An in-depth exploration of electromagnetism focusing on the relationship between electric currents and magnetic fields, featuring the Biot-Savart Law, Ampere's Law, and the Lorentz Force.
Label B: Field lines are always perpendicular to the current vector.
Label C: Field strength is uniform at a fixed distance \(r\).
III. Ampere's Law: Solving for Field Magnitude
In AP Physics C, we use **Ampere's Law** to determine the magnitude of the magnetic field \(B\). It states that the circulation of the magnetic field around any closed loop is proportional to the current passing through it.
Note: Observe how \(B\) decreases linearly with distance. This \(1/r\) relationship is characteristic of infinite line sources.
IV. Interaction: The Lorentz Force Law
Magnetic fields exert forces on electric charges, but only if they are **moving**. A stationary charge in a magnetic field experiences zero magnetic force. This interaction is the foundation of electric motors and particle accelerators.
Force on a Moving Charge
\[ \vec{F}_B = q(\vec{v} \times \vec{B}) \]
The force is always perpendicular to both the velocity (\(\vec{v}\)) and the magnetic field (\(\vec{B}\)). Its magnitude is \(F_B = |q|vB \sin\theta\).
Force on a Wire Segment
\[ \vec{F}_B = I (\vec{L} \times \vec{B}) \]
Used for macroscopic wires where \(I\) is current and \(\vec{L}\) is the length vector in the direction of the current.
The Second Right Hand Rule (RHR-2)
Step-by-Step Force Direction:
1 Fingers: Point straight in direction of \(\vec{v}\) or \(I\).
2 Curl: Curl fingers toward the field lines \(\vec{B}\).
3 Thumb: Points in the direction of force \(\vec{F}_B\) for \(+\) charges.
CRITICAL: If the charge is negative (like an electron), the force is in the OPPOSITE direction of your thumb.
Diagram 4.1: Color-Coded Interaction
v
F
V. Synthesis: Particle Motion and Parallel Wires
Magnetic Centripetal Force
Because the magnetic force is always perpendicular to velocity, it performs no work on the particle (since \(W = \int \vec{F} \cdot d\vec{r}\) and \(\vec{F} \perp d\vec{r}\)). Instead, it acts as a centripetal force, causing the particle to move in a circle.
\[ F_B = F_c \implies qvB = \frac{mv^2}{R} \]
\[ R = \frac{mv}{qB} \]
The "Cyclotron Radius" Formula
B-field (\(\otimes\))
Constant force toward center = Uniform Circular Motion
Forces Between Parallel Currents
When two wires carry current, wire 1 produces a field that exerts a force on wire 2, and vice versa.
A
Parallel (Same Direction)
Wires ATTRACT each other.
Using RHR-1, wire 1's field is \(\otimes\) at wire 2. Using RHR-2, current up in field \(\otimes\) pulls left toward wire 1.
B
Anti-Parallel (Opposite)
Wires REPEL each other.
The directions of the forces reverse, pushing the conductors away from one another.
Blueprint Summary Checklist
✅ Direction of field around wire: RHR-1
✅ Direction of force on charge: RHR-2
✅ Field Magnitude (Wire): \(\frac{\mu_0 I}{2\pi r}\)
✅ Force Magnitude (Charge): \(qvB\sin\theta\)
✅ Force Magnitude (Wire): \(ILB\sin\theta\)
✅ Negative charge force: Reverse thumb
END OF BLUEPRINT ARTICLE • REVISION 2.0 • AP PHYSICS C
CURRENT OUT (\(\odot\))
Counter-Clockwise Field
Reversed Polarity
CASE B
CURRENT IN (\(\otimes\))
Clockwise Field
The Reversal Principle
If the direction of the current vector \(\vec{I}\) is reversed (\(\vec{I} \rightarrow -\vec{I}\)), the magnetic field vector \(\vec{B}\) at every point in space is also reversed (\(\vec{B} \rightarrow -\vec{B}\)). This is a direct consequence of the linear relationship between current and field magnitude.
Current Direction
Positive \(z\)
Negative \(z\)
Field Geometry
Counter-Clockwise
Clockwise
IV. Interaction: The Lorentz Force Law
Magnetic fields do not only surround currents; they also interact with them. A magnetic field exerts a force on a moving charge or a current-carrying conductor, known as the Lorentz Force.
