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Show that the enerty of a charged capaci...

Show that the enerty of a charged capacitor is `(1)/(2)CV^(2)`. Also express this in other forms.

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To charge a capacitor, an external agent has to do work against the electrostatic forces due to the charges already present on the plates of the capacitor.
Let C be the capacitance of the capacitor. Let Q and V be the final charge and the potential difference respectively when the capacitor is charged. Let q be the charge on the capacitor at an intermediate stage during the charging and v, the corresponding potential difference between the plates. The work done in bringing additional small charge dp from infinity and depositing it on the capacitor is
dW = potential difference `xx` charge = vdq
But, `C=(q)/(v) " " therefore v=(q)/(C) " " therefore dW=(q)/(C)dq`
The total work done in charging the capacitor is
`W=int dW =underset(0)overset(Q)int (qdq)/(C) =(1)/(C)[(q^(2))/(2)]_(0)^(Q) =(1)/(2) (Q^(2))/(C)`
Now, `Q=CV " " therefore W=(1)/(2)CV^(2)=(1)/(2)QV`
This work is stored in the form of potential energy, in the electric field in the medium between the plates of the capacitor.
`therefore` Energy stored in a charged capacitor `=(1)/(2)(Q^(2))/(C)=(1)/(2)CV^(2)=(1)/(2)QV`
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