The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Inductance, R-C Circuits, L-C Circuits, and L-R-C Circuits – Lecture 13 (University Physics II)

Overview

This lecture turns Faraday's law back on circuits themselves. A changing current in one coil induces an emf in a nearby coil (mutual inductance), and a changing current in a single coil induces an emf in that same coil (self-inductance). Both effects are captured by a single constant that depends only on geometry, and both make the inductor a circuit element that resists changes in current while storing energy in its magnetic field. With the inductor's voltage $L\,di/dt$ in hand, the lecture solves the RL circuit as a close parallel to the RC circuit, shows that an LC circuit is an exact electrical analogue of a mass on a spring, and closes with the damped oscillations of an LRC circuit.

Key Concepts

Mutual inductance

A current $i_1$ in coil 1 produces a field that threads coil 2. The field everywhere is proportional to $i_1$ , so the flux through each turn of coil 2, $\Phi_{B2}$ , is too. The constant of proportionality is the mutual inductance:

\begin{aligned} N_2 \Phi_{B2} &= M_{21} \, i_1 \\ M &= \frac{N_2 \Phi_{B2}}{i_1} = \frac{N_1 \Phi_{B1}}{i_2} \end{aligned}

$M$ depends only on the geometry — the coils' sizes, numbers of turns, orientation, and separation — and not on the current. Once the coils are fixed in place it is a fixed number.
Remarkably, $M_{21} = M_{12}$ for any pair of coils, whatever their shapes. One coil can never be aimed to affect the other more than the other affects it, so a single symbol $M$ serves both directions.
Applying Faraday's law to the total flux $N_2\Phi_{B2} = M i_1$ gives the induced emfs:

\begin{aligned} \mathcal{E}_2 &= -M \frac{di_1}{dt} \\ \mathcal{E}_1 &= -M \frac{di_2}{dt} \end{aligned}

The unit of inductance is the henry: $1\ \text{H} = 1\ \text{Wb/A} = 1\ \text{V}\cdot\text{s/A}$ . A henry is a large inductance; typical coil pairs are in the microhenry range.
Application: classic metal detectors drive a changing current in a coil; the field induces eddy currents in a passing piece of metal, and those currents in turn induce a detectable current back in the detector coil.

Self-inductance and inductors

A changing current in a single coil changes the flux through that same coil, so the coil induces an emf in itself — a counter-emf that opposes the change. Raising the current is resisted; once the current is established, trying to stop it is resisted too.
The self-inductance is defined just like $M$ , but with the coil acting on itself:

\begin{aligned} L &= \frac{N \Phi_B}{i} \\ \mathcal{E} &= -L \frac{di}{dt} \end{aligned}

An inductor is a circuit element (drawn as a coil) built to have significant inductance. It is closer in spirit to a capacitor than to a resistor: a capacitor stores energy in an electric field, an inductor stores energy in a magnetic field. Magnetic material in the core plays a role much like a dielectric, enhancing the field.
Because an inductor fights every change in current, it smooths out a fluctuating current.

Voltage across an inductor

Kirchhoff's loop rule assumed a conservative field, $\oint \vec{E}\cdot d\vec{l} = 0$ . In general, Faraday's law gives $\oint \vec{E}\cdot d\vec{l} = -d\Phi_B/dt$ , which is nonzero whenever an inductor is present.
Even so, treating the inductor as if it had a voltage drop $V_{ab} = L\,di/dt$ (in the direction of the current) and applying the loop rule gives correct results. For a battery, resistor, and inductor in series:

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

Why this works. Along the wire of an ideal (zero-resistance) inductor, the total electric field must be zero. The changing flux produces a non-conservative field $\vec{E}_n$ , so surface charges arrange themselves along the wire to produce a conservative field $\vec{E}_c$ that cancels it — but only along the path of the wire, not in the empty space inside the coil. Since $\vec{E}_n$ is nonzero only within the inductor, the loop integral reduces to the stretch from $a$ to $b$ :

\begin{aligned} \oint \vec{E}_n \cdot d\vec{l} &= \int_a^b \vec{E}_n \cdot d\vec{l} = -L\frac{di}{dt} \\ V_{ab} &= -\int_a^b \vec{E}_c \cdot d\vec{l} = L\frac{di}{dt} \end{aligned}

The voltage across an inductor is therefore a real potential difference, created by real static charges on its surface.

Inductance from geometry

Solenoid (length $l$ , area $A$ , $N$ turns), with $B = \mu_0 N i / l$ inside:

L = \frac{N}{i}\left(\frac{\mu_0 N i}{l}\right)A = \frac{\mu_0 N^2 A}{l}

Toroidal solenoid (mean radius $r$ ), with $B = \mu_0 N i / 2\pi r$ :

L = \frac{\mu_0 N^2 A}{2\pi r}

This is the solenoid result with $l$ replaced by the circumference $2\pi r$ — the length of the toroid if it were cut and straightened.

