Question

Find out the time at which the potential across the capacitor is equal to that across the
resistor, when switch S is closed at t=0, the capacitor is charged through the resistor.
Consider the value of capacitance of a capacitor= 12mF and resistance of resistor is 25ohm.

Answer 1

The voltage across a capacitor during charging can be describe with the following formula:

$$U_C = U_0(1 - e^{-\frac{t}{RC}})$$

where $U_0$ is the voltage of the battery, $t$ is time, $C = 12\times 10^{-3}F$ is the capacitance of the capacitor and $R = 25\Omega$ is the resistance of the resistor.

On the other hand, the voltage across the resistor is:

$$U_R = U_0 - U_C = U_0 - U_0(1 - e^{-\frac{t}{RC}}) = U_0e^{-\frac{t}{RC}}$$

The required condition is:

$$U_C = U_R\\$$

$$U_0(1 - e^{-\frac{t}{RC}}) = U_0e^{-\frac{t}{RC}}\\ e^{-\frac{t}{RC}} = \dfrac12$$

Expressing the time from the last equation, get:

$$t = -RC\ln\dfrac12 = -25\cdot 12\cdot 10^{-3}\ln \dfrac12\approx 0.21s$$

Answer. 0.21 s.