Energy in a Charging RC Circuit

Picture a battery of EMF V charging a capacitor C through a resistor R. The battery pushes a total charge Q = CV through the circuit, so its total work is W = QV = CV^2. Once charging finishes, the capacitor holds energy U = (1/2)CV^2. That leaves (1/2)CV² unaccounted for — and that missing half is exactly the heat dissipated in the resistor. This 50-50 split holds no matter what R is: a bigger resistor just stretches out the time constant RC, making charging slower, but the same fraction of energy still turns into heat because the resistor is the only place energy can leak out along the way.

The reason is baked into how charging actually happens. Early on, the capacitor is empty and the full battery voltage falls across the resistor, driving a large initial current and burning energy fast. As charge builds up, the capacitor's voltage rises, the voltage across the resistor shrinks, and the current trickles to zero. Integrating i^2R over the entire exponential charging process always yields (1/2)CV², independent of R, because R only controls the timescale, not the total mismatch between source and capacitor voltage that current has to fight through.

This connects directly to why charging in stages wastes less energy: each smaller step reduces the voltage gap the resistor has to resolve at any instant, so less energy gets dissipated per step. Charging in infinitely many infinitesimal steps is the ideal limit, approaching zero heat loss — the same idea that shows up later comparing reversible and irreversible processes.