Energy stored in an inductor

The power delivered to an inductor is $P = V_{ab}\, i = L i \, di/dt$ , so $dU = L i \, di$ . Integrating from zero current up to $I$ :

U = \int_0^I L i \, di = \frac{1}{2} L I^2

This parallels the capacitor, $U = Q^2/2C$ , and resembles kinetic energy $\tfrac{1}{2}mv^2$ with $L$ in the role of mass and current in the role of velocity.
Magnetic energy density. Dividing the solenoid's energy by its volume $Al$ and substituting $L = \mu_0 N^2 A / l$ :

\begin{aligned} u &= \frac{U}{Al} = \frac{\mu_0 N^2 I^2}{2l^2} = \frac{1}{2\mu_0}\left(\frac{\mu_0 N I}{l}\right)^2 \\ u &= \frac{B^2}{2\mu_0} \end{aligned}

This result holds for any magnetic field in vacuum, not just a solenoid's. Inside a magnetic material, $\mu_0$ is replaced by the material's permeability $\mu$ , giving $u = B^2/2\mu$ .

RC circuit recap

For a battery charging a capacitor through a resistor, $\mathcal{E} = iR + q/C$ with $i = dq/dt$ . Separating variables and using $q(0) = 0$ :

\begin{aligned} q(t) &= C\mathcal{E}\left(1 - e^{-t/RC}\right) = Q_f\left(1 - e^{-t/RC}\right) \\ i(t) &= \frac{\mathcal{E}}{R}e^{-t/RC} = I_0 e^{-t/RC} \end{aligned}

The time constant $\tau = RC$ depends only on the circuit's components. After one time constant the charge has reached $1 - 1/e \approx 63\%$ of its final value and the current has fallen to $1/e \approx 37\%$ of its initial value.

RL circuit

Current growth. With an inductor in place of the capacitor, $\mathcal{E} = iR + L\,di/dt$ . The initial current is zero (the inductor blocks a sudden change), and the final current is $\mathcal{E}/R$ (once $i$ stops changing, the inductor drops no voltage). Solving:

\begin{aligned} i(t) &= \frac{\mathcal{E}}{R}\left(1 - e^{-(R/L)t}\right) \\ \frac{di}{dt} &= \frac{\mathcal{E}}{L}e^{-(R/L)t} \end{aligned}

The current changes fastest at $t = 0$ , which is why very large rates such as $10^6\ \text{A/s}$ are realistic over very short intervals. The time constant is $\tau = L/R$ .
Power. Multiplying the loop equation by $i$ : $\mathcal{E}i = i^2R + Li\,di/dt$ . The battery's output splits between heat in the resistor and energy stored in the inductor.
Current decay. If the battery is removed and the circuit closed, $0 = iR + L\,di/dt$ , and the inductor now supplies the energy instead of absorbing it:

i(t) = I_0 e^{-(R/L)t}

LC circuit and simple harmonic motion

Mass on a spring (recap). Newton's second law with Hooke's law gives $d^2x/dt^2 + (k/m)x = 0$ , solved by $x = A\cos(\omega t + \phi)$ with $\omega = \sqrt{k/m}$ . Any equation of the form $\ddot{f} + (\text{const}) f = 0$ has this solution, with $\omega$ the square root of the constant. From initial conditions, $\phi = \tan^{-1}\!\left(-v_0/\omega x_0\right)$ and $A = \sqrt{x_0^2 + v_0^2/\omega^2}$ .
Qualitative picture. A charged capacitor is connected to an inductor. Charge flows off the plates, the current builds, and energy moves from the capacitor's electric field into the inductor's magnetic field. When the capacitor is empty the current is at its maximum; the inductor then keeps the current going, recharging the capacitor with the opposite polarity, and the cycle repeats in reverse. The charge and current oscillate out of phase, and since $E \propto q$ and $B \propto i$ , energy sloshes back and forth between the two fields.
Equation and solution. Around the loop, $L\,di/dt + q/C = 0$ with $i = dq/dt$ :

\begin{aligned} \frac{d^2q}{dt^2} + \frac{1}{LC}q &= 0 \\ q(t) &= Q\cos(\omega t + \phi), \qquad \omega = \frac{1}{\sqrt{LC}} \\ i(t) &= -\omega Q \sin(\omega t + \phi) \end{aligned}

Energy is conserved. Substituting $q$ and $i$ and using $\omega^2 = 1/LC$ :

\begin{aligned} \frac{q^2}{2C} + \frac{1}{2}Li^2 &= \frac{Q^2}{2C}\cos^2(\omega t + \phi) + \frac{L\omega^2 Q^2}{2}\sin^2(\omega t + \phi) \\ &= \frac{Q^2}{2C}\left[\cos^2(\omega t + \phi) + \sin^2(\omega t + \phi)\right] = \frac{Q^2}{2C} \end{aligned}

The analogy. $q \leftrightarrow x$ , $i \leftrightarrow v$ , $L \leftrightarrow m$ , and $1/C \leftrightarrow k$ . The capacitor's energy $q^2/2C$ plays the role of the spring's potential energy $\tfrac{1}{2}kx^2$ , and the inductor's $\tfrac{1}{2}Li^2$ plays the role of kinetic energy $\tfrac{1}{2}mv^2$ .
More fundamentally, energy lives in the fields themselves: in a circuit, energy flows in the direction of $\vec{E} \times \vec{B}$ , not by electrons pushing one another along the wire.

