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Question
Obtain the expression for the energy stored in a capacitor connected across a dc battery. Hence define energy density of the capacitor
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Solution
A capacitor is connected across the terminals of a d.c. battery.
The energy stored on a capacitor is equal to the work done by the battery.
The work required to transport a small amount of charge (dQ) from the negative to positive plates of a capacitor is equal to V dQ, where V represents the voltage across the capacitor.
dU = V dQ
= `Q/C dQ`
∴ Energy stored (U) = ∫V dQ
= `1/C int Q dQ`
= `1/2 Q^2/C`
= `1/2 CV^2` ...(i)
Energy density is defined as the total energy per unit volume of the capacitor.
For a parallel plate capacitor,
C = `(A epsilon_0)/d`
Putting in eqn. (i),
U = `1/2 (A epsilon_0)/d V^2`
= `epsilon_0/2 Ad(V/d)^2`
= `epsilon_0/2 Ad E^2` ...[Putting `V/d` = E]
A × d = Volume of space between plates
So, energy is stored per unit volume.
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