On a Point Charge
\[ \vec{F}_B = q(\vec{v} \times \vec{B}) \]
Force depends on charge (\(q\)), velocity (\(v\)), and field strength (\(B\)).
On a Current Segment
\[ \vec{F}_B = I (\vec{L} \times \vec{B}) \]
Force depends on current (\(I\)), length (\(L\)), and field strength (\(B\)).
Right Hand Rule 2 (Force Direction)
1
Point fingers in direction of velocity (\(v\)) or current (\(I\)).
2
Curl fingers toward the magnetic field (\(B\)).
3
Your thumb points in the direction of the magnetic force (\(F_B\)).
NOTE: If the charge is negative (e.g., an electron), the force direction is exactly opposite to your thumb.
v (Velocity)
B (Field)
Force Out
V. Real-World Application: Forces on Parallel Wires
When two parallel wires carry currents, they each produce a magnetic field that exerts a force on the other. This phenomenon is used to define the unit of current, the Ampere.
Case 1: Parallel Currents
ATTRACTION
The magnetic field from wire 1 curls "into" the page at wire 2. RHR-2 indicates a force toward wire 1.
Case 2: Anti-Parallel Currents
REPULSION
Reversing current in one wire reverses the field or the force direction, leading to repulsion.
Mastery Checklist
Field Direction: Defined by RHR-1 (Thumb = Current, Fingers = Field).
3D Vectors: \(\otimes\) is in, \(\odot\) is out.
Current Reversal: Reversing \(I\) reverses the direction of all \(\vec{B}\) vectors.
Magnetic Force: Defined by RHR-2 (Fingers = v, Curl = B, Thumb = Force).
END OF BLUEPRINT ARTICLE • REVISION 3.0 • AP PHYSICS C
Current Out (\(\odot\))
Field: Counter-Clockwise
Case B: Downward Flow
REVERSED
3D Perspective
Current In (\(\otimes\))
Field: Clockwise
The Symmetry Principle
Reversing the current (\(I \rightarrow -I\)) forces the entire magnetic field to undergo a 180° rotation at every point. This linear dependence is a fundamental property of electromagnetism in AP Physics C: the field is a direct mathematical consequence of the moving charges. If the charges move the other way, the field must reflect that reversal.
IV. Deep Dive: The Magnetic Lorentz Force
The magnetic force \(\vec{F}_B\) acts differently than any other force you have studied. While gravity pulls directly toward a mass and electric forces push along field lines, the magnetic force acts perpendicularly to both the velocity of the charge and the magnetic field.
Magnitude & Geometric Constraints
Scalar Form
\[ F_B = |q| v B \sin\theta \]
Maximum Force: Occurs when \(\vec{v} \perp \vec{B}\) (\(\theta = 90^\circ\)).
Zero Force: Occurs when \(\vec{v} \parallel \vec{B}\) or \(\vec{v} \text{ antiparallel } \vec{B}\) (\(\theta = 0^\circ\) or \(180^\circ\)).
A Force That Does No Work
In AP Physics C, one of the most critical theoretical takeaways is that magnetic forces do zero work on moving point charges. Since the force is always perpendicular to velocity (\(\vec{F}_B \cdot \vec{v} = 0\)), it cannot change the kinetic energy or speed of the particle.
\( P = \vec{F}_B \cdot \vec{v} = 0 \)
\( W = \int \vec{F}_B \cdot d\vec{r} = 0 \)
Resulting Particle Motion
Because the force only changes the direction of velocity without changing its magnitude, a particle moving perpendicularly into a uniform field will undergo Uniform Circular Motion.
Circular: If \(\vec{v}\) is perfectly perpendicular to \(\vec{B}\).
Helical: If \(\vec{v}\) has a component parallel to \(\vec{B}\) (the parallel component remains constant).
RHR-2: The Cross Product Visualized
1
Fingers: Direction of Velocity (\(\vec{v}\)).
2
Curl: Direction of B-Field (\(\vec{B}\)).
