Inductance and RL/RLC Circuits
An inductor is what Unit 208's induction becomes when a coil's own changing current creates a changing flux through itself, inducing a "back-EMF" that opposes the change — not the current, the change in current. This gives circuits a second way to store energy (in the magnetic field, alongside the capacitor's electric-field storage from Unit 204) and, when an inductor and capacitor share a circuit, produces LC oscillation: energy sloshing back and forth between the two fields, a direct electrical analog of a mass on a spring.
What you'll learn
- Calculate the self-inductance of a solenoid from its geometry.
- Calculate the EMF induced across an inductor by a changing current through it.
- Analyze the growth and decay of current in an RL circuit using exponential functions.
- Calculate the energy stored in an inductor's magnetic field.
- Analyze the oscillation of an LC circuit, including its angular frequency.
- Describe qualitatively how adding resistance to an LC circuit changes its behavior.
1. Self-Inductance
A coil's own current creates its own magnetic flux through itself. When that current changes, the resulting changing flux induces an EMF back across the coil, opposing the change — this is self-inductance: EMF=−L dI/dt, where L (in henries) is a purely geometric property, exactly parallel to a capacitor's C.
For a solenoid, L=μ₀n²Al — note the n² dependence, stronger than the single power of n that appeared in Unit 207's B=μ₀nI, since each turn both carries current and links flux from every other turn.
2. RL Circuits
An inductor and resistor in a loop produce exponential current growth or decay, exactly mirroring Unit 205's RC circuit with current standing in for charge: I(t)=I_max(1−e^(−t/τ)) for growth, I(t)=I₀e^(−t/τ) for decay, with τ=L/R.
The physical difference from RC is what's "sluggish": a capacitor resists changes in voltage (charge takes time to build up), while an inductor resists changes in current itself — at the instant a switch closes, current in an RL circuit starts at zero and ramps up, rather than jumping immediately to what Ohm's law alone would predict.
3. LC Oscillation
An inductor and capacitor together, with no resistance, form a circuit that oscillates indefinitely: energy sloshes between the capacitor's electric field and the inductor's magnetic field, exactly like a frictionless mass on a spring exchanges kinetic and potential energy. The oscillation's angular frequency is ω=1/√(LC), and total energy (½Q²/C + ½LI²) stays constant throughout, even as it continuously trades forms.
Adding resistance (an RLC circuit) introduces energy dissipation each cycle, causing the oscillation's amplitude to gradually decay — damped oscillation, the same qualitative behavior a real (frictional) mass-spring system shows.
Key equations
- EMF = −L dI/dt — The EMF an inductor generates in response to a changing current through it — depends entirely on the *rate of change* of current, not the current's own size. A steady current, however large, induces zero EMF.
- L = μ₀n²Al — Self-inductance of a solenoid, from its geometry alone — the inductor's analog of a capacitor's C=ε₀A/d.
- U = ½LI² — Energy stored in an inductor's magnetic field — the magnetic parallel to a capacitor's U=½CV².
- I(t) = I_max(1−e^(−t/τ)) [growth], I(t) = I₀e^(−t/τ) [decay], τ=L/R — Current in an RL circuit — the exact structural mirror of Unit 205's RC circuit, with current here playing the role charge played there.
- ω = 1/√(LC) — An LC circuit (no resistance) oscillates indefinitely, exchanging energy between the capacitor's electric field and the inductor's magnetic field — the electrical analog of a frictionless mass-spring system, with L playing the role of mass and 1/C playing the role of spring stiffness.