\)) provides more paths for electrons, decreasing resistance.
Calculus Application: Variable Cross-Sections
If the resistivity or the cross-sectional area is not constant along the length of the conductor, we must use the differential form of the resistance formula: \[ dR = \rho(x) \frac{dx}{A(x)} \] The total resistance is found by integrating along the length: \[ R = \int_0^L \frac{\rho(x)}{A(x)} dx \]
Ohm's Law: The Macroscopic Version
By combining the physical definition of resistance with the microscopic form of Ohm's Law, we arrive at the familiar macroscopic version: \[ V = IR \] An ohmic material is one where \(R\) is constant regardless of the applied voltage \(V\). In many electronic devices, such as diodes or transistors, the relationship is non-linear—these are non-ohmic devices.
Page 5 of 12
24.7 Energy and Power in Circuits
As a charge \(dQ\) moves through a potential difference \(V\), the electric field does work on it. The change in potential energy is \(dU = V dQ\). Power (\(P\)) is the rate at which this work is done: \[ P = \frac{dU}{dt} = V \frac{dQ}{dt} = IV \] This equation (\(P = IV\)) is universal—it applies to batteries, motors, resistors, and lightbulbs.
Power Dissipation in Resistors
In a resistor, the work done by the field is converted entirely into thermal energy via collisions (Joule heating). Substituting Ohm's Law into the power formula gives: \[ P = I^2 R = \frac{V^2}{R} \] The \(I^2 R\) form is particularly useful for analyzing power loss in transmission lines, while the \(V^2/R\) form is useful for analyzing devices connected to a constant voltage source (like household appliances).
Internal Resistance of EMF Sources
A real battery is not an ideal EMF source. It has its own internal resistance (\(r\)). When the battery delivers a current \(I\), there is a potential drop \(Ir\) inside the battery. The terminal voltage (\(V\))—what the rest of the circuit actually sees—is: \[ V = \mathcal{E} - Ir \]
Page 6 of 12
24.8 Combining Resistors
Resistors can be combined into networks to achieve specific total resistances. These combinations are governed by the conservation of charge and energy.
1
Series Combination
In series, the current through each resistor is the same. The total potential difference is the sum of individual drops.
\( R_{eq} = \sum R_i \)
"The path is one, the resistance grows."
2
Parallel Combination
In parallel, the potential difference across each resistor is the same. The total current is the sum of branch currents.
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
"Many paths shared, the resistance falls."
Physical Interpretation
Think of series resistors as extending the length \(L\) of a conductor—hence \(R\) increases. Think of parallel resistors as increasing the cross-sectional area \(A\) of the total flow path—hence \(R\) decreases.
Page 7 of 12
24.9 Kirchhoff's Rules
Many complex circuits cannot be reduced using simple series and parallel rules. For these "networks," we apply two fundamental principles first formalized by Gustav Kirchhoff.
The Junction Rule (Node Rule)
The algebraic sum of the currents into any junction is zero. This is a direct consequence of the Conservation of Charge.
\( \sum I_{in} = \sum I_{out} \)
The Loop Rule
The algebraic sum of the potential differences around any closed loop must be zero. This is a consequence of the Conservation of Energy in a conservative electric field.
\( \sum \Delta V = 0 \)
The loop rule is essentially the circuit equivalent of saying: "If you walk in a circle in a hilly landscape and return to your starting point, your net change in altitude must be zero."
Page 8 of 12
Applying the Loop Rule: Sign Conventions
The most common errors in circuit analysis are sign errors. To use the Loop Rule, you must first choose a direction to traverse the loop (clockwise or counter-clockwise).
Resistor Traverse
With current: Potential drops.
\(\Delta V = -IR\)
Against current: Potential rises.
\(\Delta V = +IR\)
EMF Traverse
From \(-\) to \(+\) terminal: Potential rises.
\(\Delta V = +\mathcal{E}\)
From \(+\) to \(-\) terminal: Potential drops.
