At t = 0 , a capacitance C , a resistance R , and an emf E are...

The charge on the capacitor at any time $t$ during the charging of an RC circuit is given by:

$q(t) = C \mathcal{E} (1 - e^{-t/RC})$

The energy stored in the capacitor at any time $t$ is:

$U(t) = \dfrac{q(t)^2}{2C}$

The maximum charge on the capacitor is reached in the steady state as $t \rightarrow \infty$:

q_\max = C \mathcal{E}

Substituting the maximum charge into the energy formula, the maximum energy stored in the capacitor is:

U_\max = \dfrac{q_\max^2}{2C} = \dfrac{(C \mathcal{E})^2}{2C} = \dfrac{1}{2} C \mathcal{E}^2