Can the Electric Field Inside a Uniformly Charged Sphere be Zero?

[elizabeth9681]

Homework Statement

A point charge Q is imbedded at the center of a uniformly charged spherical distribution of charge Q'= -Q with radius 'a'. Write down the volume charge density for this negatively charged sphere. Calculate the electric field and the potential inside, and outside, the atom.

Homework Equations

I'm concerned that I do not have the correct electric field. I believe that the electric field inside the sphere should be zero, but I do not completely understand why. Though I do understand that when the E-field is zero, then the Potential will be constant.

The Attempt at a Solution

For the volume charge density of the the negatively charged sphere:

-Q = rho*volume of the sphere, where rho is the volume charge density
-Q = rho*4/3*pi*a^3
thus rho = -Q/(4/3*pi*a^3)

then to find the E-field of the sphere considering the point charge, Q, and the negatively charged sphere I found the total charge Q(r):

Q(r) = Q + rho * 4/3*pi*r^3 , where r is some arbitrary radius, r could be greater than or less than 'a'
Q(r) = Q + [-Q/(4/3*pi*a^3) * 4/3*pi*r^3
thugs Q(r) = Q - Q(r^3/a^3)

Then for the E-field:
Integrating E*dA = Q(r)/eo; (where eo is the constant epsilon not)
I get E = Q/(4*pi*eo)[1/r^2 - r/a^3] for r>a and then when r<a then E=0 right?

From here I'm pretty sure I can find the potential, (when E is not zero) but I'm just not sure if I have it right. Thank you for your help!

 

[Shooting Star]

I'm concerned that I do not have the correct electric field. I believe that the electric field inside the sphere should be zero, but I do not completely understand why.

Why should it be zero?

For the volume charge density of the the negatively charged sphere:

-Q = rho*volume of the sphere, where rho is the volume charge density
-Q = rho*4/3*pi*a^3
thus rho = -Q/(4/3*pi*a^3)

The volume charge density is to be calculated over infinitesimal volumes, and so need not be constant.

then to find the E-field of the sphere considering the point charge, Q, and the negatively charged sphere I found the total charge Q(r):

Q(r) = Q + rho * 4/3*pi*r^3 , where r is some arbitrary radius, r could be greater than or less than 'a'
Q(r) = Q + [-Q/(4/3*pi*a^3) * 4/3*pi*r^3
thugs Q(r) = Q - Q(r^3/a^3)

Then for the E-field:
Integrating E*dA = Q(r)/eo; (where eo is the constant epsilon not)
I get E = Q/(4*pi*eo)[1/r^2 - r/a^3] for r>a and then when r<a then E=0 right?

No; the other way round. For points outside the sphere, the total charge enclosed by a Gaussain surface is zero.

 

[elizabeth9681]

Thanks so much! I thought I had it backwards but a study partner was adamant that it was E=0 for r<a.

 

[cartonn30gel]

electric field is zero when there is no charge enclosed (Gauss' Law)
what I understand from "uniformly charged spherical distribution" is a non-conducting sphere with charge on every piece of it. So electric field inside cannot be zero.

 

[Shooting Star]

electric field is zero when there is no charge enclosed (Gauss' Law)

In this case, would it hold outside the body if the charge distribution on the nonconducting body is uniform but the shape of the body has been changed into an ellipsoid or some other shape?