Rise and fall of current in RC circuit

Circuit: battery ε, resistor R, capacitor C in series. The time constant is τ = RC.

        +---- R ----+
        |           |
      ε =|         ===  C
        |           |
        +---- S ----+

Charging (switch to battery)

1. Loop rule at time t: ε = iR + q/C, with i = dq/dt.
2. dq/(Cε − q) = dt/(RC). Integrating from q = 0 at t = 0: ln[(Cε − q)/Cε] = −t/RC
3. q = Cε (1 − e^(−t/RC))
4. Current: i = dq/dt = (ε/R) e^(−t/RC) = i₀ e^(−t/τ), with i₀ = ε/R

The current starts at its maximum i₀ and decays to zero as the capacitor charges.

Discharging (battery removed, plates joined through R)

1. Loop rule: the capacitor drives the current through R, so q/C − iR = 0, with i = −dq/dt (the charge decreases).
2. dq/q = −dt/RC. With q = q₀ at t = 0: q = q₀ e^(−t/RC)
3. Current: i = (q₀/RC) e^(−t/RC) = i₀ e^(−t/τ), with i₀ = q₀/RC = V₀/R, in the direction opposite to the charging current.

Graphs

  i                     i
 i0|\                  i0|\
   | \  charging         | \  discharging
   |  `.                 |  `.
   |    ` .  .           |    ` .  .
   |_________ t          |_________ t
   0  tau  2tau            0  tau  2tau

Charging current i = i₀e^(−t/τ) (flows in the charging direction); discharging current has the same exponential shape but flows in the reverse direction.

Explanation

• At t = 0 the uncharged capacitor acts like a short circuit and current is ε/R.
• After t = τ = RC, the current falls to 36.8% (1/e) of i₀.
• After about 5τ, the current is practically zero.
• A larger R or C increases τ and slows the process.