An insulating hollow sphere has inner radius a and outer radius b. Within the in

Question

An insulating hollow sphere has inner radius a and outer radius b. Within the insulating material the volume charge density is given by p(r)=alpha/r, where alpha is a positive constant. In terms of alpha and a, what is the magnitude of the electric field at a distance r from the center of the shell, where a<r<b? A point charge q is placed at the center of the hollow space, at r=0. In terms of alpha and a, what value must q have (sign and magnitude) in order for the electric field to be constant in the region a<r<b, and what then is the value of the constant field in this region?

Explanation / Answer

apllying gauss law for r<a

charge inside=0

=>E=0

(alpha)

for a<r<b

charge inside=ar 4*pi*r^2*/r dr

==ar 4*pi*r* dr=2*pi*r^2 |ar

=2*pi*r^2-2*pi*a^2

surface area=4*Pi*r^2

E=Qin/S=0.5*(1-(a/r)^2)/

in an extra charge is placed

the Eext=q/4*pi**r^2

E=0.5*(1-(a/r)^2)/+ q/4*pi**r^2 =constant

=>-0.5a^2/r^2 +q/4*pi**r^2 =0

=>q/4*pi**r^2=0.5a^2/r^2

=>q=2*pi*a^2

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