Storing Energy in a Capacitor

The energy stored on a capacitor can be expressed in terms of the work done by the battery. Voltage represents energy per unit charge, so the work to move a charge element dq from the negative plate to the positive plate is equal to V dq, where V is the voltage on the capacitor. The voltage V is proportional to the amount of charge which is already on the capacitor.

Element of energy stored:

If Q is the amount of charge stored when the whole battery voltage appears across the capacitor, then the stored energy is obtained from the integral:

More detail Calculation

This energy expression can be put in three equivalent forms by just permutations based on the definition of capacitance C=Q/V.

Storing Energy in a Capacitor

When the switch is closed to connect the battery to the capacitor, there is zero voltage across the capacitor since it has no charge buildup. The voltage on the capacitor is proportional to the charge

Storing energy on the capacitor involves doing work to transport charge from one plate of the capacitor to the other against the electrical forces. As the charge builds up in the charging process, each successive element of charge dq requires more work to force it onto the positive plate. Summing these continuously changing quantities requires an integral.

Calculation More detail on integral

Note that the total energy stored QV/2 is exactly half of the energy QV which is supplied by the battery, independent of R!

Capacitor Energy Integral

Transporting differential charge dq to the plate of the capacitor requires work

But as the voltage rises toward the battery voltage in the process of storing energy, each successive dq requires more work. Summing all these amounts of work until the total charge is reached is an infinite sum, the type of task an integral is essential for. The form of the integral shown above is a polynomial integral and is a good example of the power of integration.