Lost Energy When Charging and Discharging Capacitors
When a charge
moves through a potential difference
the energy change is
If the voltage is constant, then the energy change is proportional to the charge transferred. However, the voltage is not generally constant. When a capacitor is being charged the voltage is increasing. The equation
obeyed by capacitors implies that the voltage is directly proportional to the charge. If we draw a graph of voltage against charge we obtain a straight line graph through the origin.
Then the work done in charging the capacitor is equal to the area under the graph is
The energy delivered when a capacitor is being discharged is also
for the same reasons.
We can show this mathematically. For a charging capacitor at any time,
and
so the work done in transferring a small amount of charge
through a potential differenceis
Then
Energy
is needed to charge the capacitor but when fully charged the capacitor only stores
The 'missing' energy is used to overcome resistance in the charging circuit.