# Lecture 12: Inductance and RL Circuits ## Physics II — Electromagnetism, Optics & Modern Physics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Define self-inductance and mutual inductance and calculate inductance for simple geometries 2. Calculate the energy stored in an inductor and the magnetic energy density 3. Describe the transient behavior of RL circuits (growth and decay of current) 4. Apply the RL time constant to analyze circuit response 5. Compare and contrast RL and RC circuit behavior --- ## Lecture Content ### I. Self-Inductance When the current through a coil changes, the resulting change in magnetic flux induces an EMF in the coil itself. This phenomenon is called **self-induction**, and the coil opposes changes in its own current. The **self-inductance** (L) quantifies this effect by relating the induced EMF to the rate of current change: EMF = -L (dI/dt). Equivalently, L = N Phi_B / I, the total flux linkage per unit current. The SI unit of inductance is the **henry** (H) = 1 V s/A = 1 Wb/A, with typical inductors ranging from millihenries to henries. An inductor is a circuit element designed to have a specific inductance. Its circuit symbol is a coil, and it opposes changes in current much like inertia opposes changes in motion. The **inductance of a solenoid** with N turns, length l, and cross-sectional area A can be derived straightforwardly. The field inside is B = mu_0 (N/l) I, giving a flux of Phi_B = BA = mu_0 NIA/l. The inductance is then L = N Phi_B / I = mu_0 N^2 A / l = mu_0 n^2 Al, depending only on the geometry and core material. For a **toroid** with N turns, inner radius a, outer radius b, and height h, the inductance is L = mu_0 N^2 h ln(b/a) / (2 pi). ### II. Mutual Inductance When a changing current in one coil induces an EMF in a nearby coil, this is **mutual induction**. The mutual inductance M relates the induced EMF in the second coil to the rate of current change in the first: EMF_2 = -M (dI_1/dt). The mutual inductance can be expressed as M = N_2 Phi_21 / I_1 = N_1 Phi_12 / I_2, and by the Neumann formula, the mutual inductance is symmetric: M_12 = M_21 = M. The SI unit is the henry. The value of M depends on the geometry, separation, and relative orientation of the two coils. For two tightly coupled coils, M = k sqrt(L_1 L_2), where k is the coupling coefficient ranging from 0 (no coupling) to 1 (perfect coupling). Transformers rely on mutual inductance with k close to 1, achieved by winding both coils on a shared iron core. ### III. Energy Stored in an Inductor Work is required to establish current through an inductor because the inductor opposes the current buildup. The energy stored is U = (1/2) L I^2, analogous to U = (1/2) C V^2 for a capacitor. This energy is stored in the **magnetic field** within the inductor. The **magnetic energy density** (energy per unit volume) is u_B = B^2 / (2 mu_0). This is a general result that applies to any magnetic field, just as the electric energy density u_E = (1/2) epsilon_0 E^2 applies to any electric field. For a solenoid, the total energy U = (1/2) L I^2 can be rewritten as B^2/(2 mu_0) multiplied by the volume of the solenoid, confirming the energy density formula. ### IV. RL Circuit: Current Growth Consider a circuit with a battery (EMF epsilon), a resistor R, and an inductor L in series, with the switch closing at t = 0. Kirchhoff's loop rule gives epsilon - IR - L(dI/dt) = 0. The solution is I(t) = (epsilon/R)(1 - e^(-t/tau_L)), where the **RL time constant** is tau_L = L/R with units of H/Omega = seconds. The voltage across the resistor is V_R(t) = epsilon(1 - e^(-t/tau_L)), and the voltage across the inductor is V_L(t) = epsilon e^(-t/tau_L). At t = 0, the current is zero (the inductor acts like an open circuit) and the full voltage appears across the inductor: V_L = epsilon. As t approaches infinity, the current reaches its steady-state value I = epsilon/R (the inductor acts like a wire) and V_L drops to zero. At t = tau_L, the current has reached 63.2% of its maximum value. After 5 tau_L, the current is within 0.7% of its final value. Two graphs for an RL circuit during current growth, stacked vertically. Top graph: Current I versus time t, starting at zero and rising exponentially toward I_max = epsilon/R (dashed asymptote). The time constant tau_L = L/R is marked, with I = 0.632 I_max at that point. The equation I(t) = (epsilon/R)(1 - e^(-t/tau)) is shown. Bottom graph: Voltage across the inductor V_L versus time t, starting at epsilon and decaying exponentially toward zero. At t = tau_L, V_L = 0.368 epsilon. The equation V_L(t) = epsilon e^(-t/tau) is shown. The circuit diagram (battery, switch, R, and L in series) is inset in the corner. ### V. RL Circuit: Current Decay When the battery is removed and the inductor discharges through the resistor, the loop rule becomes -IR - L(dI/dt) = 0. The solution is I(t) = I_0 e^(-t/tau_L), with V_R(t) = I_0 R e^(-t/tau_L) and V_L(t) = -I_0 R e^(-t/tau_L). The negative sign on V_L indicates that the inductor is opposing the decreasing current, consistent with Lenz's law. The same time constant tau_L = L/R governs the decay. The inductor maintains current flow as long as possible, and the energy stored in its magnetic field is gradually dissipated as heat in the resistor. An important practical warning: if the circuit is broken abruptly (effectively making R very large), the term L(dI/dt) can become enormous, producing a large voltage spike that may cause sparking. Flyback diodes are commonly used to protect circuits from these inductive voltage spikes. ### VI. Comparison of RC and RL Circuits RC and RL circuits are duals of each other, sharing the same mathematical structure but with complementary roles for voltage and current. The RC time constant is tau = RC, while the RL time constant is tau = L/R. In an RC circuit, the capacitor charges up (voltage builds), storing energy in the electric field and opposing changes in voltage. In an RL circuit, the inductor builds up current, storing energy in the magnetic field and opposing changes in current. The behavior at the extremes of time is reversed: at t = 0, a capacitor acts as a short circuit while an inductor acts as an open circuit. At t = infinity, a capacitor acts as an open circuit while an inductor acts as a short circuit. Both circuits exhibit exponential approach to steady state with the same mathematical form. | Property | RC Circuit | RL Circuit | |---|---|---| | Time constant | tau = RC | tau = L/R | | Energy storage | U = (1/2)CV^2 | U = (1/2)LI^2 | | Opposes changes in | Voltage | Current | | At t = 0 (charging) | Short circuit | Open circuit | | At t = infinity | Open circuit | Short circuit | Side-by-side comparison of RC and RL circuit transient responses. Left: RC charging circuit diagram and graph showing V_C rising exponentially with time constant tau = RC. Right: RL current growth circuit diagram and graph showing I rising exponentially with time constant tau = L/R. Both curves have the same mathematical form: (1 - e^(-t/tau)). Below each graph, the corresponding energy storage formula is shown: U = (1/2)CV^2 for RC and U = (1/2)LI^2 for RL.