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Question
The plates of a parallel plate capacitor have an area of 90 cm2 each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply.
(a) How much electrostatic energy is stored by the capacitor?
(b) View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u. Hence arrive at a relation between u and the magnitude of electric field E between the plates.
Solution
Area of the plates of a parallel plate capacitor, A = 90 cm2 = 90 × 10−4 m2
Distance between the plates, d = 2.5 mm = 2.5 × 10−3 m
Potential difference across the plates, V = 400 V
(a) Capacitance of the capacitor is given by the relation,
C = `(in_0"A")/"d"`
Electrostatic energy stored in the capacitor is given by the relation, `"E"_1 = 1/2 "CV"^2`
= `1/2 (in_0"A")/"d""V"^2`
Where,
`in_0` = Permittivity of free space = 8.85 × 10−12 C2 N−1 m−2
∴ `"E"_1 = (1 xx 8.85 xx 10^-12 xx 90 xx 10^-4 xx (400)^2)/(2 xx 2.5 xx 10^-3) = 2.55 xx 10^-6 "J"`
Hence, the electrostatic energy stored by the capacitor is `2.55 xx 10^-6 "J"`
(b) Volume of the given capacitor,
`"V'" = "A" xx "d"`
= `90 xx 10^-4 xx 25 xx 10^-3`
= `2.25 xx 10^-4 "m"^3`
Energy stored in the capacitor per unit volume is given by,
`"u" = "E"_1/("V'")`
= `(2.55 xx 10^-6)/(2.25 xx 10^-4) = 0.113 "J m"^-3`
Again, u = `"E"_1/("V'")`
= `((1/2)"CV"^2)/("Ad") = ((in_0"A")/(2"d")V^2)/("Ad") = 1/2in_0("V"/
"d")^2`
Where,
`"V"/"d"` = Electric intensity = E
∴ `"u"=1/2 in_0"E"^2`
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