Question

1. A total charge of Q is distributed throughout a spherical volume defined by R < 8 cm. This charge results in an electric flux density given by: D 7.5 x 10-10 R C/m2 for R 3 8 cm An equal and opposite charge, -Q, is uniformly distributed over an infinitely thin spherical shell at R 16 cm. You may assume that E Eo throughout all space for this problem a) Determine the charge density within the volume R < 8 cm, and the value of the charge, Q. 3 12 b) Determine the electric field intensity for the two regions, 8 cm R 16 cm, and R > 16 cm.

can someone please explain how to do 1 c) in details, especially how they found v(r) for r<8cm, which formula is used and why did it that way? thanks

Answer 1

Gauss law Net electric flux linked with any closed Imaginary surface equal to total charge enclosed by that surface integral D^rightarrow ds^rightarrow = Q_enclosed integral E^rightarrow ds^rightarrow = Q_enclosed/epsilon_0 D^rightarrow = electric flux density (c/m^2) E^rightarrow = electric field intensity (a) Total charge enclosed by spherical volumes r < 8 cm D^rightarrow = 7.5 times 10^- 10 c/m^2 Q_Total = integral D^rightarrow ds^rightarrow = integral (7.5 times 10^- 10) ds^rightarrow = 7.5 times 10^- 10 integral ds^rightarrow = (7.5 times 10^- 10) (4 pi r^2) = (7.5 times 10^- 10) (4 pi) (8/100)^2 = 60.3 times 10^- 12 coulomb Q_Total = 60.3 pc charge density with in volume R < 8 cm divergence theorem say contourintegral A^rightarrow ds^rightarrow = integral (nabla A^rightarrow) dv contourintegral nabla^rightarrow ds^rightarrow = integral (nabla D^rightarrow) dv Q = integral s_v dv] rightarrow nabla D^rightarrow = f_v c/m^2 diverges of spherical coordinate nabla A^rightarrow = 1/r^2 [partial differential A_r (r^2)/partial differential r] = rho_v rho_v = 1/r^2 partial differential D^-_r r^2/partial differential r = 7.5 times 10^- 10/r^2 partial differential r^2/dr rho_v = 7.5 times 10^- 10/r^2 (2 r) = rho_v = 15 times 10^- 10/r c/m^3 rightarrow r < 8 cm