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Calculate E Field between 4 cylinders

  1. Feb 16, 2016 #1
    Hi,

    I have four cylinder with circular cross section. Each has the same potential (+V) applied to it. They are all the same size, length, diameter etc. and each are spaced an equal distance apart so that they all touch on to an imaginary circle with a distance x from the origin.

    How to calculate the E field in between these cylinders?
     
  2. jcsd
  3. Feb 16, 2016 #2

    mfb

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    Are the cylinders very long compared to their diameter? Then the field will be negligible. If not, you probably need a three-dimensional numerical approximation.
     
  4. Feb 17, 2016 #3
    Thank you mfb for ur reply. yes - they are very long compared to the diameter. but i was thinking that an analytical solution may b possible. if one were to treat this as say a 2d dimensional problem. so in say an x-y plane (assuming we are at the mid-point along the cylinders) we have 4 circles each with potential V applied.
    any ideas?
     
  5. Feb 17, 2016 #4

    mfb

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    If something is surrounded by areas of constant voltage, the electric field is zero. You only get an electric field from the regions at the ends of the cylinders, something a two-dimensional analysis (x/y) misses.
     
  6. Feb 17, 2016 #5
    Thanks! I take your point. Nevertheless in the space in between the cylinders there exists an region which wud invoke a force on a unit positive test charge. So if say we assumed dat a unit positive test charge entered the region between the 4 cylinders how wud it behave? if we assumed say constant axial velocity and ignored the effects from the cylinders ends. what sort of motion wud the charge exhibit. can we describe that?
     
  7. Feb 17, 2016 #6

    mfb

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    As long as the cylinders are kept at their potential, there would be no force on charged objects.
     
  8. Feb 18, 2016 #7
    I get that the field is zero But what if

    a charged particle was moving axially in between the cylinders but very close to one of the cylinder surfaces. Do you mean 2 say that it would not be repelled more by that particular cylinder. . . Intuitively I am struggling to grasp this. Can any1 offer some help? Scientific, anecdotal or otherwise pls

    Also lets say a charged particle wss again moving axially in between the cylinders but had some angular momentum which directed it towards a cylinder surface. .. surely in this case there must be a force exerted which will push the particle towards the 'central axis'? Any help wud b gladly received
     
  9. Feb 18, 2016 #8

    mfb

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    If the cylinders have the same potential everywhere, nothing happens. Why should there be a force?
     
  10. Feb 19, 2016 #9
    Due to proximity. if i have 2 opposite charges and i try to bring them together the force will increase as they get closer, this is coulombs law as well all know. but i am trying to get a real grasp on this. just because the potential is the same on all the cylinders doesnt mean that a test charge will feel the same force,? is that true. i can grasp your point if u say that the test charge is exactly in the center. but what if that test charge moves closer to one of the cyinders, surely its influence is now greater than that of the other 3.
    sorry if this sounds dumb but any sort of illustration you or any1 else can offer may help me
    tks
     
  11. Feb 19, 2016 #10
    just incase i am not on the same page as u. this is what i was thinking of in the attahed png image. the black dot is the positive test charge. left is 3d view, right is a 2d cross section somewher along the middle of the cylinders
    tks
     

    Attached Files:

  12. Feb 19, 2016 #11
    Voltage isn't the same thing as charge. Charge has an absolute meaning. Voltage is only relative. You need to specify a zero point for your potential.
     
  13. Feb 19, 2016 #12

    mfb

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    I interpreted post #1 as touching cylinders. If they do not touch, there will be some electric field. Yes, this field will be a bit stronger close to the cylinders, and a positive charge would be repelled by positively charged cylinders.
     
  14. Feb 19, 2016 #13
    Thanks. So is there a way I could determine by an equation motion of a charged particle moving axially between the cylinders? On the face it seems to me that there might be. If we assume the test charge has constant axial velocity and the cylinders are all the same with same Volts.

    How would I begin to solve this?

    And are your previous statements still correct for my drawing?

    Tks
     
  15. Feb 19, 2016 #14

    mfb

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    No, with such a setup you have an electric field.
    A two-dimensional numerical simulation will work. Make a fine grid, solve for the potential at every point with the Poisson equation, then calculate the electric field as the derivative.
     
  16. Feb 20, 2016 #15
    How can u be so sure that no analytical solution exists. How can u tell

    Tks
     
  17. Feb 20, 2016 #16

    mfb

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    I am not sure, but it would surprise me.
     
  18. Feb 20, 2016 #17
    ok. i thought maybe u had a way of telling.

    so solving this numerically. for sake of making this easier for me to understand let us just say we only want to solve the potential at 1 point on the grid. i know the poisson equation. how do does someone actually use the poisson equation in practise? for my case with the 4 cylinders - or circles for this 2d case in question - can u give me any hints or tips or point me in the right direction.

    tks
     
  19. Feb 20, 2016 #18

    mfb

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    That doesn't help, unfortunately.

    For cartesian coordinates with a rectangular grid, the Poisson equation simplifies to "every cell has a potential that is the average of the 4 cells around it". Fix the potential of cells that are in your cylinders, fix the potential "far away" (borders of the simulation, far away from the cylinders), iterate the calculation of averages until the system does not change any more.

    Edit: quick and dirty with excel:
    potential.png
     
    Last edited: Feb 20, 2016
  20. Feb 21, 2016 #19
    very nice how do u do it. how long did it take to calculate? i never thought of using excel.
    few questions if u can pls.
    what is far away and to what potential do u fix it?
    is it possible that u can share the excel formula with the corresponding theory. that wud be helpful to see how u did it.

    tks
     
  21. Feb 21, 2016 #20

    mfb

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    As described. I fixed some cells (in four roughly circular patterns) to 1 and the boundary to 0, then I let each cell to be the mean of the four surrounding cells and let excel solve it iteratively.
    Every finite distance will introduce some error, but you cannot calculate infinite grids of course. Something that is far away compared to the size of the cylinder structure will keep the error small.

    I attached the file. I changed the central potential values to 9 so the 1-digit display has a bit more details than "1" and "0", Everything changes linearly, so the absolute values are quite meaningless anyway.
     

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