Short on time? This page is the recap. For the derivation, Figures 1 and 2 and the worked examples, use the full study guide. This topic is in the calculus-based course.
Recap
- An inductor’s emf opposes the change in current: ℰ_L = −L dI/dt.
- Its current cannot jump. Just after a switch moves, it has its old value; an inductor with zero current acts like an open branch.
- The induced emf just after switching is equal in size and opposite in direction to the potential difference applied across the inductor’s branch.
- After a long time (several τ), the current is steady and the inductor acts as a wire with zero resistance.
- Current, inductor voltage and stored energy all change exponentially between these two states.
- The resistor dissipates the energy the inductor stored.
Key relationships
| Situation | Result |
|---|---|
| Loop rule, series ℰ, R, L | ℰ = IR + L dI/dt |
| Time constant | τ = L/R (H/Ω = s) |
| Growth from zero | I = (ℰ/R)(1 − e^(−t/τ)); V_L = ℰe^(−t/τ) |
| Initial rate of growth | dI/dt = ℰ/L; at this rate I would reach ℰ/R at t = τ |
| Decay from I₀ (no battery) | I = I₀e^(−t/τ) |
| After one τ | growth: 0.63 of final; decay: 0.37 of start |
| Energy stored | U_L = ½LI²; during decay U ∝ e^(−2t/τ) |
| Energy balance | ℰI = I²R + d/dt(½LI²) |
| Energy dissipated in a decay | ∫I²R dt = ½LI₀² |
Assumptions behind the results
- Ideal inductor (no resistance in its windings) unless a resistance is given; a real coil’s resistance adds to R.
- Ideal battery (no internal resistance) unless stated.
- Straight connecting wires have negligible inductance.
Mistakes to avoid
- Letting the inductor current jump. It is continuous; other branch currents can jump.
- τ = RL or R/L. It is L/R.
- “Final value after one τ.” Only 63%; allow about 5τ.
- Same τ for growth and decay. Use the resistance in the loop the inductor’s current actually flows through.
- Forgetting direction. When a switch opens, the inductor current can reverse the current in a parallel resistor.
- Energy decays with τ. It decays with τ/2, because U ∝ I².
Quick self-check
- L = 0.50 H and R = 25 Ω. What is τ? (20 ms)
- A series LR circuit has ℰ = 6.0 V and R = 30 Ω. What is the current a long time after switching on? (0.20 A)
- A current of 2.0 A decays through a resistor. What is it after 2τ? (2.0e⁻² ≈ 0.27 A)
- Just after closing the switch in a series LR circuit, what is the potential difference across the inductor? (ℰ, the full battery emf)
Next: practice questions.