The physics
Through a resistor, a capacitor charges along a curve, not a straight line. The time constant τ = RC sets the pace.
q = CE (1 − e−t/RC) i = (E / R) e−t/RC
At the first instant, an empty capacitor acts like a plain wire. After a long time, a full one carries no current. Its branch then acts like a break in the wire.
Emptying through R, the charge falls as q₀ e−t/RC. In one τ it falls to 1/e, about 37%. The stored energy goes with q², so it falls twice as fast.
U = U₀ e−2t/RC
Charging from empty, half the battery's work becomes heat in the resistor. This is true for any R.
In a bigger circuit, take the capacitor out. Find the voltage across the gap. That is where it aims. For its τ, make the battery a wire. Then use the resistance between the capacitor's two ends.