In charging singe loop R - C circuit, after how many time constant, kthe potential energy in the capacitor will be 25% of its maximum value.

Explanation

The potential energy of the capacitor at time t is
`U = (q^(2))/(2C)` where `q = C epsilon(1-e^(t//RC))`
Then `U = (C epsilon)/(2) (1 - e^(-t//RC))^2` where `(C epsilon^(2))/(2) = U_(0)`
Putting `U = (U_(0))/(4) = (C epsilon^(2))/(8)`, we have
`(1- e^(-t//RC))^(2) = (1)/(4)`
or `1=e^(-t//RC) = (1)/(2)` or `e^(-t//RC) = (1)/(2)`
or `(t)/(RC) In e = - In 2 or tau = (t)/(RC) = ln2`