Question

Derive an expression for the energy stored in charged capacitor

Answer 1

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: $dU = Vdq = \frac{q}{C}dq$

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:

$U = \int\limits^Q_0 \frac{q}{C} \ dq = \frac{1}{2} \frac{Q^{2}}{C}$

Answer 2

"The work done in charging a capacitor is stored in the capacitor in the form of electric energy."
suppose a capacitor of capacitance C initially possess no charge,being connected to a battery. Suppose that it charges to a potential V and q is charge on the plate of the capacitor,then
q= CV
Suppose that battery supplies a small charge dq to the capacitor at constant potential V, then the small amount of work done by the battery is given by,
dw = V dq = (q/C)dq
therefore,amount of work done in delevering charge q to the capacitor is given by,
q q
W = ∫  (q/C) dq = 1/C ∫  q dq
° q °
= 1/C (q^2/2)
°
= 1/2( q^2/C)
and this work done is stored in the capacitor as electrical energy,
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[ W = U = 1/2(q^2/C) = 1/2 (CV^2) = 1/2 qV. ]
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