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A parallel plate capacitor with plate area 'A' and distance of separation 'd' is filled with a dielectric. What is the capacity of the capacitor when the permittivity of dielectric varies as: ε(x) = ε₀ + kx, for (0 ≤ x ≤ d/2) ε(x) = ε₀ + k(d−x), for (d/2 ≤ x ≤ d)

1. (ε₀ + kd/2)^(2/kA)
2. kA / (2 ln((2ε₀ + kd)/(2ε₀)))
3. 0
4. (kA/2) ln((2ε₀)/(2ε₀ − kd))

Correct answer: kA / (2 ln((2ε₀ + kd)/(2ε₀)))

Solution

Treat the dielectric as infinitesimal slabs in series. For 0 to d/2, 1/C1 = (1/(kA)) ln((e0+kd/2)/e0); the second half is identical by symmetry. So 1/C = (2/(kA)) ln((2e0+kd)/(2e0)), giving C = kA / (2 ln((2e0+kd)/(2e0))).

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