\(\Delta V = -\mathcal{E}\)
When setting up equations for a multiloop circuit with \(N\) junctions, you need \(N-1\) independent junction equations and enough loop equations to match the number of unknown branch currents.
Page 9 of 12
24.10 Systematic Analysis: The Matrix Method
For advanced problems, Kirchhoff's rules lead to a system of linear equations. Consider a 3-branch circuit with currents \(I_1, I_2, I_3\). The equations might look like: \[ \begin{cases} I_1 - I_2 - I_3 = 0 \\ R_1 I_1 + R_3 I_3 = \mathcal{E}_1 \\ R_2 I_2 - R_3 I_3 = -\mathcal{E}_2 \end{cases} \] In matrix form (\(AI = B\)): \[ \begin{bmatrix} 1 & -1 & -1 \\ R_1 & 0 & R_3 \\ 0 & R_2 & -R_3 \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix} = \begin{bmatrix} 0 \\ \mathcal{E}_1 \\ -\mathcal{E}_2 \end{bmatrix} \]
This systematic approach is essential for computer modeling of circuits (e.g., SPICE simulations). While manual algebraic substitution works for simple systems, matrix methods provide a robust framework for any arbitrary network topology.
Page 10 of 12
24.11 Ammeters and Voltmeters
All measurements in a circuit involve adding a device that itself has resistance. This "loading effect" must be understood to minimize measurement error.
Ammeters
Measures current; must be in series.
Requirement:
Extremely low resistance (\(R_A \approx 0\)) so it does not significantly decrease the branch current.
Voltmeters
Measures potential; must be in parallel.
Requirement:
Extremely high resistance (\(R_V \approx \infty\)) so it does not draw significant current away from the component.
The Potentiometer and Bridge Circuits
Advanced measurement techniques, such as the Wheatstone Bridge, use a "null method" to measure resistance with extreme precision. By balancing the bridge until no current flows through the central galvanometer, the measurement becomes independent of the meter's own internal resistance.
Page 11 of 12
Chapter Synthesis
Foundational Physics
Micro-to-Macro: How the chaotic motion of \(10^{28}\) electrons per \(m^3\) results in a steady, predictable current.
Resistance: Not just a constant, but a measure of scattering frequency and lattice density.
Systemic Principles
Energy & Charge: The twin pillars of Kirchhoff's Rules (\(\sum I = 0, \sum V = 0\)).
Thermodynamics: The inevitable conversion of ordered electrical work into disordered thermal heat (\(I^2 R\)).
\( \vec{J} = \sigma \vec{E} \)
Field-Current Link
\( R = \int \frac{\rho(x) dx}{A(x)} \)
Geometry Definition
\( P = IV \)
Energy Transfer Rate
"The ability to control the flow of charge is what transformed physics from a philosophical curiosity into the engine of modern civilization."
Page 12 of 12
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery + - \(\mathcal{E}\) r Terminal Voltage V
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is measured across nodes \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
Page 11
Resultant
\( R_{eq} = R_1 + R_2 + \dots \)
Resistance increases as more paths are blocked.
Resistors in Parallel
Conservation Rules
Voltage is Constant
Current is Additive
Resultant
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
"Adding paths decreases total resistance."
R₁ R₂
Kirchhoff's Laws of Network Analysis
Junction Rule
"Conservation of Charge"
\( \sum I_{in} = \sum I_{out} \)
Loop Rule
"Conservation of Energy"
\( \sum \Delta V = 0 \)
Crucial for solving complex, non-reducible circuits via systems of equations.
Scaling to Complexity
When circuit networks involve multiple loops and junctions, manual substitution becomes inefficient. We utilize Mesh Analysis to structure the physics into linear algebra.
The Matrix Form
\( \mathbf{V} = \mathbf{R} \cdot \mathbf{I} \)
Symmetry Shortcut
Theoretical Mastery
Advanced problems (like the infinite resistor grid or resistor cube) are solved using Symmetry Arguments to identify points of Equipotential.
If two points are symmetric relative to the source and sink, their potential difference is zero. You can "short" or "cut" these connections to simplify the topology.
