Laplace's equation in presence of a dipole (perfect or physical)

[Ahmed1029]

TL;DR

I'm wondering if the laplacian of the electrostatic potential function will still be zero at the location of a dipole.

Does Laplace's equation hold true for electrostatic potential at the location of a dipole? Or should poisson's equation be used?

 

[Orodruin]

You need to use Poisson’s equation. However, just as for a point charge, you need to be wary of what charge distribution you put in. In the case of a point charge, the charge distribution is a three-dimensional delta distribution. In the case of an idealised dipole it is a delta distribution in two directions and a derivative of a delta distribution in the direction of the dipole:
$$
\rho = \vec p \cdot \nabla \delta^{(3)}(\vec x).
$$

This means the potential will satisfy the Laplace equation everywhere except at the dipole.

 

[LCSphysicist]

You need to use Poisson’s equation. However, just as for a point charge, you need to be wary of what charge distribution you put in. In the case of a point charge, the charge distribution is a three-dimensional delta distribution. In the case of an idealised dipole it is a delta distribution in two directions and a derivative of a delta distribution in the direction of the dipole:
$$
\rho = \vec p \cdot \nabla \delta^{(3)}(\vec x).
$$

This means the potential will satisfy the Laplace equation everywhere except at the dipole.

Shouldn't we have a minus sign here?

 

[Orodruin]

Shouldn't we have a minus sign here?

Possibly, I did not think too much about signs.

(As Feynman allegedly said: Factors of 2, pi, and i are only for publication purposes )

 

[Ahmed1029]

so consider the following problem :

"A point dipole p is imbedded at the center of a sphere of linear
dielectric material (with radius R and dielectric constant e). Find the electric po-
tential inside and outside the sphere."

How can I solve it only using Laplace's equation? Do I use the superposition principle ?

 

[LCSphysicist]

so consider the following problem :

"A point dipole p is imbedded at the center of a sphere of linear
dielectric material (with radius R and dielectric constant e). Find the electric po-
tential inside and outside the sphere."

How can I solve it only using Laplace's equation? Do I use the superposition principle ?

Solve Laplace equation outside.
Solve Poisson equation inside.
Reject terms that blows up as r goes to infinity.
Check the boundary conditiions at the sphere.
Realize that you have made an algebric mistake.
Returns to step 1
Realize you have made another algebraic mistake.
Returns to step 1.
Get the right answer.

That't the recipe

 

[Delta2]

This means the potential will satisfy the Laplace equation everywhere except at the dipole.

I think it is much better to say that the potential will satisfy the Poisson equation everywhere, with source charge density the one you defined.

 

[Ahmed1029]

I think it is much better to say that the potential will satisfy the Poisson equation everywhere, with source charge density the one you defined.

Got it, thanks!