Inductance

The Complete Guide to Inductance and RL Circuits

I. Introduction

In mechanics, mass is the measure of inertia—resistance to changes in motion. In circuits, the inductor serves the same role for current. An inductor possesses self-inductance (L), which causes it to oppose any change in current.

An inductor opposes changes in current by storing energy in a magnetic field. This behavior is rooted in Faraday's Law and Lenz's Law.

Physical Basis

A changing magnetic flux induces an EMF (\(\mathcal{E}\)) in a loop:

\mathcal{E} = -N \frac{d\Phi_B}{dt}

Lenz's Law: The induced EMF creates a current whose field opposes the original flux change.

II. Defining Inductance (L)

When current i flows through a coil, it creates a magnetic field. We define the constant of proportionality as the Inductance (L):

L = \frac{N\Phi_B}{i}

The SI unit is the Henry (H). 1 H = 1 Wb/A.

The Solenoid Model

For an ideal solenoid of length l, area A, and turns N:

\[ L = \mu_0 \frac{N^2 A}{l} \]

Inductance depends only on geometry.

III. The Model RL Circuit

A proper RL circuit model includes a two-way switch for both charging and discharging.

Operational Logic

Position 1: Charge (Build field).
Position 2: Decay (Release field).

Figure 1.2: Complete RL Model

Immediate (t=0)

Inductor produces a back-EMF opposing the battery. Current is zero.

Steady (t→∞)

Current is constant. Back-EMF is zero. Current is \(\mathcal{E}/R\).

IV. The Loop Equation

Kirchhoff's Rule

Summing potential differences clockwise through the loop:

\mathcal{E} - iR - L \frac{di}{dt} = 0

Key Insights

As i increases, iR increases.
Consequently, L(di/dt) must decrease.
The rate of current growth slows down over time.

V. Solving for i(t)

The Calculus Path

1. Separate Differentials

\frac{di}{\mathcal{E} - iR} = \frac{1}{L} dt

2. Exponential Growth Form

\[ i(t) = \frac{\mathcal{E}}{R} \left( 1 - e^{-\frac{Rt}{L}} \right) \]

VI. The Time Constant \( (\tau) \)

τ = L / R

Speed of Response

I_max = \(\mathcal{E}/R\)

Steady State

Current Profile over 5\(\tau\)

After 1 time constant, current reaches ~63.2% of its max value. By 5\(\tau\), the circuit is effectively at steady state.

\[ i(\tau) \approx 0.632 I_{max} \]

VII. Voltage Profiles

Inductor (V_L)

V_L(t) = \mathcal{E} e^{-\frac{t}{\tau}}

Resistor (V_R)

V_R(t) = \mathcal{E} ( 1 - e^{-\frac{t}{\tau}} )

VIII. Energy Stored

Potential Energy (U_L)

\[ U_L = \frac{1}{2} L I^2 \]

Energy Density (u_B)

Energy stored per unit volume in a magnetic field:

\[ u_B = \frac{B^2}{2\mu_0} \]

IX. Current Decay

The Discharging Phase

When battery is removed:

\[ i(t) = I_0 e^{-\frac{Rt}{L}} \]

Physical Mechanism

The collapsing magnetic field induces an EMF in the same direction as the original current, attempting to maintain the flow. This energy is eventually dissipated as heat in the resistor.

X. Summary Comparisons

INDUCTOR VS CAPACITOR

Trait	Inductor	Capacitor
Opposes	Δ i	Δ V
Energy	1/2 LI²	1/2 CV²

MASTER EQUATIONS

\[ \tau = L/R \]
\[ V_L = -L \frac{di}{dt} \]

Watchdog Principle

"The inductor allows steady currents to pass without complaint, but barks at any attempt to change them."

End of Guide • AP Physics C

Inductance Guide

Inductance Guide

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