Source: University Physics: Elec & Mag, UIUC
Tags: inductor, self-inductance, RL circuit, Faraday's law, time constant, solenoid, exponential decay, tau, L/R, EMF
Difficulty: Intermediate Prerequisites: Faraday's law basics, Kirchhoff's voltage rule (KVR), RC circuits (for analogy).
Big picture: Inductors are the magnetic counterpart to capacitors. Where a capacitor resists sudden changes in voltage by storing energy in an electric field, an inductor resists sudden changes in current by storing energy in a magnetic field. This topic builds directly on Faraday's law and is the gateway to understanding LC oscillations, RLC damping, and eventually AC circuits. If you are comfortable with RC circuit analysis (exponential charging/discharging, time constants), the RL circuit maths will feel familiar.
An inductor opposes changes in current through it by generating a back-EMF proportional to the rate of current change. In an RL circuit, current rises or falls exponentially with a time constant τ = L/R. At t = 0 the inductor holds whatever current it had before the switch flipped; after a long time it behaves like a plain wire.
Self-inductance (L)
The ratio of total magnetic flux through a coil to the current producing that flux: L = Φ_B / I. Measured in henrys (H). In simple terms, it tells you how strongly a coil resists changes in current. A larger L means more resistance to change.
EMF (electromotive force, ε)
The voltage induced across an inductor when the current through it changes. Given by ε = −L (dI/dt). Think of it as the inductor's "pushback" against whatever is trying to alter its current.
Time constant (τ)
For an RL circuit, τ = L/R. It sets the timescale for exponential current growth or decay. After about 5τ, the circuit has essentially reached its steady state. Think of it as the circuit's response speed: large τ means slow to change, small τ means quick.
Solenoid
A coil of wire wound in a helix. Its self-inductance is L = μ₀ n² π r² ℓ, where n is the number of turns per unit length, r is the radius, and ℓ is the total length.
Faraday's law: ε = −dΦ_B / dt
Since Φ_B = LI (with L constant for a given coil), this becomes:
ε = −L (dI/dt)
The negative sign means the induced EMF opposes the change in current (Lenz's law).
Key consequence: an inductor prevents discontinuous (instantaneous) current changes. Current through an inductor is always continuous.
L = μ₀ n² π r² ℓ
μ₀ = permeability of free space (4π × 10⁻⁷ T·m/A)
n = number of turns per unit length (n = N/ℓ)
r = radius of the solenoid
ℓ = length of the solenoid
Circuit: battery (V_batt), inductor (L), and resistor (R) in series.
Apply KVR: −V_batt + V_L + V_R = 0, so −V_batt + L(dI/dt) + IR = 0.
At t = 0: current is unchanged from its value just before the switch closes. If the inductor was previously carrying no current, I(0) = 0.
The inductor initially takes the full battery voltage: V_L = V_batt, V_R = 0.
Over time, current grows exponentially toward its maximum value.
After a long time (t >> τ): dI/dt → 0, so V_L → 0 and V_R → V_batt. The inductor behaves as a plain wire.
Steady-state current: I = V_batt / R
If the inductor has been carrying steady current I₀ = V_batt / R and the battery is then removed (switch opens or rerouted), the inductor drives current through R on its own.
At t = 0 (switch opens): I_L is unchanged. I(0) = I₀.
KVR for the decay loop: −L(dI/dt) + IR = 0, which gives V_L = V_R.
Current decays exponentially: I(t) = I₀ e^(−t/τ), where τ = L/R.
A larger time constant τ means the inductor takes longer to discharge its stored current.
