A static charge distribution exists in vacuum inside a sphere of radius a with a

Question

A static charge distribution exists in vacuum inside a sphere of radius a with a known uniform volume charge density row_v C/m3.

Show using a symmetry argument that the electric field is purely radial in the spherical co-ordinate system centered at the center of the spherical charge distribution.

Derive an expression for the electric field produced by this charge distribution throughout all space, i.e., inside and outside the charge sphere.

A uniform surface charge distribution is now placed on the spherical surface r=2a such that no electric field exists outside of this spherical surface. What is the charge density row_s C/m2 of this charge distribution?

Explanation / Answer

a) Suppose the electric field is not radial, and say it is at an angle a with the radial vector.Then, since the sphere is spherically symmetric, we can say ask why the electric field is not directed at an angle -a with the radial vector.

So, by symmetry the electric field exists in a direction where a = -a.

So, a=0. So, the electric field is in radial direction.

b)
Consider a spherical gaussian surface of radius r>a.

Since the electric field is radial, it is everywhere perpendicular to the surface of the sphere and by symmetry its magnitude is same everywhere at a radius r.Let magnitude = E at radius r

So, the flux through the sphere = integral E*ds*cos 0 = E*4pir^2

According to gauss law, E*4pir^2 = Qin/e

E*4pir^2 = p*(4/3*pi*r^3)/e

So, E = p*r/3e

This is for r>a

For r<a,

Going along same lines, we get

E*4pir^2 = Qin/e

= p*(4/3*pi*r^3)/e

So, E = p*r/3e

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