3
Thumb: Resultant Force direction (\(\vec{F}_B\)).
v
B
Force (Out)
V. Real-World Application: Forces on Parallel Wires
When two parallel wires carry currents, they each produce a magnetic field that exerts a force on the other. This phenomenon is used to define the unit of current, the Ampere.
Case 1: Parallel Currents
ATTRACTION
The magnetic field from wire 1 curls "into" the page at wire 2. RHR-2 indicates a force toward wire 1.
Case 2: Anti-Parallel Currents
REPULSION
Reversing current in one wire reverses the field or the force direction, leading to repulsion.
Mastery Checklist
Field Direction: Defined by RHR-1 (Thumb = Current, Fingers = Field).
3D Vectors: \(\otimes\) is in, \(\odot\) is out.
Current Reversal: Reversing \(I\) reverses the direction of all \(\vec{B}\) vectors.
Magnetic Force: Defined by RHR-2 (Fingers = v, Curl = B, Thumb = Force).
END OF BLUEPRINT ARTICLE • REVISION 4.0 • AP PHYSICS C
IV. Lorentz Force Part 1: Moving Point Charges
The fundamental interaction between a magnetic field and an electric charge is described by the Lorentz Force Law. Crucially, the magnetic force only acts on charges that are in motion relative to the field.
Vector Definition
\[ \vec{F}_B = q(\vec{v} \times \vec{B}) \]
Where \(q\) is charge, \(\vec{v}\) is velocity vector, and \(\vec{B}\) is magnetic field vector. The cross product dictates that \(\vec{F}_B\) is perpendicular to both velocity and field.
Key Properties:
Zero Work: Since \(\vec{F}_B \perp \vec{v}\), the magnetic force does zero work. It changes the particle's direction but never its speed.
Circular Motion: A charge entering a uniform field perpendicularly will undergo uniform circular motion with radius \(R = \frac{mv}{qB}\).
Diagram 4.1: Point Charge Force
+q
v F
Positive charge, v-right, B-in \(\rightarrow\) Force UP (RHR-2)
V. Lorentz Force Part 2: Wire Segments
When many charges move through a conductor (a current), the individual magnetic forces on each charge sum up to a macroscopic force on the wire itself. This is often called the Motor Effect.
Wire Force Definition
\[ \vec{F}_B = I(\vec{L} \times \vec{B}) \]
Where \(I\) is current, \(\vec{L}\) is a vector in the direction of current with magnitude equal to the length of the segment, and \(\vec{B}\) is the external magnetic field.
Application Checklist:
Magnitude: \(F = ILB \sin\theta\). Maximum force when wire is perpendicular to field.
Direction: Use RHR-2. Point fingers with current \(I\), curl with \(\vec{B}\), thumb is \(\vec{F}\).
Non-Uniform Fields: For curved wires, use integration: \(d\vec{F} = I(d\vec{l} \times \vec{B})\).
Diagram 5.1: Wire Segment Force
N
S
Current Up (\(I\)), B-field Right (\(\vec{B}\)) \(\rightarrow\) Force OUT (\(\odot\))
END OF BLUEPRINT ARTICLE • REVISION 5.0 • AP PHYSICS C
IV. Lorentz Force Magnitude: \(F = qvB\sin\theta\)
The magnetic force acting on a single moving point charge is defined by the vector cross product. This relationship results in a specific mathematical formula for the magnitude of the force.
MAGNITUDE FORMULA
\[ F_B = |q| v B \sin(\theta) \]
Variables Defined:
\(q\): Net electric charge (Coulombs).
\(v\): Velocity of the charge (m/s).
\(B\): Magnetic field strength (Tesla).
\(\theta\): The angle between \(\vec{v}\) and \(\vec{B}\).
The Role of \(\theta\):
The sine function dictates that force depends heavily on relative orientation:
\(\theta = 90^\circ\)Max Force (\(qvB\))
\(\theta = 0^\circ\)Zero Force
\(\theta = 180^\circ\)Zero Force
V. Lorentz Force on Wires: \(F = BIL\sin\theta\)
For a macroscopic wire, the individual forces on moving charges sum up to a total force on the conductor. This is the derivation of the \(BIL\) formula from the point charge formula.
