Proving Poisson's Equation

ozone
Sorry guys I have a lot of trouble with proofs and could use your help with this one.

This is right out of Griffiths (problem 2.29) I have a solution manual, but I'd like to try to get a nudge in the right direction before I turn to it.

Poisson's eq:

$\nabla ^2 V = - \frac{ \rho }{\epsilon }$

Using an identity out of Griffiths we have $\nabla ^2\left(\frac{1}{r}\right) = -4\pi \delta ^3(r)$

finally we know that $V(r) = \frac{1}{4\pi \epsilon }\int \frac{p(r')}{r} \, dr'$

distributing the gradient into our function will yield
$V(r) = -\frac{1}{\epsilon }\int p(r')\delta ^3(r)dr'$

But I do not know what to do with the dirac delta in the final function we have arrived at.

Answers and Replies

Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
Sorry guys I have a lot of trouble with proofs and could use your help with this one.

This is right out of Griffiths (problem 2.29) I have a solution manual, but I'd like to try to get a nudge in the right direction before I turn to it.

Poisson's eq:

$\nabla ^2 V = - \frac{ \rho }{\epsilon }$

Using an identity out of Griffiths we have $\nabla ^2\left(\frac{1}{r}\right) = -4\pi \delta ^3(r)$
The ##r## on the lefthand side is the distance between two points whereas the ##r## in the argument of the delta function should be the vector ##\vec{r}##.

finally we know that $V(r) = \frac{1}{4\pi \epsilon }\int \frac{p(r')}{r} \, dr'$
The potential V isn't a function of only radial distance. It's a function of all three coordinates. In my edition of Griffiths, he uses the notation V(P) to indicate it's the potential at point P. He doesn't write V(r), which implies something much different. The symbols ##p## and ##\rho## look similar, but they aren't the same symbol. Be consistent. You should also define exactly what you mean by ##r## inside the integral. How is it related to the variables of integration? Finally, ##dr'## should actually be the volume element ##d\tau##.

distributing the gradient into our function will yield
$V(r) = -\frac{1}{\epsilon }\int p(r')\delta ^3(r)dr'$

But I do not know what to do with the dirac delta in the final function we have arrived at.
The lefthand side should not be the potential V.

Electromagnetism is a fairly math-intensive course, so you should really make an effort to be precise with your notation. The sloppiness that you could get away with in earlier courses will really start to hinder you now. Clean it up so you can avoid making dumb errors and confusing yourself.