Calculating Potential Distribution of a Dielectric Cylinder in Electric Field

[hectoryx]

How to calculation the potential distribution for a dielectric cylinder placed in electric field of point charge?

The height of the dielectric cylinder is H, while its section radius is R. The dielectric constant of the cylinder is ε1.

http://i1021.photobucket.com/albums/af335/hectoryx/11111.jpg"


I found that it is hard for me to write its Laplace Equation and the boundary conditions.

Maybe, a cylindrical coordinate system should be used in the analysis.

Could anyone give me some help？

Thanks very much!

p.s. I wrote a half-baked Laplace Equation for the system in the following URL.
http://i1021.photobucket.com/albums/af335/hectoryx/11112.jpg"

 

[gabbagabbahey]

Hi hectoryx, welcome to PF!

It can sometimes take quite a while for site admin to find time to look through attachments and appove them. Until they are approved, they aren't visible to other posters (also many people are unwilling to open .doc files since they often contain viruses), so it will save time if instead of posting attachments, you upload your images to a free image hosting site (like imageshack.us) and post a link to it, and type out your attempted solutions using $\LaTeX$ (There is a nice introduction to using $\LaTeX$ on these forums in this thread)

 

[hectoryx]

Hi gabbagabbahey, thanks for your advices, I have already changed my post.

 

[gabbagabbahey]

I notice that you seem to be unsure as to whether the potential outside the cylinder $u_2$ satisfies Laplace's equation or not. I assume that the fact that there is a point charge out there is throwing you off?

If so, just realize that charges or no charges, the electrostatic potential will always satisfy Poisson's equation. Outside the cylinder, the charge density will be zero everywhere except for the exact location of the point charge, so unless you want to calculate $u_2$ at that exact location, you can say that the potential will satisfy Laplace's equation.

Now, looking at your boundary conditions I see that while your proposed boundary condition $u_2\to 0$ at infinity is correct, it doesn't really tell you anything about there being a point charge in that region...instead of looking at $\rho\to\infty$, why not look a little closer to the charge...if you are very far away from the cylinder, but not so far away that the potential due to the point charge is zero, what would you expect the potential to be?

 

[hectoryx]

Thanks for your reply.

I change the equation to a new one and post on the following URL.
http://i1021.photobucket.com/albums/af335/hectoryx/11113.jpg
http://i1021.photobucket.com/albums/af335/hectoryx/11113.jpg"

Actually, I find an oversight on the boundary conditions that on z=0 and z=H.
Could you be kind enough to review it again?

Furthermore, I do not totally understand the last paragraph of your post. Do you mean that I should find a boundary condition on a place that not so far away from the point charge?

At last, I find that it really difficult to resolve the equation. In many examples on electromagnetic field theory, the placements of charge and cylinder or ball are under certain conditions. As a result, one in the three variables of the Laplace equation can be eliminated because of the symmetry. For instance, a metal ball in electric field of a point charge or a cylinder in external uniform electric field, it will be easier to resolve Laplace equation in the two conditions. Could you give me some suggestions?

Thanks very much!

 

[gabbagabbahey]

Actually, I find an oversight on the boundary conditions that on z=0 and z=H.
Could you be kind enough to review it again?

You might want to explicitly add "for $0\leq z\leq H$" to your boundary conditions at $\rho=R$, and "for $0\leq \rho\leq R$" to your boundary conditions at $z=0$ and $z=H$

Furthermore, I do not totally understand the last paragraph of your post. Do you mean that I should find a boundary condition on a place that not so far away from the point charge?

Yes, $u_2\to 0$ for $\rho\to 0$ will be true whether there are no point charges, 1 point charge, or several point charges outside the cylinder. Subsequently, that boundary condition doesn't really tell you any information about there being a point charge Q. In order to build that information into your boundary conditions, just look at the far-field region $\rho\gg R$...

In this region, the monopole term of $u_2$ should dominate, and since the only electric monopole in this problem is the point charge Q, you expect $u_2\approx \frac{1}{4\pi\epsilon_0}\frac{Q}{|\textbf{r}-\textbf{r}_Q|}$ for $\rho\gg R$

At last, I find that it really difficult to resolve the equation. In many examples on electromagnetic field theory, the placements of charge and cylinder or ball are under certain conditions. As a result, one in the three variables of the Laplace equation can be eliminated because of the symmetry. For instance, a metal ball in electric field of a point charge or a cylinder in external uniform electric field, it will be easier to resolve Laplace equation in the two conditions. Could you give me some suggestions?

You can still use Separation of Variables, and the general solution in cylindrical coordinates is well known (See for example, Jackson's Classical Electrodynamics) and involves Bessel functions.

 

[hectoryx]

Thanks for your reply. I really appreciate your help.It is really important for me!

New question is coming. The dielectric cylinder remains unchanged. However, it is not placed in free space, but stand with a certain distance (h) high on the Earth ground.

Besides, the field source is not point charge any more, but two parallel metal plates which are placed paralleled to the z axis and close to the cylinder. The distance between the two plates is d2, and the distance between the cylinder and the plate which is closer to the cylinder is d1. The voltage between the two plates is ΔU.
The new placement can be seen in the following URL,
http://i1021.photobucket.com/albums/af335/hectoryx/cylinder.jpg

How to calculation the potential distribution? I have written a preliminary Laplace Equation for this problem. In my equation, for simple, I assume that the metal plates are arc and totally parallel with the surface of the cylinder (so the sizes of the two plates are not extremely same). You can see it in the following URL,
http://i1021.photobucket.com/albums/af335/hectoryx/equation.jpg

However, I do not know how to take the Earth ground into account. Could you give me some suggestions?

 

[gabbagabbahey]

Everything you have so far looks good to me!

However, you can combine your two boundary conditions for the parallel plates into one single condition:

$$u_2(\rho=R+d_2+d_1)-u_2(\rho=R+d_1)=-\Delta U$$

and you can (without loss of generality) choose your coordinate system so that the angle you call $\Phi$ is zero. And if you know the length of the parallel plates, a little trigonometry will allow you to express $\Delta\phi$ in terms of R and the length.

As for the Earth ground, that's simple, just choose your coordinate system so that the Earth is at z=0, then $u_2(z=0)=0$ is all you need.