WIRE MAGNITUDE FORMULA
\[ F_B = B I L \sin(\theta) \]
Variables Defined:
\(B\): External magnetic field strength (Tesla).
\(I\): Conventional current (Amperes).
\(L\): Length of the wire segment (meters).
\(\theta\): Angle between wire direction and \(\vec{B}\).
Derivation Insight:
"Consider \(N\) charges moving at drift velocity \(v_d\) in a wire of length \(L\). The total charge \(Q = Nq = It = I(L/v_d)\). Substituting \(Qv_d\) into \(F = Qv_dB\) gives \(F = (I L / v_d) v_d B = BIL\)."
VI. The Second Right Hand Rule (RHR-2): The Force Rule
While RHR-1 finds the source of a field, RHR-2 determines the interaction (force). Follow these three specific steps using your Right Hand Only:
1
The Fingers (Velocity/Current)
Point your four fingers straight in the direction of the velocity vector \(\vec{v}\) (for a charge) or the current \(I\) (for a wire).
2
The Curl (Magnetic Field)
Curl your fingers towards the direction of the magnetic field vector \(\vec{B}\). You must be able to curl comfortably (no straining).
3
The Thumb (Resultant Force)
Your thumb now points in the direction of the Magnetic Force \(\vec{F}_B\) acting on a positive charge.
RHR-2 Hands-On Guide
v / I
B
FORCE (OUT)
Rule for Electrons: Use the same steps, then simply flip the result of your thumb 180°.
VII. Case Study: Point Charge in Uniform \(\vec{B}\)
Consider a positive charge \(+q\) moving to the right (\(+x\)) into a region with a uniform magnetic field pointing into the page (\(\otimes\)).
+q
Velocity
Force Direction (UP)
"By RHR-2: Fingers right, curl into page, thumb points UP. Since velocity is always perpendicular to force, the particle follows a counter-clockwise circular arc."
Geometric Summary
Entry Angle (\(\theta\))
If \(\theta=90^\circ\), the path is a Circle.
Pitch Angle
If \(\theta \neq 90^\circ\), the path is a Helix.
Parallel
If \(\theta = 0^\circ\), the path is a Straight Line.
VIII. Case Study: Force Between Two Parallel Wires
This setup is a classic AP Physics C application of both RHR-1 and RHR-2. Wire 1 creates a field that acts on Wire 2.
Step 1: The Field (\(\vec{B}_1\))
\(I_1\)
Field from \(I_1\) is \(\otimes\) at \(I_2\)
By RHR-1, Wire 1 creates a magnetic field that enters the page at the location of Wire 2.
Step 2: The Force (\(\vec{F}_{12}\))
\(I_2\)
Force Pulls Left
By RHR-2 (Fingers up, curl IN), the force on Wire 2 points toward Wire 1. Result: Attraction.
A proton (\(q=1.6\times 10^{-19}\) C) moves at \(5.0\times 10^6\) m/s at a \(30^\circ\) angle to a 0.20 T magnetic field. Calculate the force magnitude.
A 20 cm wire carries 4.0 A of current. It is placed in a region where the B-field is 0.5 T and perpendicular to the wire. Calculate the force.
1. \(F = BIL\sin(90^\circ)\)
2. \(F = (0.5)(4.0)(0.20)(1.0)\)
3. \(F = 0.4 \text{ Newtons}\)
X. Final Review & Synthesis
The Electromagnetism Flowchart
SOURCE
Current \(I\) creates \(\vec{B}\) loops. Use RHR-1.
SYMMETRY
Reversed Current = Reversed Field Loops.
INTERACTION
Field acts on Charge/Wire. Use RHR-2.
Crucial Takeaways for the AP Exam
RHR-1: Thumb is \(I\), Fingers are \(\vec{B}\).
RHR-2: Fingers are \(v\), Curl is \(\vec{B}\), Thumb is \(\vec{F}\).
B-Field Units: Measured in Tesla (T). 1 T = 1 N/(A·m).
Lorentz Work: Always zero work on point charges.
Negative Charges: Direction of force is reversed.
Magnetic field: Forms closed loops, never starts or ends.
