- #1
roam
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Homework Statement
A sphere of radius R has a volume charge distribution ##\rho(r)## given by:
##\rho(r)= \rho_0 \left( \frac{r}{R_0} \right) \ \ for \ 0 <r<R##
##\rho(r)=0 \ \ elsewhere##
Where ##\rho_0## is a constant. Use Gauss's law to find E field outside the sphere.
Homework Equations
Integral form of Gauss's law
The Attempt at a Solution
I am not sure how to find the enclosed charge. Here's what I did:
##\int \rho \ dV = \int (0) (r^2 \ sin \theta dr d \theta d\phi)##
##\int^R_0 (0) . r^2 dr \int^\pi_0 sin \theta \ d \theta \int^{2 \pi}_0 d \phi = C 4 \pi##
Did I do the integration correctly?
I'm not sure what to do with the constant, I tried to find ##E_{in}## and use boundary conditions:
##\frac{\rho_0}{R} \int r.r^2 \ sin \theta dr d \theta d \phi = \frac{\rho_0 r^4 \pi}{R_0} \implies E_{in} = \frac{\rho_0 r^2}{\epsilon_0 R_0}##
At the boundary R:
##4\pi C = \frac{\rho_0 R}{\epsilon_0} \implies C=\frac{\rho_0 R}{4 \pi \epsilon_0} \implies Q_{enc} = \frac{\rho_0 R}{\epsilon_0}##
So this gives:
##\oint E_{out} da= |E| 4 \pi r^2= \frac{\rho_0 R}{\epsilon_0^2} \implies \therefore E_{out} = \frac{\rho_0 R}{4 \pi \epsilon_0^2 r^2}##
It doesn't look right. Am I using the correct method here to solve the problem? Any explanation would be appreciated.