Question

P.3-8 A spherical distribution of charge \(\rho=\rho_{0}\left[1-\left(R^{2} / b^{2}\right)\right]\) exists in the region \(0 \leqslant R \leqslant b\). This charge distribution is concentrically surrounded by a conducting shell with inner radius \(R_{i}(>b)\) and outer radius \(R_{o} .\) Determine \(\mathbf{E}\) everywhere.

P.3-9) Two infinitely long coaxial cylindrical surfaces, \(r=a\) and \(r=b(b>a)\), carry surface charge densities \(\rho_{s a}\) and \(\rho_{s b},\) respectively.

a) Determine \(\mathbf{E}\) everywhere.

b) What must be the relation between \(a\) and \(b\) in order that \(\mathbf{E}\) vanishes for \(r>b ?\)

Answer 1

Radius of the sphere is b. Volume charge density of the so here is s = S_0 [1 - (R^2/b^2)] First we calculate of the sphere we draw a elemental sphere of radius R and width dR concentric with the sphere where (0 < R < b) volume of this elemental sphere is dv = 4 pi R^2 dR now change of this elemental strip is dq = S. dv \r\n dq = So (1 - R^2/b^2) times 4 pi R^2 dR dq = 4 pi S_0 (1 - R^2/b^2) R^dR dq = = 4 pi S_0(R^2 - R^4/b^2) dR Now we calculate electric field in the region R < b we draw a Gaussian surface is q_in = integral_0^R dq q_in = integral^R_0 4 pi integral_0 Vertical-bar R^2 - R^4/d^2) dR \r\n q_in = 4 [I S_0 (R^3/3 - R^5/5b^2)^R_0 q_in = 4 pi S_0 (R^3/3 - R^5/5b^2) Now from Gauss�s law E^rightarrow A^rightarrow = q_in/E_0 E 4 pi R^2 = 4 pi s_0 (R^3/3 - R^5/5b^2)/E_0