REFERENCE: Halliday & Resnick Mastery END OF MASTER BLUEPRINT ARTICLE Page 10 of 10
IV. Lorentz Force Magnitude: \(F = qvB\sin\theta\)
The magnetic force acting on a single moving point charge is defined by the vector cross product. This relationship results in a specific mathematical formula for the magnitude of the force.
MAGNITUDE FORMULA
\[ F_B = |q| v B \sin(\theta) \]
Variables Defined:
\(q\): Net electric charge (Coulombs).
\(v\): Velocity of the charge (m/s).
\(B\): Magnetic field strength (Tesla).
\(\theta\): The angle between \(\vec{v}\) and \(\vec{B}\).
The Role of \(\theta\):
The sine function dictates that force depends heavily on relative orientation:
\(\theta = 90^\circ\)Max Force (\(qvB\))
\(\theta = 0^\circ\)Zero Force
\(\theta = 180^\circ\)Zero Force
V. Lorentz Force on Wires: \(F = BIL\sin\theta\)
For a macroscopic wire, the individual forces on moving charges sum up to a total force on the conductor. This is the derivation of the \(BIL\) formula from the point charge formula.
WIRE MAGNITUDE FORMULA
\[ F_B = B I L \sin(\theta) \]
Variables Defined:
\(B\): External magnetic field strength (Tesla).
\(I\): Conventional current (Amperes).
\(L\): Length of the wire segment (meters).
\(\theta\): Angle between wire direction and \(\vec{B}\).
Derivation Insight:
"Consider \(N\) charges moving at drift velocity \(v_d\) in a wire of length \(L\). The total charge \(Q = Nq = It = I(L/v_d)\). Substituting \(Qv_d\) into \(F = Qv_dB\) gives \(F = (I L / v_d) v_d B = BIL\)."
VI. The Second Right Hand Rule (RHR-2): The Force Rule
While RHR-1 finds the source of a field, RHR-2 determines the interaction (force). Follow these three specific steps using your Right Hand Only:
1
The Fingers (Velocity/Current)
Point your four fingers straight in the direction of the velocity vector \(\vec{v}\) (for a charge) or the current \(I\) (for a wire).
2
The Curl (Magnetic Field)
Curl your fingers towards the direction of the magnetic field vector \(\vec{B}\). You must be able to curl comfortably (no straining).
3
The Thumb (Resultant Force)
Your thumb now points in the direction of the Magnetic Force \(\vec{F}_B\) acting on a positive charge.
RHR-2 Hands-On Guide
v / I
B
FORCE (OUT)
Rule for Electrons: Use the same steps, then simply flip the result of your thumb 180°.
VII. Case Study: Point Charge in Uniform \(\vec{B}\)
Consider a positive charge \(+q\) moving to the right (\(+x\)) into a region with a uniform magnetic field pointing into the page (\(\otimes\)).
+q
Velocity
Force Direction (UP)
"By RHR-2: Fingers right, curl into page, thumb points UP. Since velocity is always perpendicular to force, the particle follows a counter-clockwise circular arc."
Geometric Summary
Entry Angle (\(\theta\))
If \(\theta=90^\circ\), the path is a Circle.
Pitch Angle
If \(\theta \neq 90^\circ\), the path is a Helix.
Parallel
If \(\theta = 0^\circ\), the path is a Straight Line.
VIII. Magnetic Centripetal Force
A fundamental consequence of the Lorentz force being always perpendicular to the velocity (\(\vec{F}_B \perp \vec{v}\)) is that it acts as a centripetal force. This forces a moving charge into a circular trajectory without changing its speed.
The Mathematical Equality
For a charge moving perpendicularly to a uniform field (\(\theta = 90^\circ\)):
\[ F_B = F_c \implies qvB = \frac{mv^2}{R} \]
Solving for the Cyclotron Radius (\(R\)):
\[ R = \frac{mv}{qB} \]
Increasing speed (\(v\)) or mass (\(m\)) results in a larger circle. Increasing field (\(B\)) or charge (\(q\)) results in a tighter, smaller circle.
Frequency & Period
The time for one revolution (\(T\)) and the frequency (\(f\)) are independent of the particle's speed!
