Is the energy supplied to a capacitor by a battery the same as the energy stored by a capacitor?

Question

If we have three capacitors in series, would the energy supplied to the system be the same as the energy that is contained in the equivalent capacitance of these three capacitors?

Answer 1

The energy conservation law tells us that the energy stored in the capacitors will be the energy supplied by the buttery, minus the energy dissipated in the wires while charging the capacitors.

Answer 2

TL;DR: The battery has to supply twice the energy that gets stored in the capacitors.

The characteristics of your power source matter.

The "three capacitors in series" don't make a difference, they behave just the same as one capacitor of appropriate value C.

In your question's title (but not in its body), you mention the power source to be a battery, and a battery is typically best described as constant-voltage source (voltage U) with a series resistor (resistance R).

The battery will supply current to the capacitor until the capacitor's voltage equals the battery voltage. During this charging process, the voltage difference between the battery and the partially-charged capacitor is the voltage drop of the resistor R, resulting in heat dissipation = energy loss.

Finally (after "infinite" time), the capacitor's voltage reaches the battery voltage. The charge necessary for this is

Q = C * U

and the energy stored in the capacitor is

Ecap = 0.5 * Q * U

The energy supplied by the battery is

Ebatt = Q * U

So, half of the battery energy goes into the capacitor, the other half gets dissipated in the resistor (wires, internal resistance of the battery or the capacitors.

If you had a controllable voltage source, capable of producing a slow voltage ramp rising from 0 to U, there would be nearly no voltage drop on the resistor, and nearly all the energy would reach the capacitor. But a battery doesn't work like that.