Growth and decay of current in LR circuit

Circuit: battery emf ε, resistor R, inductor L in series with a switch. The inductive time constant is τ = L/R.

        +----- R -----+
        |             |
      ε =|            L (coil)
        |             |
        +---- S ------+

Growth of current (switch to the battery)

1. Kirchhoff's loop rule: ε − iR − L di/dt = 0
2. Rearrange: di/(ε/R − i) = (R/L) dt
3. Integrate from i = 0 at t = 0: −ln[(ε/R − i)/(ε/R)] = Rt/L
4. i = (ε/R)(1 − e^(−t/τ)) = i₀(1 − e^(−t/τ)), where i₀ = ε/R

Decay of current (battery removed, circuit closed through R)

1. Loop rule with ε = 0: iR + L di/dt = 0
2. di/i = −(R/L) dt. Integrate from i = i₀ at t = 0:
3. i = i₀ e^(−t/τ)

Graph

  i      Growth                    i      Decay
 i0 |- - - - - - - _____         i0 |.
    |         . '                   | `.
    |      .'                       |   `.
    |    .'                         |     ` .
    |  .'                           |         ` .  .  .
    |.'_____________________ t      |_____________________ t
    0   tau   2tau   3tau           0   tau   2tau   3tau

The growth curve rises towards i₀ and the decay curve falls towards zero; the two are mirror images.

Explanation

• Time constant τ = L/R (seconds): the time in which the current grows to 63.2% of i₀ (1 − e⁻¹), or decays to 36.8% (e⁻¹) of i₀.
• At t = 0 the inductor opposes any change: growth starts with i = 0 and slope ε/L.
• After about 5τ the current is within 1% of its final value.
• A large L or small R gives a larger τ, so the current changes more slowly.