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

In summary, the potential at the location of the dipole will satisfy the Laplace equation, but only if you use the Poisson equation.
  • #1
Ahmed1029
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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?
 
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  • #2
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.
 
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  • #3
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?
 
  • #4
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 😉)
 
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  • #5
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 ?
 
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  • #6
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
 
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  • #7
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.
 
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  • #8
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!
 
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1. What is Laplace's equation in the presence of a dipole?

Laplace's equation in the presence of a dipole refers to a mathematical equation that describes the electric potential in a region of space where a dipole is present. It is a special case of the more general Laplace's equation, which describes the electric potential in a region of space where there are no charges present.

2. How is Laplace's equation in the presence of a dipole different from the general Laplace's equation?

Laplace's equation in the presence of a dipole includes an additional term that takes into account the presence of the dipole. This term is proportional to the distance from the dipole and the strength of the dipole moment. In the general Laplace's equation, this term is not present.

3. What is a dipole in the context of Laplace's equation?

A dipole is a pair of equal and opposite charges that are separated by a small distance. In the context of Laplace's equation, this dipole creates an electric field that can be described by the equation. It is a common model used to describe the behavior of electric fields in many physical systems.

4. What is a perfect dipole in Laplace's equation?

A perfect dipole in Laplace's equation refers to a theoretical dipole with a perfectly symmetrical distribution of charges and a perfectly aligned dipole moment. This means that the dipole has no net charge and the positive and negative charges are located at equal distances from the center of the dipole.

5. How is Laplace's equation in the presence of a dipole used in scientific research?

Laplace's equation in the presence of a dipole is used in many areas of scientific research, including physics, engineering, and biology. It is particularly useful in understanding the behavior of electric fields in complex systems, such as biological membranes or electronic circuits. It is also used in the development of new technologies, such as sensors and detectors.

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