Key Takeaways
01
Current is defined by charge flux; current density connects to carrier micro-mechanics.
02
Resistance derives from lattice scattering (Drude Model) and depends on material resistivity and geometry.
03
Network reduction and Kirchhoff's Rules provide a systematic framework for any DC circuit.
"Circuits are the architecture of electrical energy flow."
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) + - r Terminal Voltage V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is measured across the output nodes.
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
Page 11
The "Black Box" Concept
From the battery's perspective, it does not "see" the internal arrangement of resistors. It only "sees" the total load. By calculating \(R_{eq}\), we can determine the Total Current delivery:
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is always the sum of individual power dissipation.
Series Power
\( P = I^2 R \)
Parallel Power
\( P = V^2 / R \)
Page 3
Result: Total resistance is always less than any individual branch.
The "Black Box" Model
From the perspective of the source terminals, the entire internal arrangement of three resistors behaves like a single lumped component. We use this to solve for current:
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Lumped Model Representation
Power in Multi-Resistor Networks
Conservation of Energy dictates that Total Power Delivered equals the sum of individual dissipations:
Series Power Focus
\( P_{total} = \sum I^2 R_i \)
Larger R = Larger P drop.
Parallel Power Focus
\( P_{total} = \sum \frac{V^2}{R_i} \)
Smaller R = Larger P drop.
Page 3
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) + - r Terminal Voltage V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is the potential difference between terminals \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks, we utilize systematic Mesh Analysis: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \]
24.14 Electrical Instruments
Ammeter
In series. Ideal \(R \to 0\).
Voltmeter
In parallel. Ideal \(R \to \infty\).
The Wheatstone Bridge
A balanced bridge allows measuring unknown resistance \(R_x\) with high precision.
Page 10
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
Page 11
Total resistance is always less than any individual branch.
The "Black Box" Model
From the perspective of the source terminals, the entire internal arrangement of three resistors behaves like a single lumped component. This simplification allows us to find total current delivery:
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is always the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R dominates power.
Parallel Power
\( P_T = V^2 (\sum \frac{1}{R_i}) \)
Lower R dominates power.
Page 3
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) + - r Terminal Voltage V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is the potential difference between terminals \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks with many loops, the number of algebraic substitutions becomes unmanageable. We utilize Mesh Analysis or Nodal Analysis to set up a matrix equation: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \] Where \(\mathbf{R}\) is the resistance matrix, \(\mathbf{I}\) is the vector of unknown currents, and \(\mathbf{V}\) is the vector of EMF sources.
This systematic approach is the foundation of modern circuit simulation software. In AP Physics C, you are expected to be able to set up these equations, even if you solve the final arithmetic with a calculator.
A Note on Symmetry
Many advanced physics problems involve high-symmetry grids (e.g., an infinite lattice of resistors or a cube of resistors). In these cases, symmetry arguments can often bypass the need for full matrix inversion by identifying points of equipotential.
Page 10
24.14 Electrical Instruments
To analyze a circuit experimentally, we must measure its properties using ammeters and voltmeters. However, the act of measurement itself can alter the circuit's behavior—a phenomenon known as the loading effect.
The Ammeter
Function: Measures the electric current (\(I\)) flowing through a branch.
Connection: Must be connected in series with the component being measured. This ensures that the same charge carriers passing through the load also pass through the meter.
Ideal Properties: To avoid changing the current it measures, an ideal ammeter has zero internal resistance (\(R_a \to 0\)). A real ammeter with finite resistance adds to the total resistance of the loop, thereby decreasing the actual current.
The Voltmeter
Function: Measures the potential difference (\(\Delta V\)) between two points.
Connection: Must be connected in parallel across the component. This ensures the meter experiences the same electric potential difference as the load.
Ideal Properties: To avoid drawing current away from the circuit, an ideal voltmeter has infinite internal resistance (\(R_v \to \infty\)). A real voltmeter draws a small amount of current, which reduces the current through the load and changes the measured voltage.
A In Series R V In Parallel
Figure 24.6: Proper connection of meters. The ammeter intercepts the current flow, while the voltmeter bridges the potential difference across the load.
