Question

How long does it take for the energy stored in a capacitor in a series RC circuit to reach 75% of its maximum value? Express answer in terms of the time constant t = RC.

Answer 1

Energy stored on a capacitor U = (1/2) * C * V²

maximum energy U₀   = 0.5 * C * V₀²

Hence 75% of maximum value U = 0.75 * U₀   = 0.5 * C * V²

Dividing 0.75 =   V² / V₀²

During charging V = V₀ * (1 - e^-t/τ) where τ (time constant) = RC

=> 0.75 = {V₀ * (1 - e^-t/τ) / V₀}²

0.75 = (1 - e^-t/τ)²

√0.75 = 1 - e^-t/τ

e^-t/τ = 0.134

e^t/τ = 1/0.134

taking log t/R*C = ln (7.464)

time taken t = 2.01 * R * C