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

The charge on the capacitor in a series RC circuit as a function of time is given by:

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

The potential difference across the capacitor at any time $t$ is:

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

The maximum potential difference is attained at steady state, which corresponds to $t \rightarrow \infty$.

At $t \rightarrow \infty$, $e^{-t/RC} \rightarrow 0$.

Therefore, the maximum potential difference across the capacitor is:

(V_C)_{\text\max} = \mathcal{E}(1 - 0) = \mathcal{E}

Alternatively, at steady state, the capacitor acts as an open circuit. The current in the circuit becomes zero, so the potential drop across the resistor is zero. By Kirchhoff's voltage law, the entire emf $\mathcal{E}$ appears across the capacitor.