Period (T)
\[ T = \frac{2\pi R}{v} = \frac{2\pi m}{qB} \]
Cyclotron Frequency (f)
\[ f = \frac{1}{T} = \frac{qB}{2\pi m} \]
This property is used in cyclotrons (particle accelerators) to accelerate ions using an oscillating electric field.
Velocity (v) Force (Fc) B field OUT (\(\odot\))
Visual Insight
In this diagram, the field is pointing OUT of the page. By RHR-2, fingers point with the velocity (right), curl toward the viewer (up), and the thumb points DOWN toward the center of the loop. As the particle moves, the force remains perpendicular, always pulling toward the center, maintaining uniform circular motion.
IX. Case Study: Force Between Two Parallel Wires
This setup is a classic AP Physics C application of both RHR-1 and RHR-2. Wire 1 creates a field that acts on Wire 2.
Step 1: The Field (\(\vec{B}_1\))
\(I_1\)
Field from \(I_1\) is \(\otimes\) at \(I_2\)
By RHR-1, Wire 1 creates a magnetic field that enters the page at the location of Wire 2.
Step 2: The Force (\(\vec{F}_{12}\))
\(I_2\)
Force Pulls Left
By RHR-2 (Fingers up, curl IN), the force on Wire 2 points toward Wire 1. Result: Attraction.
A proton (\(q=1.6\times 10^{-19}\) C) moves at \(5.0\times 10^6\) m/s at a \(30^\circ\) angle to a 0.20 T magnetic field. Calculate the force magnitude.
A 20 cm wire carries 4.0 A of current. It is placed in a region where the B-field is 0.5 T and perpendicular to the wire. Calculate the force.
1. \(F = BIL\sin(90^\circ)\)
2. \(F = (0.5)(4.0)(0.20)(1.0)\)
3. \(F = 0.4 \text{ Newtons}\)
XI. Final Review & Synthesis
The Electromagnetism Flowchart
SOURCE
Current \(I\) creates \(\vec{B}\) loops. Use RHR-1.
SYMMETRY
Reversed Current = Reversed Field Loops.
INTERACTION
Field acts on Charge/Wire. Use RHR-2.
Crucial Takeaways for the AP Exam
RHR-1: Thumb is \(I\), Fingers are \(\vec{B}\).
RHR-2: Fingers are \(v\), Curl is \(\vec{B}\), Thumb is \(\vec{F}\).
B-Field Units: Measured in Tesla (T). 1 T = 1 N/(A·m).
Lorentz Work: Always zero work on point charges.
Centripetal: Magnetic force acts as \(F_c = mv^2/R\).
Radius: \(R = mv/qB\). Radius is speed dependent.
REFERENCE: Halliday & Resnick Mastery END OF MASTER BLUEPRINT ARTICLE Page 11 of 11
III. Anti-Parallel Currents: Mutual Repulsion
When currents flow in opposite directions, the magnetic interaction changes polarity, resulting in a force that pushes the conductors apart.
Step 1: Wire 1 Field
Using RHR-1: Thumb up (\(I_1\)), fingers curl. At Wire 2's location, the field \(\vec{B}_1\) points INTO the page (\(\otimes\)). (This step is identical to Case A).
Step 2: Force on Wire 2
Using RHR-2: Fingers point DOWN with \(I_2\), curl IN with \(\vec{B}_1\). Your thumb now points RIGHT, away from Wire 1.
Result: REPULSION
\(I_1\)
\(\vec{B}_1\) at Wire 2
Force (\(\vec{F}_{21}\))
\(I_2\)
Summary of Wire Rules
Like Currents (\(\uparrow\uparrow\)) \(\rightarrow\) Attract
To find the magnitude of the force per unit length, we combine the formula for the magnetic field of a long wire with the Lorentz force formula for a wire segment.
1. Magnetic Field from Wire 1: \[ B_1 = \frac{\mu_0 I_1}{2\pi d} \]
2. Force on Wire 2 segment: \[ F_{21} = I_2 L B_1 \sin(90^\circ) \]
3. Final Expression (Force per Length): \[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \]
The Official Definition of the Ampere
Interestingly, the SI unit for current—the **Ampere**—was historically defined based on this magnetic interaction between parallel wires rather than the flow of charge directly.
