What Is the Charge Distribution p(rho) for This Electric Field?

[jajay504]

Homework Statement

The Electric field E produced by an unknown charge distribution p (rho) is E(r)= (constant)*((exp(-ar))/r^2)*(r_hat).
a.) Use Gauss' law in differential for to determine p(rho)
b.) Find the total charge q_tot by directly integrating p(rho), and show that it is 0.
c.) Find q_tot again, except this time use Gauss' law in integral form.
d.) Make a sketch of p(rho)

Homework Equations

The Attempt at a Solution

 

[vela]

According to the forum rules, you need to show some effort at working the problem yourself before you can receive help.

 

[jajay504]

Hey! Sorry for that... I did some work but I got stuck.

I used divergence E = p/e0

a.) div E = (-a/r^2 - 2/r^3) * exp(-a*r) = p/e0. So I know what p is.
b.) Then I tried to integrate over p(rho) to get q_tot

q_tot = Integral [pdV]
I got --> exp(-a*r)/r^2 - 2*exp(-a*r)/r^3 = q_tot

I'm still trying to figure out what is this surface

 

[jajay504]

According to the forum rules, you need to show some effort at working the problem yourself before you can receive help.

Sorry for that! I put down the work I had

 

[vela]

Hey! Sorry for that... I did some work but I got stuck.

I used divergence E = p/e0

a.) div E = (-a/r^2 - 2/r^3) * exp(-a*r) = p/e0. So I know what p is.

You calculated the divergence incorrectly. Look up the formula for the divergence in spherical coordinates. Your book should have a list of the various vector operators for different coordinate systems.

b.) Then I tried to integrate over p(rho) to get q_tot

q_tot = Integral [pdV]
I got --> exp(-a*r)/r^2 - 2*exp(-a*r)/r^3 = q_tot

I'm still trying to figure out what is this surface

It's not a surface; it's a volume. What is dV in spherical coordinates? What limits should you be integrating over?

Please show more details of your work.