# Chargedistribution from a given potential

1. Aug 10, 2009

### eXorikos

The following potential is given

The question is what the charge distribution is. The middle part is a charged dielectric. The two discontinuous points are the result of a charge accumulated in one point. And after that point the potential doesn't vary. So my thoughts are that the physical situation is a charged dielectric between two charged plated, with the charges of the dielectric oposite to the charge on the plate it faces.

I think I'm right so far. But now I want to calculate the charge distribution. The hint was to use delta-function and I can see why, but I don't know how. Can any of you help me?

PS: My paint skills suck, but I hope it's clear that the middle parabolic and the left potential is higher than the right one.

2. Aug 10, 2009

### Bob_for_short

Use the equation ∆φ ~ ρ. In a 1D case the second derivative of your potential will give the charge density.

3. Aug 10, 2009

### eXorikos

But what is the equation for such a potential?

4. Aug 10, 2009

### Bob_for_short

Sorry, I should have written it as ρ ~ ∆φ (Gauss law) or ρ(x) ~ (d²/dx²)φ(x) in your case.

ρ is a charge density and φ is the electrostatic potential. Depending on units, the equation may contain 4π, etc.

5. Aug 10, 2009

### eXorikos

I know how to solve a laplacian, but I can't find the equation for the potential.

6. Aug 10, 2009

### Bob_for_short

The equation is the following: ρ(x) ~ (d²/dx²)φ(x) in your case. All you have to do is to differentiate twice your potential given in your figure.

7. Aug 10, 2009

### eXorikos

I know all that. I've studied my book (Introduction to Electrodynamics), but I need the equation for the potential. That's my problem...

8. Aug 10, 2009

### Bob_for_short

You mean an analytical formula for your curve in the figure? Approximate it with something differentiable and you will obtain an approximate charge density.

The differential equation for a potential is the Gauss law ∆φ ~ ρ.

If the charge density ρ is given, you have to integrate this differential equation to find the potential φ.

If the potential φ is given, you have to differentiate it to find the density ρ.