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

The current in an RC series circuit during charging is given by $I = \dfrac{\mathcal{E}}{R} e^{-t/RC}$.

The power converted into heat is the power dissipated across the resistor, which is given by $P = I^2 R$.

Substituting the expression for current, we get:

$P = \left( \dfrac{\mathcal{E}}{R} e^{-t/RC} \right)^2 R = \dfrac{\mathcal{E}^2}{R} e^{-2t/RC}$

The power is maximum when the exponential term $e^{-2t/RC}$ is maximum. For $t \ge 0$, the maximum value occurs at $t = 0$, where $e^{-2t/RC} = 1$.

Therefore, the maximum power converted into heat is:

P_\max = \dfrac{\mathcal{E}^2}{R}