Poisson's equation: Calculating the Laplacian of an electric potential

[threeonefouronethree]

Homework Statement

We have a sphere with radius R and charge density p=ar (a is a constant, r is the distance from the midpoint of the sphere). There is no charge outside of the sphere.

Relevant Equations

Calculate the electric potential V(r) outside the sphere (in terms of total charge Q).

First I calculated the electric fields outside of the sphere in terms of the total charge Q.

total charge Q:
Q = aπR^4

electric field outside: (r>R)
E(r) = (1/4πε) Q/r^2 (ε is the vacuum permittivity)

electric potential outside: (r>R)
V(r) = (1/4πε) Q/r

This was no problem for me (at least if my answer is right),
I wanted to check my answer with poisson's equation:

ΔV = -p/ε = -ar/ε

However i don't know how to calculate the laplacian of an electric potential which has 1/r in it.
by blindly doing the Laplacian of spherical coordinates I obviously got 0, so i checked that:

Δ(1/r) = -4πδ

However there is no intergral so i can't get rid of the dirac delta, and I also have constants so how do i deal with those?

 

[BvU]

Hello 31413, !

There is no question in your homework statement and no equation in your 'relevant equations'.
Also

First I calculated the electric fields outside of the sphere in terms of the total charge Q

Looks weird, because the impresssion you give is that that is all that is asked from you.
So with

V(r) = (1/4πε) Q/r

You are done with this exercise.

 

[TSny]

Y

I wanted to check my answer with poisson's equation:

ΔV = -p/ε = -ar/ε

However i don't know how to calculate the laplacian of an electric potential which has 1/r in it.
by blindly doing the Laplacian of spherical coordinates I obviously got 0

Your expression for V is only valid for points outside the sphere where the charge density is zero. So, Poisson's equation implies that the Laplacian of V should be zero outside the sphere. That's what you got.

 

[rude man]

Yes, and be glad you didn't try to check for the field inside the sphere by Poisson's eq as you started out with.
Can be done, is still a linear equation, but you have to solve the Euler-Cauchy eq. I actually tried it & it works!