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(a) Describe briefly the process of transferring the charge between the two plates of a parallel plate capacitor when connected to a battery. Drive an expression for the energy stored in a capacitor. 

(b) A parallel-plate capacitor is charged by a battery to a potential difference V. It is disconnected of the same capacitance. Calculate the ratio of the energy stored in the combination of the initial energy on the single capacitor.

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(a) When the two plates of a parallel-plate capacitor are connected to a battery, it pulls electrons from the positive plate of the capacitor and gives them to the negative plate.

Thus, the battery transfers positive charge from negative to positive plate the work done in this transfer process is stored in form of electrostatic energy in the capacitor.

Energy stored in capacitor: Let charge on the capacitor at any instant be q. Then, the potential of capacitor is q c/ , where c is the capacitance. Suppose an additional charge dq is given to capacitor, then the work done in giving this additional charge is

Total work done in giving charge Q to capacitor is

which is the form of energy stored. Using Q = CV, we have

which is the expression for energy stored in a capacitor. 

(b) When a parallel-plate capacitor is charge by a battery to a potential difference V is disconnected from battery and connected to an uncharged capacitor of same capacitance, charge flows from the charged capacitor to the uncharged capacitor till their potentials become equal. The equal potential of the two capacitors is called common potential. Initially: 

• Charge on capacitor 1 is q1 = q 

• Potential difference V1 is V. 

• Capacitance C1 is C.

• Charge on capacitor 2 is q2 = 0. 

• Potential difference V2 is 0. 

• Capacitance C2 is C.

The energy stored in capacitor 1 is

When two capacitors are combined:

Charge on combination q' is q1 + q2 =q

Capacitance C' is  C1 + C2 = C + C =2C.

Potential difference is

Energy stored in combination is

The ratio of energy stored in the combination to initial energy on single capacitor is 1/2.

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