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Calculating the Coulomb Force between 2 charges in space

  1. Mar 5, 2005 #1
    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 [itex]Q_{1}[/itex] and [itex]Q_{2}[/itex] 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

    [tex]F = \frac{1}{4\pi\epsilon_{0}K}\frac{Q_{1}Q_{2}}{r^2}[/tex]

    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 [itex]d_{1}[/itex] and [itex]d_{2}[/itex] and dielectric constants [itex]K_{1}[/itex] and [itex]K_{2}[/itex] such that [itex]d_{1}+d_{2} = d[/itex].

    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 = [itex]r\sqrt{K}[/itex] in air to keep the force of interaction constant. This gives the effective distance they must be placed at in air, as [itex]\frac{r}{2} + \frac{r\sqrt{K}}{k}[/itex]. How do I use this for the second problem?

    Advice/help is greatly appreciated.

    Thanks and cheers
    vivek
     
    Last edited: Mar 5, 2005
  2. jcsd
  3. Mar 5, 2005 #2
    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:

    [tex]\frac{d_{1}+d_{2}}{K} = \frac{d_1}{k_1}+\frac{d_2}{k_2}[/tex]

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

    Thanks,
    vivek
     
  4. Mar 7, 2005 #3
    To everyone: a reminder....

    To the Moderator: please relocate this if you feel the same.
     
  5. Mar 8, 2005 #4

    Doc Al

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    Staff: Mentor

    To derive this equation for the effective dielectric constant of the two slabs, consider two parallel plate capacitors in series:
    [tex]1/C = 1/C_1 + 1/C_2[/tex]
    [tex]1/(k_{eff}/d) = 1/(k_1/d_1) + 1/(k_2/d_2)[/tex]
    ... etc ([itex]d = d_1 + d_2[/itex])
     
  6. Mar 8, 2005 #5
    Are the slabs of dielectric stacked on top of one another or is it that the capacitor has a dielectric [itex]K_1[/itex] on the left half and a dielectric [itex]K_2[/itex] on the right half?
     
  7. Mar 9, 2005 #6

    reilly

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    Science Advisor

    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
     
  8. Mar 9, 2005 #7
    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). :smile:

    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
     
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