In a charging RC circuit, determine the number of time constants that will elapse before the energy stored in the capacitor attains half of its equilibrium value.

Step-by-step solution

The energy stored in a capacitor at any time t is given by U = q^2 2C , where q is the charge on the capacitor at time t . The maximum (equilibrium) energy stored in the capacitor is U₀ = q₀^2 2C , where q₀ is the maximum charge. Given that the energy attains half of its equilibrium value: U = U₀ 2 q^2 2C = 1 2 q₀^2 2C q = q₀ 2 For a charging RC circuit, the charge as a function of time is: q = q₀(1 - e^ -t/ ) Substituting q = q₀ 2 : q₀ 2 = q₀(1 - e^ -t/ ) 1 - e^ -t/ = 1 2 e^ -t/ = 1 - 1 2 = 2 -1 2 e^ t/ = 2 2 -1 =