A spherical shell has inner radius Rin and outer radius Rout. The shell contains

Question

A spherical shell has inner radius Rin and outer radius Rout. The shell contains total charge Q, uniformly distributed. The interior of the shell is empty of charge and matter.
(a) Find the electric field outside the shell, at r = Rout.
(b) Find the electric field in the interior of the shell, at r = Rin. (c) What is the charge density (? = charge per unit volume) of the shell? (d) What is the volume of the part of the shell extending from Rin
to r, where r is inside the shell, i.e., Rin = r = Rout?
(e) Find the electric field within the shell, at Rin = r = Rout. (f) Show that your three solutions match at both boundaries. (g) Draw a graph of E versus r

Explanation / Answer

a) according to Gauss's law. Q/epsilon0 = E*4pir^2 so E=Q/(4pir^2*epsilon0)=kQ/r^2. b)at r = Rin, according to Gauss's law, the charge inside this Gaussian surface is zero. so E=0. c) volume of the shell: V=4pi*(Rout^3-Rin^3)/3 so charge density P=Q/V=3Q/(4pi*(Rout^3-Rin^3)) ---- d) that is V=4pi*(r^3-Rin^3)/3 e) In that gaussian surface, the charge Q'=P*V(r)=Q*(r^3-Rin^3)/(Rout^3-Rin^3) E=Q'/epsilon0*4pir^2 =Q*(r^3-Rin^3)/((Rout^3-Rin^3)epsilon0*4pi*r^2) =kQ*(r^3-Rin^3)/((Rout^3-Rin^3)*r^2) f) plug r=Rin and r=Rout to see that they are matched.

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