0
$\begingroup$

I'm following Griffith's Intro to Electrodynamics, and when discussing Laplace's Equation, it states

Very often we are interested in finding the potential in a region where $\rho = 0$.

So, if you had a sphere with equal but opposite charge densities distributed over the northern and southern hemispheres, would Laplace's equation hold only over the entire sphere? But you couldn't analyze a hemisphere using the same method?

$\endgroup$
1
$\begingroup$

The Laplace equation is a differential equation which tells us how the potential varies locally at a point, so is the volume charge density $\rho$.

One cannot say because the total net charge inside a sphere is zero, so we can apply the equation inside the sphere.

It tells you that if you have $\rho=0$ at a point, then the potential satisfies Laplace equation at this point.

For your case, you cannot apply the Laplace equation at any point inside the sphere, because $\rho \ne 0$ at every point. Instead, the potential satisfies the Poisson's equation.

$\endgroup$
  • $\begingroup$ The only place you would have $\rho = 0$ at a single point would be places where there is absolutely no charge- is that right? Does that mean Laplace's equation describes the potential in open space? $\endgroup$ – A4Treok Oct 16 '17 at 22:06
  • 1
    $\begingroup$ The Laplace equation is satisfied at a point where there is no net charge. $\endgroup$ – velut luna Oct 16 '17 at 22:07
  • $\begingroup$ Can two (opposite) charges occupy the same space? Wouldn't it be required that there is no charge at all? $\endgroup$ – A4Treok Oct 16 '17 at 22:10
  • 1
    $\begingroup$ Classical EM is an effective theory for macroscopic scales. $\endgroup$ – velut luna Oct 16 '17 at 22:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.