Question

The electric current in a charging R−C circuit is given by i = i₀ e^−t/RC where i₀, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) t = 0, (b) t = RC, (c) t = 10 RC.

Answer 1

Given: i = i₀ e^−t/RC
∴ Rate of change of current $=\frac{di}{dt}$
$=i_0\left(\frac{-1}{RC}\right)e^{-t/RC}$
$=\frac{-i_0}{RC}\times e^{-t/RC}$
On applying the conditions given in the questions, we get:
(a) At $t=0,\frac{di}{dt}=\frac{-i_0}{RC}\times e^0=\frac{-i_0}{RC}$
(b) At $t=RC,\frac{di}{dt}=\frac{-i_0}{RC}\times e^{-1}=\frac{-i_0}{RCe}$
(c) At $t=10RC,\frac{dt}{di}=\frac{-i_0}{RC}\times e^{10}$ $=\frac{-i_0}{RC{e}^{10}}$