Finding the Electric field at a point OUTSIDE a charged dielectric sphere with rho?

khfrekek1992

1. The problem statement, all variables and given/known data

There is a charged dielectric sphere of radius R. I center it at the origin of an x,y,z cartesian system. It has a charge density rho=3Qcos(theta)/4*Pi*R^3. Q and R are given.

I need to find the Electric field at a point that is outside the sphere.

2. Relevant equations

int(E*dA)=Q_in/Epsilon0


3. The attempt at a solution

int(E*dA)=Q_in/Epsilon0
E(4*Pi*r^2)=(rho)(4/3)*Pi*r^3)/epsilon0
E=3Qcos(theta)r/12*Pi*epsilon0*R^3 in the r-hat direction

Is this right? Dont I have to figure out the field in the x-hat, y-hat, and z-hat directions? How does this work if my E is in the r-hat direction?

Thanks in advance!

SammyS

It appears that you are trying to use Gauss's Law to find the electric field. The charge density depends on θ, so there isn't enough symmetry for Gauss's Law to be of help here.

It looks like you will have to use direct integration to find either electric field, E, or electric potential, V.

khfrekek1992

Thanks for the reply. I tried integrating rho directly earlier, but always got 0 because sin evaluated from 0 to Pi is always 0. :(

khfrekek1992

1. The problem statement, all variables and given/known data

There is a charged dielectric sphere of radius R. I center it at the origin of an x,y,z cartesian system. It has a charge density rho=3Qcos(theta)/4*Pi*R^3. Q and R are given.

I need to find the Electric field at a point that is outside the sphere.


2. Relevant equations

E=-(grad)V
V=kq/r

3. The attempt at a solution

I integrated rho over the volume of the sphere to get q=(Pi/2)Qcos(theta)
then used V=q/4*pi*epsilon0r to get v=(Qcos(theta))/(8epsilon0*r)

How do i split this voltage into x,y, and z components so i can take the gradient to get E at a certain (x,y,z) point?

Thanks in advance

SammyS

This looks like the same problem that you posted in another thread.

khfrekek1992

Yeah haha it is the same but im doing the problem completely different now(from your suggestion) and am now stuck at a different part :/

SammyS

Yes, the net charge is indeed zero. Integrating the cosine from 0 to π does give zero.

That doesn't mean that the electric potential and/or the electric field are zero. (The same can be said for an electric dipole.)

khfrekek1992

Oh okay that makes sense, thank you! So I found the potential by taking kq/r, and now im taking -(grad)V to get E. So I have E=-(d/dxx-hat+d/dyy-hat+d/dzz-hat)*(3Qcos(theta))/4*Pi*R^3Epsilon0

is that right?

khfrekek1992

oh wait if i take the gradient of that I get 0 in every direction..

SammyS

How did you find V ?

khfrekek1992

See below

khfrekek1992

Actually, I guess I should Multiply the charge density by the volume of the sphere to get Q, then use that in V=kQ/R to get V=(QR^2/repsilon0)cos(theta) Where everything except for r and theta is a known constant. Then I need to take the negative gradient to get E. I just dont know how to take the x,y, and z derivatives of that voltage, because they would all be 0? :/

berkeman

Please do not multiple post. It is against the PF rules.

Two threads merged into one.