Page 11
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Ammeters must be in series (\(R \to 0\)); Voltmeters in parallel (\(R \to \infty\)).
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
\( P = IV = I^2 R \)
Page 12
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) + - r Terminal Voltage V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is the potential difference between terminals \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks with many loops, the number of algebraic substitutions becomes unmanageable. We utilize Mesh Analysis or Nodal Analysis to set up a matrix equation: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \] Where \(\mathbf{R}\) is the resistance matrix, \(\mathbf{I}\) is the vector of unknown currents, and \(\mathbf{V}\) is the vector of EMF sources.
This systematic approach is the foundation of modern circuit simulation software. In AP Physics C, you are expected to be able to set up these equations, even if you solve the final arithmetic with a calculator.
A Note on Symmetry
Many advanced physics problems involve high-symmetry grids (e.g., an infinite lattice of resistors or a cube of resistors). In these cases, symmetry arguments can often bypass the need for full matrix inversion by identifying points of equipotential.
Page 10
24.14 Electrical Instruments
To analyze a circuit experimentally, we must measure its properties using ammeters and voltmeters. However, the act of measurement itself can alter the circuit's behavior—a phenomenon known as the loading effect.
A
The Ammeter
How to connect:
Must be connected in SERIES. This forces the current to flow through the meter, allowing it to "count" the charge passing per second.
Why:
Ideal \(R_a \to 0\). If it had significant resistance, it would increase the loop's total resistance and decrease the very current it is trying to measure.
V
The Voltmeter
How to connect:
Must be connected in PARALLEL. This allows the meter to "bridge" two points and measure the potential difference between them.
Why:
Ideal \(R_v \to \infty\). If it had low resistance, it would provide an alternate path for current, drawing it away from the load and changing the potential drop.
A SERIES R V PARALLEL current flow
Figure 24.6: Correct meter placement. The ammeter measures current "through" the circuit path; the voltmeter measures potential "across" the load.
Page 11
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Ammeters must be in series (\(R \to 0\)); Voltmeters in parallel (\(R \to \infty\)).
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
\( P = IV = I^2 R \)
"The ability to measure and model charge flow is the bridge between theoretical physics and the electronic reality of our modern age."
Page 12
Page 2
24.3 Current Density
Current is a macroscopic scalar quantity. To describe the flow of charge at a specific point in space, we define the current density (\(\vec{J}\)), which is a vector quantity. The magnitude of \(\vec{J}\) is the current per unit cross-sectional area.
The Flux Definition of Current
The net current through a surface is the flux of the current density vector:
\( I = \int \vec{J} \cdot d\vec{A} \)
For a uniform current flowing through an area \(A\) perpendicular to the flow, \(J = I/A\). In SI units, current density is measured in Amperes per square meter (\(A/m^2\)).
24.4 The Microscopic Model of Conduction
To understand what drives current, we must look at the microscopic level. In a conductor, "free" electrons move randomly due to thermal energy at speeds of approximately \(10^6 \text{ m/s}\). This motion is chaotic and averages to zero net displacement over time.
When an external electric field (\(\vec{E}\)) is applied, it exerts a force \(\vec{F} = -e\vec{E}\) on each electron. This force causes the electrons to accelerate. However, they do not accelerate indefinitely; they constantly collide with the stationary ions of the metal lattice.
Page 3
Drift Velocity (\(v_d\))
The result of collisions is a slow, net motion in the direction opposite to the electric field. This average velocity is called the drift velocity.
Consider a conductor with \(n\) charge carriers per unit volume, each with charge \(q\). The relationship between current density and drift velocity is: \[ \vec{J} = n q \vec{v}_d \] In metals, \(q = -e\), so \(\vec{J} = -n e \vec{v}_d\). Note that because \(e\) is negative, \(\vec{J}\) and \(\vec{v}_d\) point in opposite directions.
24.5 What is Resistance?
Resistance is the macroscopic manifestation of microscopic collisions between charge carriers and the material's internal structure.