Historical SI Definition
"One Ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section, placed one meter apart in vacuum, would produce between these conductors a force equal to exactly \(2 \times 10^{-7}\) Newtons per meter of length."
Note: This definition implies \(\mu_0 = 4\pi \times 10^{-7}\) exactly. In 2019, the definition was updated to be based on the elementary charge \(e\), but the physical relationship remains a cornerstone of the field.
Mastery Check
• If you double both currents, the force quadruples (\(2 \times 2 = 4\)).
• If you double the distance, the force is halved.
Using RHR-2: Fingers point DOWN with \(I_2\), curl IN with \(\vec{B}_1\). Your thumb now points RIGHT, away from Wire 1.
Result: REPULSION
\(I_1\)
\(\vec{B}_1\) at Wire 2
Force (\(\vec{F}_{21}\))
\(I_2\)
IV. Mapping the Fields: Individual Contributions
To truly understand why the force occurs, we must examine the magnetic field geometry created by each wire independently. Every wire acts as a source of circular field loops. When two wires are near each other, their fields overlap at every point in space.
Magnetic Field of Wire 1 (\(\vec{B}_1\))
OUT (\(\odot\))
IN (\(\otimes\))
Wire 1 creates field loops that are "out" on its left and "in" on its right.
Magnetic Field of Wire 2 (\(\vec{B}_2\))
OUT (\(\odot\))
IN (\(\otimes\))
Wire 2 also creates its own field loops, independent of Wire 1's field.
The Region Between the Wires
In the space directly between the two wires, the fields from Wire 1 and Wire 2 interact. This is the key to understanding the net interaction:
Case 1: Parallel Currents
Between the wires, \(\vec{B}_1\) is IN (\(\otimes\)) and \(\vec{B}_2\) is OUT (\(\odot\)). These fields point in opposite directions and partially cancel each other out. This creates a region of low field density, and the wires are "pushed" into it from the high-density outer regions.
Case 2: Anti-Parallel Currents
Between the wires, if \(I_2\) is reversed, its field \(\vec{B}_2\) is now IN (\(\otimes\)). Since \(\vec{B}_1\) is also IN (\(\otimes\)), they point in the same direction and reinforce each other. This creates a region of high field density, and the wires are "pushed" away from each other.
V. Quantitative Mastery & SI Definitions
Calculating the Magnitude
To find the magnitude of the force per unit length, we combine the formula for the magnetic field of a long wire with the Lorentz force formula for a wire segment.
1. Magnetic Field from Wire 1: \[ B_1 = \frac{\mu_0 I_1}{2\pi d} \]
2. Force on Wire 2 segment: \[ F_{21} = I_2 L B_1 \sin(90^\circ) \]
3. Final Expression (Force per Length): \[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \]
The Official Definition of the Ampere
Historical SI Definition
"One Ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section, placed one meter apart in vacuum, would produce between these conductors a force equal to exactly \(2 \times 10^{-7}\) Newtons per meter of length."
Mastery Check
• Individual Fields: Each wire creates its own B-field regardless of the other.
• Cancellation: Same-direction currents have fields that cancel between the wires \(\rightarrow\) Attraction.
• Reinforcement: Opposite-direction currents have fields that reinforce between the wires \(\rightarrow\) Repulsion.
Explain why a stationary charge placed next to a wire carrying a high current experiences zero magnetic force, while an identical charge moving parallel to the wire experiences a significant force. Use the concept of the Lorentz Force Law in your explanation.
(b) Identify a point on the \(x\)-axis between the wires (other than infinity) where the net magnetic field is zero. Calculate its coordinate \(x\).
(c) Determine an expression for the magnitude of the force per unit length \((F/L)\) exerted by Wire 1 on Wire 2. Is this force attractive or repulsive?
(d) A third wire carrying current \(3I\) in the \(-y\) direction is placed at the location identified in part (b). Determine the magnitude and direction of the net force per unit length acting on this third wire.
(e) If the current in Wire 1 were reversed, describe how the location of the zero-field point on the \(x\)-axis would change. Would it still be between the wires?