Starting from L(dI/dt) + IR = 0:
(1/I) dI = −(R/L) dt
Integrate both sides: ln(I) = −(R/L)t + ln(I₀)
Therefore I = I₀ e^(−Rt/L) = I₀ e^(−t/τ)
Quantity | Formula |
|---|---|
Self-inductance | L = Φ_B / I |
Solenoid inductance | L = μ₀ n² π r² ℓ |
Inductor EMF | ε = −L (dI/dt) |
Time constant | τ = L / R |
Current decay (RL) | I(t) = I₀ e^(−t/τ) |
Inductors are everywhere in power supplies and signal filtering. The RL time constant governs how quickly relays switch, how spark plugs fire in engines, and how fast current ramps up in electromagnets. Transformers, which you will meet later, rely entirely on mutual inductance between coils.
Students often assume that current through an inductor can jump instantaneously when a switch is flipped. It cannot. The inductor enforces continuity of current at every instant.
Confusing the RL time constant (τ = L/R) with the RC time constant (τ = RC). Both describe exponential behaviour, but the formulas are different. For RL, a larger R means a faster decay (smaller τ). For RC, a larger R means a slower decay (larger τ).
Thinking the inductor "blocks" current permanently. It only opposes changes. Once the current is steady, an ideal inductor is just a wire with zero resistance.
Forgetting the sign convention: V_L = L(dI/dt) when writing KVR, but the induced EMF is ε = −L(dI/dt). Getting these mixed up leads to sign errors in circuit equations.
⚠️ At t = 0, the inductor's current is unchanged from its value just before the switch event. This is the most common starting condition tested.
⚠️ After a long time (t >> L/R), V_L = 0 and the inductor acts as a wire. Know how to find steady-state current from this.
⚠️ Be ready to derive the exponential decay by separation of variables. This derivation appears frequently on exams.
⚠️ Watch the direction of V_L in KVR: the sign depends on whether current is increasing or decreasing.
True or false: The current through an inductor can change instantaneously.
Fill in the blank: The time constant of an RL circuit is τ = ______.
True or false: After a very long time, the voltage across an ideal inductor in a DC RL circuit is zero.
Fill in the blank: The self-inductance of a solenoid depends on μ₀, the square of ______, the cross-sectional area, and the length.
True or false: Doubling the resistance in an RL circuit doubles the time constant.
Answers: 1. False. 2. L/R. 3. True. 4. The number of turns per unit length (n). 5. False (it halves it).
Q: A 50 mH inductor is in series with a 200 Ω resistor and a 10 V battery. What is the time constant, and what is the current after a very long time?
A: τ = L/R = 0.050/200 = 0.25 ms. After a long time, I = V/R = 10/200 = 50 mA.
Q: In an RL circuit, the switch has been closed for a long time and then opens at t = 0. Describe the current and inductor voltage at t = 0⁺.
A: At t = 0⁺, the current is unchanged from its steady-state value I₀ = V_batt/R. The inductor voltage jumps to V_L = I₀R (equal and opposite to V_R), driving current through the resistor as the stored energy dissipates.
Q: Why does an ideal inductor behave like a wire after a long time in a DC circuit?
A: Because dI/dt = 0 at steady state, so V_L = L(dI/dt) = 0. With no voltage drop across it, the inductor is indistinguishable from a short piece of wire.
Q: Derive the expression for current decay in an RL circuit after the battery is disconnected.
A: From KVR: L(dI/dt) + IR = 0. Rearrange: dI/I = −(R/L)dt. Integrate: ln(I/I₀) = −(R/L)t. Therefore I(t) = I₀ e^(−Rt/L) = I₀ e^(−t/τ).
This connects directly to RC circuits: the mathematical structure is identical (exponential approach/decay), but the roles of the components are swapped. Understanding RL circuits is essential for the LC and RLC oscillation topics that follow, where the interplay between inductor and capacitor produces oscillatory behaviour. It also underpins transformer theory and AC circuit analysis later in the course.
Related Terms / Search Tags: inductor, self-inductance, henry, RL circuit, time constant tau, L/R, Faraday's law, Lenz's law, solenoid inductance, exponential decay, KVR, back-EMF, magnetic flux, PHY 212, UIUC, midterm 3, inductor as wire, current continuity