Microscopic Scattering Model
Collision Point
Figure 24.3: Electrons lose kinetic energy and momentum during collisions with lattice ions. This energy is transferred to the lattice as heat (Joule Heating).
Page 4
24.6 The Drude Model
Paul Drude derived the relationship between current and electric field by assuming electrons gain a drift velocity between collisions. Let \(\tau\) be the mean free time.
The drift velocity is \(v_d = (e E / m) \tau\). Substituting this into \(\vec{J} = n e \vec{v}_d\): \[ \vec{J} = \left( \frac{n e^2 \tau}{m} \right) \vec{E} \]
Microscopic Ohm's Law
Defining Resistivity (\(\rho\)):
\( \vec{E} = \rho \vec{J} \)
Resistivity from microscopic constants:
\( \rho = \frac{m}{n e^2 \tau} \)
This derivation connects macroscopic resistivity to microscopic constants. Resistivity depends on carrier density (\(n\)) and the frequency of collisions (\(1/\tau\)).
Page 5
24.7 The Nature of Ohm's Law
Ohm's Law (\(V=IR\)) is an empirical relationship—it is a descriptive property of materials rather than a fundamental law of physics.
Ohmic vs. Non-Ohmic Materials
A material is Ohmic if its resistance remains constant over a range of voltages, leading to a linear \(I\)-\(V\) curve.
V I Ohmic Diode
Figure 24.4: I-V characteristics for linear and non-linear components.
Differential Resistance
For non-linear components, we define differential resistance as the derivative: \[ r = \frac{dV}{dI} \]
Page 6
24.8 Resistance and Geometry
Resistance (\(R\)) is an extrinsic property that depends on the geometry of the conductor. For a uniform wire: \[ R = \rho \frac{L}{A} \]
Integral Form for Variable Geometry
For non-uniform cross-sections \(A(x)\), the total resistance is:
\( R = \int_0^L \frac{\rho}{A(x)} dx \)
24.9 Energy and Power
Power (\(P\)) is the rate at which electric potential energy is converted into other forms: \[ P = IV \] In a resistor, this energy is dissipated as heat: \[ P = I^2 R = \frac{V^2}{R} \]
Page 7
24.10 Internal Resistance
Every real-world EMF source, such as a chemical battery or a generator, has some inherent internal resistance (\(r\)). This resistance arises from the physical materials and chemical processes inside the source.
a b Real Battery Model \(\mathcal{E}\) + - r Terminal Voltage V = \(\mathcal{E}\) - Ir
Figure 24.5: A real battery modeled as an ideal EMF \(\mathcal{E}\) in series with an internal resistance \(r\). The terminal voltage \(V\) is the potential difference between terminals \(a\) and \(b\).
When the battery delivers a current \(I\), a potential drop \(Ir\) occurs internally. Consequently, the terminal voltage (\(V\)), or the voltage available to the external circuit, is less than the ideal EMF: \[ V = \mathcal{E} - Ir \]
Page 8
24.11 Network Reduction
Series
\( R_{eq} = \sum R_i \)
Parallel
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_i} \)
24.12 Kirchhoff's Rules
Kirchhoff's Rules are essential for analyzing complex networks.
The Junction Rule
\( \sum I_{in} = \sum I_{out} \)
Conservation of Charge
The Loop Rule
\( \sum \Delta V = 0 \)
Conservation of Energy
Page 9
24.13 Matrix Methods
For complex networks with many loops, the number of algebraic substitutions becomes unmanageable. We utilize Mesh Analysis or Nodal Analysis to set up a matrix equation: \[ \mathbf{R} \cdot \mathbf{I} = \mathbf{V} \] Where \(\mathbf{R}\) is the resistance matrix, \(\mathbf{I}\) is the vector of unknown currents, and \(\mathbf{V}\) is the vector of EMF sources.
This systematic approach is the foundation of modern circuit simulation software. In AP Physics C, you are expected to be able to set up these equations, even if you solve the final arithmetic with a calculator.
