Finding potential of a dipole outside of a sphere

In summary, the study explores the electric potential generated by a dipole positioned outside a conducting or non-conducting sphere. It analyzes the influence of the dipole's orientation and distance on the potential distribution around the sphere. The findings indicate that the dipole's effect diminishes with distance and that the sphere's geometry modulates the potential landscape, leading to unique field patterns. This research has implications for understanding electric fields in various physical contexts, including antenna design and electrostatics.
  • #1
nicholasBourbaki
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TL;DR Summary
What is the potential of a dipole outside of a sphere when the orientation of the dipole is tangent to the sphere?
Hey guys! I am currently tackling the following problem: An electric dipole P is kept close to a conducting grounded sphere of radius A -assume a distance B between the center of the sphere and dipole. What is the potential of the dipole outside of the sphere when the orientation of the dipole is tangent to the sphere?

I tried finding the green function of the structure and then tried to find the electric dipole in terms of two volume charged density and finally tried to find the potential using these two values. However that did not get me anywhere. I would appreciate if someone could point me to the right direction.
 
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  • #2
Welcome to PF.

Just an idea, but I would look at the reflection of the dipole in the conductive-mirror surface of the sphere. Analyse the dipole and its image. The image of the dipole is distorted by the surface curvature, and will be reversed.

There may be a simplification, but you have not specified the relative length of the dipole to the diameter of the sphere.
 
  • #3
If you model the dipole as a plus charge and a minus charge, a distance d apart, you can find the field of each charge by the usual image calculation.
 

FAQ: Finding potential of a dipole outside of a sphere

What is the basic concept of finding the potential of a dipole outside of a sphere?

The basic concept involves calculating the electric potential at a point outside a spherical conducting or insulating object due to a dipole placed at a certain distance from the sphere. This requires understanding the superposition principle, boundary conditions, and the method of images or other mathematical techniques to solve the problem.

How does the method of images help in solving this problem?

The method of images simplifies the problem by replacing the sphere with an imaginary dipole inside the sphere. This imaginary dipole is positioned such that the boundary conditions on the surface of the sphere are satisfied. By calculating the potential due to both the real dipole and the imaginary dipole, the total potential outside the sphere can be determined.

What are the boundary conditions for this problem?

The boundary conditions depend on whether the sphere is conducting or insulating. For a conducting sphere, the potential on the surface must be constant (usually set to zero). For an insulating sphere, the normal component of the electric displacement field must be continuous across the boundary. These conditions ensure that the solution is physically accurate.

How do you place the image dipole inside the sphere?

The image dipole is placed along the line connecting the center of the sphere to the real dipole. The distance of the image dipole from the center of the sphere depends on the radius of the sphere and the distance of the real dipole from the center. For a sphere of radius R and a dipole at distance d from the center, the image dipole is placed at a distance R²/d from the center along the same line.

What is the mathematical expression for the potential outside the sphere?

The potential at a point outside the sphere is the sum of the potentials due to the real dipole and the image dipole. If the real dipole has a moment p and is located at a distance d from the center, and the image dipole has a moment p' and is located at distance R²/d from the center, the potential at a point outside the sphere can be expressed using the superposition principle and the standard formula for the potential due to a dipole.

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