Short on time? This page is the recap. For the derivations, Figure 1 and the worked examples, use the full study guide. This topic is in the calculus-based course.
Recap
- Inductance is a conductor’s tendency to oppose a change in its own current (self-induction, from Faraday’s and Lenz’s laws).
- It depends on the conductor’s physical form. Straight wires are modelled as having zero inductance; a solenoid is an inductor.
- A solenoid’s inductance depends on the number of turns, the length, the cross-sectional area and the permeability of the core.
- The self-induced emf depends on the rate of change of current, not on the current.
- An inductor stores energy in its magnetic field. The energy can later be dissipated in a resistor or passed to a capacitor, with total energy conserved.
- μ₀ = 4π × 10⁻⁷ T·m/A; 1 H = 1 V·s/A = 1 Wb/A = 1 Ω·s.
Key relationships
| Quantity | Expression | Notes |
|---|---|---|
| Definition of inductance | L = NΦ_B/I | flux linkage per ampere |
| Long solenoid | L = μ_core N²A/ℓ | μ_core = μ₀ for air; L ∝ N² |
| Self-induced emf | ℰ = −L dI/dt | minus sign = Lenz’s law |
| Power into the inductor | P = LI dI/dt | positive while the current rises |
| Stored energy | U = ½LI² | from integrating P dt |
| Energy released, I₁ → I₂ | ½L(I₁² − I₂²) | not ½L(I₁ − I₂)² |
Assumptions behind the results
- The solenoid is long compared with its radius, so the field inside is uniform and the field outside is negligible.
- The core’s permeability is constant (L does not depend on I).
- An “ideal” inductor has no resistance.
Mistakes to avoid
- “An inductor opposes current.” It opposes a change in current.
- Using I instead of dI/dt for the emf.
- Forgetting N² in the solenoid formula.
- Energy released as ½L(ΔI)². Subtract the two energies instead.
- Wrong emf direction. Against the current while it rises; along it while it falls.
Quick self-check
- The current in a 2.0 H inductor rises at 5.0 A/s. What is the size of the self-induced emf? (10 V, opposing the current)
- How much energy does a 0.50 H inductor store at 4.0 A? (4.0 J)
- A solenoid’s turns are tripled with nothing else changed. By what factor does L change? (× 9)
Next: practice questions.