Calculating Field Outside Polarized Sphere: A Dipole Problem

[erore]

I am having difficulty with calculating field outside of a polarized sphere. It is a field of a dipole. What I get always stuck with is the calculation of the area charge density. I attach a picture. I need the distance of the speres surfaces to calculate the density from the volume charge density. Well I call the distance $$\Delta$$R. And in every source I find that $$\Delta$$R = cos$$\Theta$$ * l.
But I have no ideat how to derive this. I tried some ways (the law of cosines for O O' A triangle) and I always got that this is just an approximation. However, none of the sources mention that it is an approximation.
Can someone help?

 

[Valerie Park]

The easiest way to calculate the area charge density is to start by assuming that the dipole is composed of two point charges, each with charge q, separated by a distance l. The electric field of this point dipole is given by E = kq/r2, where r is the distance from the center of the dipole. Now consider a thin spherical shell centered on the dipole. The electric field on the shell is equal to the electric field of the point dipole at the center of the shell, since the fields of the two charges cancel out. Therefore, the electric field normal to the shell is given by E = kq/R2, where R is the radius of the shell.The area charge density on the shell is then given by σ = E/ε, where ε is the permittivity of the medium. Thus, the area charge density is given by:σ = kq/R2εSince the distance between the two charges is l, the radius of the shell is R = l cosθ, where θ is the angle between the line connecting the two charges and the normal to the shell. This gives us the desired result:σ = kq/[l cosθ]2ε

 

[astrokreng]

One possible approach to calculating the field outside of a polarized sphere is to use the method of images. This involves creating a hypothetical image dipole on the opposite side of the polarized sphere, with the same magnitude and opposite direction as the original dipole. This creates a symmetrical situation, allowing for easier calculation of the electric field.

In terms of the area charge density, it is important to note that this is not a constant value throughout the sphere. It varies with distance from the center, and can be calculated using the volume charge density and the distance from the center of the sphere. The distance from the center, \DeltaR, can be determined using the law of cosines, as you mentioned, or by considering the geometry of the problem and using trigonometric relationships.

It is also important to keep in mind that these calculations are based on certain assumptions and approximations, and may not perfectly match experimental results. However, they can provide a good estimate and understanding of the behavior of the electric field outside of a polarized sphere.

If you are still having trouble with the calculations, I suggest consulting with a colleague or seeking additional resources for guidance. It may also be helpful to break down the problem into smaller steps and approach it systematically. Good luck with your research!