A capacitor of capacitance C is connected to a circuit with…

A capacitor of capacitance $C$ is connected to a circuit with voltage $E$ and resistance $R$. Determine the charge $q$ on the capacitor at time $t$ after connection (uncharged at $t=0$).

Explanation of Method

With $i=dq/dt$, the circuit equation $E=Ri+q/C$ becomes a linear ODE for $q(t)$, solved with the integrating factor $e^{t/(RC)}$.

Step 1: Set up and solve the linear equation

$R\dfrac{dq}{dt}+\dfrac{q}{C}=E$. With $\mu(t)=e^{t/(RC)}$: $\dfrac{d}{dt}\!\left(e^{t/(RC)}q\right)=\dfrac{E}{R}e^{t/(RC)}$, so $q=CE+Ke^{-t/(RC)}$.

Step 2: Apply the initial condition

At $t=0$, $q=0$: $0=CE+K\Rightarrow K=-CE$.

Result

Final answer:

\[\boxed{\,q(t)=CE\left(1-e^{-t/(RC)}\right).}\]

The charge rises to its maximum $CE$ with time constant $\tau=RC$.