Reasoning: The wire is placed at a point where the net magnetic field is zero. Since \(F = ILB_{net}\) and \(B_{net}=0\), the force must be zero.
(e) Reversal Impact: [2 points]
Result: Zero field point moves outside the wires, to the left of Wire 1 (\(x < -a\)). [1 pt] Reasoning: Between the wires, both fields now point in the same direction (\(\otimes\)), so they can never cancel. Outside to the left, \(B_1\) is \(\odot\) and \(B_2\) is \(\otimes\). [1 pt]
Part IV: Synthesis & Wire Interactions (Pages 9-11)
11. Parallel Wire Mechanism (Page 9): Breaking it down into two steps...
A) What determines the direction of the field that Wire 1 creates at the location of Wire 2?
B) Once that field is known, what determines the direction of the force that Wire 2 feels?
12. Summarize the "Mastery Checklist" on Page 11. Which of these concepts do you find most counter-intuitive, and why?
Final Technical Sketch
Draw a diagram of two anti-parallel wires. Label the current in each, the field created by wire 1, and the resulting force on wire 2. Show that they repel.
Deflect: Ions enter uniform \(B_0\). They follow circular arcs of radius \(R\).
Detect: Position of impact gives \(R\), allowing for the calculation of mass \(m\).
III. Torque on a Current Loop
Moving from point charges to macroscopic systems, we consider a rectangular wire loop of width \(a\) and height \(b\), carrying a current \(I\) in a uniform magnetic field \(\vec{B}\). This system is the precursor to the electric motor.
Diagram 3.1: The Force Distribution
Axis
F1
F2
"Field \(\vec{B}\) points right. Forces on top/bottom cancel. Forces on vertical sides create a couple that rotates the loop."
Analyzing the Force Pairs
Vertical Sides (Height \(b\))
Side 1 carries current \(I\) up. Side 2 carries current \(I\) down. Since both are perpendicular to \(\vec{B}\), they feel forces of magnitude \(F = IbB\). These forces point in opposite directions, creating torque about the central axis.
Horizontal Sides (Width \(a\))
The forces on these sides are parallel to the axis of rotation. While they exert tension on the wire, they contribute zero torque because their line of action passes through the axis.
IV. The General Torque Formula
To find the total torque, we look at the loop from the top down (the side view of width \(a\)). Let \(\theta\) be the angle between the magnetic field \(\vec{B}\) and the area vector \(\vec{A}\) (perpendicular to the loop's surface).
The torque depends on the orientation of the loop relative to the field:
• Maximum Torque (\(\theta=90^\circ\)): When the loop is parallel to the field, the moment arm is maximized.
• Zero Torque (\(\theta=0^\circ\)): When the loop is perpendicular to the field, the forces pull outward, but do not rotate.
Area Vector \(\vec{A}\)
Top View: Side "a"
V. The Magnetic Dipole Moment (\(\mu\))
In AP Physics C, we generalize the torque formula using a new vector quantity called the Magnetic Dipole Moment. This allows us to treat any current loop—regardless of shape—using a single property.
Definition of \(\mu\)
\[ \vec{\mu} = N I \vec{A} \]
Where \(N\) is the number of turns in the coil, \(I\) is current, and \(\vec{A}\) is the area vector (direction defined by RHR-1).
General Torque Formula
\[ \vec{\tau} = \vec{\mu} \times \vec{B} \]
The cross product naturally accounts for the \(\sin\theta\) relationship. The torque always acts to align \(\vec{\mu}\) with \(\vec{B}\).
Magnetic Potential Energy
Just like an electric dipole in an electric field, a magnetic dipole has potential energy based on its orientation:
\[ U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos\theta \]
Lowest Energy (\(\theta=0^\circ\)): \(\mu\) is aligned with \(B\). Stable equilibrium.
Highest Energy (\(\theta=180^\circ\)): \(\mu\) is anti-aligned. Unstable equilibrium.
Precison & Power Summary
MASS SPEC:
Separates by mass using \(m = qBR/v\). Heavier = Larger Radius.
VELOCITY SELECTOR:
Filters ions using \(v = E/B\). Forces must balance.
MAGNETIC TORQUE:
Causes rotation \(\tau = \mu \times B\). Basis for all DC motors.
