Question

derive an expression for the energy stored in a capacitor. show that whenever two conductor share charges by bringing them into electrical contact, there is a loss of energy?

Answer 1

The energy stored in the capacitors can be expressed in terms of work done by the battery. So, the work done to move the charge "dq" from the positive terminal to the negative terminal of the battery is equal to "Vdq" where V is the voltage of the capacitor. This voltage is directly proportional to the current present on the capacitor.
Energy store on the capacitor = dU = V.dq
Energy store on the capacitor = Q/C.dq    ​(since V = Q/C)
If "Q" is the amount of charge stored on the capacitor then the total amount of energy on the capacitor is calculated by the integral.
Hence,
Energy store on the capacitor = U = (integral from 0 to Q) Q/C.dq
Energy store on the capacitor = U = (1/2) . Q²/C
This energy expression can be expressed in three different forms as:
U = (1/2) . Q²/C
U = (1/2) . QV    (∵V = Q/C)
U = (1/2) . CV²   (∵ Q = CV)
Whenever two conductor share charges by bringing them into electrical contact, there is always some amount of loss of energy. Thus energy is scattered in the sharing of charges and is disappeared in the form of heat.

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