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

• I
• Ahmed1029
In summary, the potential at the location of the dipole will satisfy the Laplace equation, but only if you use the Poisson equation.f

#### Ahmed1029

TL;DR Summary
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?

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.

• Delta2
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?

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 )

• • malawi_glenn and Delta2
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 ?

• Delta2
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.

That't the recipe

• • Delta2 and Ahmed1029
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!

• Delta2