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13.5 Circuits with Resistors and Inductors (LR Circuits)

Part of Electromagnetic Induction in AP® Physics C: Electricity and Magnetism. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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An LR circuit contains a resistor and an inductor. When connected to a battery, the current does not jump instantly to its final value because the inductor opposes changes in current. Applying the loop rule to a series LR circuit connected to emf E gives

E−IR−LdIdt=0\mathcal{E} - IR - L\frac{dI}{dt} = 0

Rewriting,

LdIdt+RI=EL\frac{dI}{dt} + RI = \mathcal{E}

This first-order differential equation has solution

I(t)=ER(1−e−tR/L)I(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-tR/L}\right)

for current growth after the switch is closed. The time constant is

τ=LR\tau = \frac{L}{R}

which sets the timescale for the change.

When the battery is removed and the inductor discharges through the resistor, the loop rule becomes

LdIdt+RI=0L\frac{dI}{dt} + RI = 0

and the solution is

I(t)=I0e−tR/LI(t) = I_0 e^{-tR/L}

The exponential form reflects the fact that the rate of current change is proportional to how far the current is from its equilibrium value. This is directly analogous to RC circuits, but here it is the current, not the charge, that changes exponentially.

In this unit

6 lessons

  1. 13.1Magnetic FluxPreview lesson
  2. 13.2Electromagnetic InductionPreview lesson
  3. 13.3Induced Currents and Magnetic ForcesPreview lesson
  4. 13.4InductancePreview lesson
  5. 13.5Circuits with Resistors and Inductors (LR Circuits)This lesson
  6. 13.6Circuits with Capacitors and Inductors (LC Circuits)Preview lesson

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