Cylindrical Capacitor: Gauss Law

Please refer to the attachment.

a) For the region a < r < b (region 1) : Let us consider a cylindrical Gaussian surface of length L shown in dotted red in the fig. and apply Gauss theorem : ∫E.ds = Q/ε where Q is the total charge enclosed by the Gaussian surface. Here, Q = charge on length L of the inner cylindrical plate = λL

Let E1 be the radially outwards electric field on the curved face of the Gaussian cylinder. The, applying Gauss theorem : ∫E.ds = E1x 2ΠrL = λL/ε1

Or E1 = λ/2Πε1r

As D = εE, D1 = λ/2Πr

Similarly for the region b < r < c (region 2) : ∫E.ds = E2x 2ΠrL = λL/ε2

Or E2 = λ/2Πε2r and D2 = λ/2Πr

We know that energy density w = ½ εE2.

Energy density in region 1 = w1 = ½ ε1E12 = ½ ε1(λ/2Πε1r)2 = λ2/8Π2ε1r2

Total energy stored in a concentric hollow cylinder of length L, radius r and wall thickness dr = dW1 = Energy density x Volume = [λ2/8Π2ε1r2] x 2Πrdr L = [λ2L/4Πε1r] dr

b
W1 = [λ2L/4Πε1]∫1/r dr = [λ2L/4Πε1] loge(b/a)
a
Similarly total energy stored in region 2 = W2 = [λ2L/4Πε2] loge(c/b)
Total energy stored in the capacitor = W = [λ2L/4Πε1] loge(b/a) + [λ2L/4Πε2] loge(c/b)

Or W = [λ2L/4Πε1ε2][ε2loge(b/a) + ε1loge(c/b)] ...........(1)

b) Energy stored in the capacitor is also given by : W = Q2/2C

Here, Q = λL. Hence, W = λ2L2/2C .........(2)

Equating (1) and (2) : λ2L2/2C = [λ2L/4Πε1ε2][ε2loge(b/a) + ε1loge(c/b)]

Or L/C = [1/2Πε1ε2][ε2loge(b/a) + ε1loge(c/b)]

Or C = 2ΠLε1ε2/[ε2loge(b/a) + ε1loge(c/b)]

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