LRC circuit: damped oscillations

An LC circuit oscillates forever only with zero resistance. Adding a resistor drains energy on every cycle, so the oscillations die away, just as friction damps a mass on a spring. Around the loop:

\frac{d^2q}{dt^2} + \frac{R}{L}\frac{dq}{dt} + \frac{1}{LC}q = 0

For small enough $R$ , the solution is a decaying oscillation at a slightly reduced angular frequency:

\begin{aligned} q(t) &= A e^{-(R/2L)t}\cos(\omega' t + \phi) \\ \omega' &= \sqrt{\frac{1}{LC} - \frac{R^2}{4L^2}} \end{aligned}

Increasing $R$ lowers $\omega'$ . When $R^2 = 4L/C$ the circuit is critically damped: $\omega' = 0$ and the charge falls to zero without oscillating. For larger $R$ the circuit is overdamped, and the charge drains away even more slowly.
Solving equations like these systematically is the subject of a differential equations course; linear algebra helps considerably with the algebra involved.

Worked Example

Mutual inductance of two coaxial solenoids. A long solenoid of length $l = 0.50\ \text{m}$ and cross-sectional area $A = 10\ \text{cm}^2 = 1.0 \times 10^{-3}\ \text{m}^2$ has $N_1 = 1000$ turns. A second coil of $N_2 = 10$ turns is wound around its middle.

(a) The mutual inductance. A current $i_1$ in the long solenoid produces $B_1 = \mu_0 N_1 i_1 / l$ inside it and essentially no field outside, so every turn of the outer coil encloses the same flux $B_1 A$ , regardless of the outer coil's own radius:

\begin{aligned} M &= \frac{N_2 \Phi_{B2}}{i_1} = \frac{N_2}{i_1}\left(\frac{\mu_0 N_1 i_1}{l}\right)A = \frac{\mu_0 N_1 N_2 A}{l} \\ &= \frac{(4\pi \times 10^{-7}\ \text{Wb/A}\cdot\text{m})(1000)(10)(1.0 \times 10^{-3}\ \text{m}^2)}{0.50\ \text{m}} \\ &= 2.5 \times 10^{-5}\ \text{H} = 25\ \mu\text{H} \end{aligned}

The currents cancel, as they must: $M$ depends only on the geometry.

(b) Flux and induced emf. Now the outer coil carries a current that rises as $i_2 = (2.0 \times 10^6\ \text{A/s})\,t$ . At $t = 3.0\ \mu\text{s}$ , the current is $i_2 = (2.0 \times 10^6\ \text{A/s})(3.0 \times 10^{-6}\ \text{s}) = 6.0\ \text{A}$ . The flux through each turn of the long solenoid is:

\begin{aligned} \Phi_{B1} &= \frac{M i_2}{N_1} \\ &= \frac{(2.5 \times 10^{-5}\ \text{H})(6.0\ \text{A})}{1000} = 1.5 \times 10^{-7}\ \text{Wb} \end{aligned}

The induced emf in the long solenoid is:

\begin{aligned} \mathcal{E}_1 &= -M\frac{di_2}{dt} \\ &= -(2.5 \times 10^{-5}\ \text{H})(2.0 \times 10^6\ \text{A/s}) = -50\ \text{V} \end{aligned}

The magnitude is $50\ \text{V}$ . The minus sign says the emf opposes the growing current, and its actual direction follows from the right-hand rule.

Summary

Mutual inductance $M = N_2\Phi_{B2}/i_1 = N_1\Phi_{B1}/i_2$ depends only on geometry, is the same in both directions, and gives $\mathcal{E}_2 = -M\,di_1/dt$ ; for coaxial solenoids, $M = \mu_0 N_1 N_2 A / l$ .
Self-inductance $L = N\Phi_B/i$ makes a coil oppose changes in its own current, and an inductor behaves as if it had a voltage $L\,di/dt$ , produced by real surface charges along its wire.
An inductor stores energy $U = \tfrac{1}{2}LI^2$ in its magnetic field, with energy density $u = B^2/2\mu_0$ in vacuum; a solenoid has $L = \mu_0 N^2 A / l$ .
An RL circuit grows toward $\mathcal{E}/R$ as $1 - e^{-(R/L)t}$ and decays as $e^{-(R/L)t}$ , with time constant $\tau = L/R$ , mirroring the RC circuit's $\tau = RC$ .
An LC circuit is simple harmonic motion with $\omega = 1/\sqrt{LC}$ : energy oscillates between the capacitor's electric field and the inductor's magnetic field, with $L$ acting like mass and $1/C$ like a spring constant.
Resistance damps the oscillation, lowering the frequency to $\omega' = \sqrt{1/LC - R^2/4L^2}$ ; at $R^2 = 4L/C$ the circuit is critically damped and no longer oscillates.