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

Question

An insulating hollow sphere has inner radius a and outer radius b. Within the insulating material the volume charge density is given by ?(r)=?r,where ? is a positive constant.

What is the magnitude of the electric field at a distance r from the center of the shell, where a<r<b?

Express your answer in terms of the variables ?, a, r, and electric constant ?0.

A point charge q is placed at the center of the hollow space, at r=0. What value must q have (sign and magnitude) in order for the electric field to be constant in the region a<r<b?

Express your answer in terms of the variables ?, a, and appropriate constants.

what then is the value of the constant field in this region?

Express your answer in terms of the variable ? and electric constant ?0.

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

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

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

=>q=2?*pi*a^2

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