I'm having a bit of trouble with this problem:
"A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball (r≤R) is E(r)=r^4Emax/R^4.
a. What is Emax in terms of Q and R?
b. Find an expression for the volume charge density ρ(r) inside the ball as a function of r."
The Attempt at a Solution
I found the answer for part (a.) with no trouble:
However, part (b) has proved to be a lot harder. Here is my attempt:
Since φ=EA and φ=Q/ε0,, I chose to use the original equation for electric field (since I have a correct answer for Emax), multiply it by the surface area of a smaller sphere with radius "r" (inside the original sphere with radius R), and set this equal to Q/ε0 as follows:
Then, I determined the charge of the small sphere with radius "r" (inside the original sphere with radius R) as follows:
Find charge of small sphere (inside the original sphere with radius R):
Q=∫ρ4πr^2dr (with the limits of the integral being 0 to r)
Then I used the equation I found earlier:
Then I input the values for Emax and Q, and I get the formula for charge density:
I tried entering the value to see if it is correct and the program tells me that I am missing or failing to add a numerical value. I've tried going through it and I can't seem to find the error. If one of you all can help me out I'd be grateful.