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

AI Thread Summary
Laplace's equation does not hold true at the location of a dipole; instead, Poisson's equation should be used to account for the dipole's charge distribution. The charge distribution for an ideal dipole involves a delta function and its derivative, indicating that the potential satisfies Laplace's equation everywhere except at the dipole itself. When solving for the electric potential in a dielectric sphere containing a dipole, one should apply Laplace's equation outside and Poisson's equation inside, while ensuring boundary conditions are met. The discussion highlights the importance of accurately defining charge distributions to avoid algebraic errors in calculations. Ultimately, the potential satisfies Poisson's equation everywhere, with the specified source charge density.
Ahmed1029
Messages
109
Reaction score
40
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?
 
Physics news on Phys.org
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.
 
  • Like
Likes Delta2
Orodruin said:
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?
 
LCSphysicist said:
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 😉)
 
  • Haha
  • Like
Likes 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 ?
 
  • Like
Likes Delta2
Ahmed1029 said:
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
 
  • Informative
  • Haha
Likes Delta2 and Ahmed1029
Orodruin said:
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.
 
  • Like
Likes Ahmed1029
Delta2 said:
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!
 
  • Like
Likes Delta2
Back
Top