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Derive the expression for energy stored ...

Derive the expression for energy stored in a charged capacitor.

Text Solution

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Writing `dW=Vdq=(q)/(C).dq`
Integrating `int_(0)^(Q)dW=int_(0)^(Q)(q)/(C).dq`
Arriving `U=(Q^(2))/(2C)=(QV)/(2)=(CV^(2))/(2)`
Detailed Answer:
Let `dU=dW=Vdq=(q)/(c)=dq`
The total increase in potential energy in charging the capacitor from `q=0` to `q=Q` is the total energy stored in the capacitor
`therefore U=intdU`
`=int_(0)^(Q)(q)/(c)dq=(1)/(c)[(q^(2))/(2)]_(0)^(Q)=(1)/(c)[(Q^(2))/(2)]=(Q^(2))/(2C)`
`U=(1)/(2)(Q^(2))/(C)`
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