Calculating the Coulomb Force between 2 charges in space

[maverick280857]

Hi

Here is an interesting problem which I have been trying to solve. For convienience of discussion, I have broken it into parts to show where I get stuck:

Two charges $Q_{1}$ and $Q_{2}$ are kept in space at a distance r from each other in a medium of dielectric constant K. The force between them is given by

$$F = \frac{1}{4\pi\epsilon_{0}K}\frac{Q_{1}Q_{2}}{r^2}$$

Problem #1: Now if instead of the original dielectric medium we have a slab of dielectric constant K and width r/2 placed somewhere between the two charges, then find the force between the two charges (this is not homework so I guess its in the right place).

Problem #2 (This is what I could not do): The space between the two charges (i.e. r) is filled with two dielectric slabs of thickness $d_{1}$ and $d_{2}$ and dielectric constants $K_{1}$ and $K_{2}$ such that $d_{1}+d_{2} = d$.

My reasoning (so far) for the first problem: If two charges are kept at a distance r in a medium of dielectric constant K then they must be kept in air at a separation = $r\sqrt{K}$ in air to keep the force of interaction constant. This gives the effective distance they must be placed at in air, as $\frac{r}{2} + \frac{r\sqrt{K}}{k}$. How do I use this for the second problem?

Advice/help is greatly appreciated.

Thanks and cheers
vivek

 

[maverick280857]

TO add to my previous post:

Is it possible to find an effective dielectric constant in such a case?

I read somewhere that I could do this using:

$$\frac{d_{1}+d_{2}}{K} = \frac{d_1}{k_1}+\frac{d_2}{k_2}$$

Obvious as this equation seems, I can't seem to extract anything out of it let alone derive it.

Thanks,
vivek

 

[maverick280857]

To everyone: a reminder...

To the Moderator: please relocate this if you feel the same.

 

[Doc Al]

TO add to my previous post:

Is it possible to find an effective dielectric constant in such a case?

I read somewhere that I could do this using:

$$\frac{d_{1}+d_{2}}{K} = \frac{d_1}{k_1}+\frac{d_2}{k_2}$$

Obvious as this equation seems, I can't seem to extract anything out of it let alone derive it.

To derive this equation for the effective dielectric constant of the two slabs, consider two parallel plate capacitors in series:
$$1/C = 1/C_1 + 1/C_2$$
$$1/(k_{eff}/d) = 1/(k_1/d_1) + 1/(k_2/d_2)$$
... etc ($d = d_1 + d_2$)

 

[Corneo]

Are the slabs of dielectric stacked on top of one another or is it that the capacitor has a dielectric $K_1$ on the left half and a dielectric $K_2$ on the right half?

 

[reilly]

Maverick... Both problems 1 and 2 are difficult, and require either an infinite series of images, or direct solutions of the boundary value problem in terms of Fourier Bessel integrals. Prob 1 can be found in Symthe's Static and Dynamic Electricity, Chapter 5, Section 5.303 Point Charge and Dialectric Plate. Prob 2. is, or should be, a relatively straightforward extension of prob 1.
Regards,
Reilly Atkinson

 

[maverick280857]

Thanks a lot Doc Al, Corneo and reilly...

I didn't realize that problem 2 would be so tough (but that's because I already knew the governing equation...maybe?). This is because they were posed as prospective problems for an engineering entrance test (called the Joint Entrance Examination in India) which requires a background of very basic physics and electrodynamics (no Fourier-Bessel Integrals, Maxwell's Equations, PDEs, images, Laplace/Poisson equations, etc--just total derivatives and an extension of the theory mentioned in books like Crane,Zemansky/Sears, etc.) I suppose I need to know much more physics/mathematics to be able to derive these equations.

Doc, I could do the effective dielectric derivation for a parallel plate capacitor but I am not sure if I can use it for two point charges. I mean, you have two point charges in space and the space between them is filled with two dielectric slabs (only the width of which is given). I am not sure if we can use the idea of induced charge in much the same way as we set up equations for a parallel plate capacitor using Gauss's Theorem.

If however, I assume that the effective dielectric equation mentioned in my third post is correct then the problem is solved (though I don't think so as I don't know where the equation came from, for the point charges).

A related problem is (which I think can be done using the images method) to find the induced charge on a conducting plate when placed near a point charge. I would be grateful if you folks could point me to some book or internet resource where I can learn some basic stuff about the method of images to be able to apply it to relatively simple situations. I have been reading Cheng but as I have little time left for my exams, I cannot really start from scratch and understand all the mathematics required for a proper treatment of the subject.

Thanks and cheers,
Vivek