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

At $t = 0$, the capacitor is uncharged, so the potential difference across it is zero, i.e., $V_C = 0$.

According to Kirchhoff's voltage law for the series RC circuit, $\mathcal{E} = V_R + V_C$.

Substituting $V_C = 0$ at $t = 0$, we get $V_R = \mathcal{E}$.

As time progresses, the capacitor charges up, increasing $V_C$ and decreasing $V_R$ since $V_R = \mathcal{E} e^{-t/RC}$.

Thus, the maximum potential difference across the resistor occurs at $t = 0$ and is equal to $\mathcal{E}$.