# Calculating the Coulomb Force between 2 charges in space

1. Mar 5, 2005

### 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 betwen 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?

Thanks and cheers
vivek

Last edited: Mar 5, 2005
2. Mar 5, 2005

### 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

3. Mar 7, 2005

### maverick280857

To everyone: a reminder....

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

4. Mar 8, 2005

### Staff: Mentor

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 ($d = d_1 + d_2$)

5. Mar 8, 2005

### 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?

6. Mar 9, 2005

### 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

7. Mar 9, 2005

### maverick280857

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

I didn't realize that problem 2 would be so tough (but thats 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