Chargedistribution from a given potential

[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.

 

[Bob_for_short]

The following potential is given... The question is what the charge distribution is.

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

 

[eXorikos]

But what is the equation for such a potential?

 

[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.

 

[eXorikos]

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

 

[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.

 

[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...

 

[Bob_for_short]

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

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.