- #1
threeonefouronethree
- 1
- 0
- 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?
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?