A Note on Symmetry
Many advanced physics problems involve high-symmetry grids (e.g., an infinite lattice of resistors or a cube of resistors). In these cases, symmetry arguments can often bypass the need for full matrix inversion by identifying points of equipotential.
Page 10
24.14 Electrical Instruments
To analyze a circuit experimentally, we must measure its properties using ammeters and voltmeters. However, the act of measurement itself can alter the circuit's behavior—a phenomenon known as the loading effect.
A
The Ammeter
How to connect:
Must be connected in SERIES. This forces the current to flow through the meter, allowing it to "count" the charge passing per second.
The Physics Logic (Why):
Ideal \(R_a \to 0\). If it had significant resistance, it would increase the loop's total resistance and decrease the very current it is trying to measure.
V
The Voltmeter
How to connect:
Must be connected in PARALLEL. This allows the meter to "bridge" two points and measure the potential difference between them.
The Physics Logic (Why):
Ideal \(R_v \to \infty\). If it had low resistance, it would provide an alternate path for current, drawing it away from the load and changing the potential drop.
A In Series R V In Parallel current I
Figure 24.6: Correct meter placement. Ammeters must be part of the loop path; voltmeters must bridge the component.
Page 11
Chapter Summary
Foundational Concepts
Closed circuits allow flow; open circuits block it.
Ohm's Law is an empirical description of carrier equilibrium.
Ammeters must be in series (\(R \to 0\)); Voltmeters in parallel (\(R \to \infty\)).
Master Equations
\( I = \int \vec{J} \cdot d\vec{A} \)
\( \vec{E} = \rho \vec{J} \)
\( \sum \Delta V = 0 \)
\( P = IV = I^2 R \)
"The ability to measure and model charge flow is the bridge between theoretical physics and the electronic reality of our modern age."
Page 12
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
The "Black Box" Model
From the battery's perspective, the three resistors behave as a single lumped component. We use this simplified model to solve for the total circuit current:
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
\( I_T = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total power delivered (\(P_T = I_T \mathcal{E}\)) is the scalar sum of the power dissipated by each individual component (\(\sum P_i\)).
Series Power
\( P_T = I^2 (R_1 + R_2 + R_3) \)
Higher R = Higher P drop.
Parallel Power
\( P_T = V^2 \sum \frac{1}{R_i} \)
Lower R = Higher P drop.
Page 3
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Unit Summary: Power and Resistance
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual resistor in the network.
Series Power Dominance
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Power Dominance
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
"Theoretical circuit mastery lies in recognizing that conservation laws dictate branch behavior, while equivalence simplifies the system as a whole."
Page 3
Black Box
lumped parameter model
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total electrical power provided (\(P = I \mathcal{E}\)) must equal the sum of the power dissipated as heat (\(\sum I^2 R_i\)) by every individual component.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Black Box
lumped parameter model
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total electrical power provided (\(P = I \mathcal{E}\)) must equal the sum of the power dissipated by every individual component.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
Total electrical power provided (\(P = I \mathcal{E}\)) must equal the sum of the power dissipated as heat (\(\sum I^2 R_i\)) by every individual component.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Black Box
lumped parameter model
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Black Box
lumped parameter model
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Black Box
lumped parameter model
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Black Box
lumped parameter model
Lumped Modeling
Reducing a network to its equivalent resistance (\(R_{eq}\)) yields the Total Current Delivery:
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
Page 3
Total Current Delivery
\( I_{total} = \frac{\mathcal{E}}{R_{eq}} \)
R_eq
Equivalent System Representation
Network Power Dissipation
The Conservation of Energy requires that the total electrical power provided by the source equals the sum of the power dissipated as heat by every individual load.
Series Dissipation
\( P_{total} = \sum I^2 R_i \)
The largest resistor dissipates the most energy.
Parallel Dissipation
\( P_{total} = \sum \frac{V^2}{R_i} \)
The smallest resistor dissipates the most energy.
Summary Result
"Adding a resistor in series increases total resistance and decreases total current. Adding a resistor in parallel decreases total resistance and increases total current."
