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2, A spherical shell is made of dielectric material with relative permittivity εR and contains a uniform free charge density pF. The inner radius of the shell is R1 and the outer radius is R2 a. What is D in the material that makes up the shell? b. What is E in the material that makes up the shell? c. What is P in the material that makes up the shell? d. What is the surface bound charge density ơb on the outside surface of the shell? the shell? makes up the shell? e. What is the surface bound charge density ơb on the inside surface of f. What is the volume bound charge density pb in the material that

Answer 1

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We use integral D^vector middot partial differential A^vector = D 4 pi r^2 = integral^r_R_1 4 pi r^2 dr rho_F [For a sphere of radius r] = 4pi/3 rho_F(r^3 - R^3_1) Hence, E^vector = D/E = rho_F(r^3 - R^3_1)/3 epsilon r^2 r^^ [where we used D = rho_F(r^3 - R^3_1)/3r^2] Next p^vector = epsilon_0 x_e E^vector = epsilon_0x_e rho_F (r^3 - R^3_1)/3 epsilon r^2 r^^ = rho_F x_e (r^3 - R^3_1)/3 epsilon_R r^2 r^^ where x_e is electric of medium. On the outer surface = p^vector middot n^^(p^vector middot r^^) = rho_F x_e(R^3_2 - R^3_1)/3 epsilon_R R^2_2 [Note r = R_2 for outer surface] on the inner surface = rho^vector middot n^vector = p^vector middot r^^ = rho_F x_e (R^3_1 - R^3_1)/3 epsilon_R R^2_1 = 0 = -nabla^vector middot P^vector = -1/r^2 partial differential/ partial differential r(r^2 times rho_F X_e(r^3 - R^3_1)/3 epsilon_Rr^2) = -1/r^2 rho_F X_e 3 r^2/3 epsilon_R = -rho x_e/epsilon_R