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Question
Obtain the expression for energy stored in the parallel plate capacitor.
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Solution
- Capacitor not only stores the charge but also stores energy.
- When a battery is connected to the capacitor, electrons of total charge -Q are transferred from one plate to the other plate.
- To transfer the charge, work is done by the battery. This work done is stored as electrostatic potential energy in the capacitor.
- To transfer an infinitesimal charge dQ for a potential difference V, the work done is given by
dW = V dQ ………….(1)
where V = `"Q"/"C"`
The total work done to charge a capacitor is
W = `int_0^"Q" "Q"/"C" "dQ" = "Q"^2/"2C"` .....(2)
This work done is stored as electrostatic potential energy (UB) in the capacitor.
`"U"_"E" = "Q"^2/"2C" = 1/2 "CV"^2` ....(3)
where Q = CV is used. - This stored energy is thus directly proportional to the capacitance of the capacitor and the square of the voltage between the plates of the capacitor.
`"U"_"E" = 1/2 ((ε_0"A")/"d")("Ed")^2 = 1/2 ε_0 ("Ad")"E"^2` ....(4)
where Ad = volume of the space between the capacitor plates. The energy stored per unit volume of space is defined `"U"_"E" = 1/2ε_0 "E"^2` .....(5) - Energy is stored in the electric field existing between the plates of the capacitor. 0nce the capacitor is allowed to discharge, the energy is retrieved.
- The energy density depends only on the electric field and not on the size of the plates of the capacitor.
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