Question

2. A charge distribution with spherical symmetry has density PoR (1) for 0< R< a, and is zero for R spherical variable. Determine a. Here po and a are constants, and R is the (a) (20 points) E everywhere. (b) (20 points) the potential, V, everywhere.

Answer 1

Volume charge density is rho_v = rho_0 R/a first we calculate the total charge of the spherical region. we cut an elemental sphere of radius R and width dR conclntric with the sphere volume of this elemental sphere is dv = 4 pi R^2 dR charge on this elemental sphere is dq = rho_v dv \r\n dq = rho_0 R/9 times 4 pi R^2 dR dq = 4 pi rho_0/a R^3 dR now total charge of the sphere is Q = integral_0^9 dq = 4 pi rho_0/9 integral_0^9 R^3 dR Q = 4 pi rho_0/9 (R^4/4)_0^9 = pi rho_0 9^4/9 Q = pi rho_0 9^3 First we calculate the electric field in the region 0 < R < 9 we draw a Gaussian surface of radius R \r\n total charge inclosed by the gaussian surface is q_in = integral_0^R dq = 4 pi rho_0/9 integral_0^R R^3 dR