Charging a capacitor does work against the charge already there, and that work is stored as energy. A graph makes the factor of one half obvious.
The energy stored in a capacitor is the area under the graph of voltage against charge, a triangle, so W = ½QV. Using Q = CV this is also ½CV² or ½Q²/C.
Vary C and V and watch the shaded triangle under the V-Q line. Its area, ½QV, is the stored energy, and it grows with the square of the voltage.
Because the voltage rises from zero to V as the charge builds up, the average voltage is ½V and the energy is ½QV, not QV. Substituting Q = CV gives the equivalent forms ½CV² and ½Q²/C. Energy depends on the square of the voltage, so doubling V quadruples the energy.
Four quick checks on the energy stored and where the factor of ½ comes from. Each correct answer earns XP and lights this skill on your star map.
The energy stored in a capacitor equals the area under a graph of:
The energy stored in a capacitor can be written as:
Doubling the voltage on a capacitor changes the energy stored by a factor of:
The factor of ½ in W = ½QV arises because:
The energy is ½QV, not QV, because the voltage climbs from zero to V as charge accumulates (the average is ½V, giving a triangle). Energy depends on the square of the voltage, so doubling V gives four times the energy, and the three forms ½QV, ½CV² and ½Q²/C are all equal.
This skill is now lit on your star map. Keep the chain going.