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(Image Theory) Boundary conditions and how many image charges do i need?

  1. Jan 4, 2012 #1
    1. The problem statement, all variables and given/known data
    An electrostatic point charge of 1 Coulomb (C) placed symmetrically
    between two infinitely/perfectly conducting parallel plates. These two infinitely large
    conducting plates are parallel to the yz plane.The region between the two plates is designated as “Region A.” Starting with this original problem,we will remove the conducting plates, we will use the image theory, and we will define image charges to obtain an equivalent problem. Conducting plates coincide with planes P1 and P2, and hence these conducting plates impose some boundary conditions on the electric field.
    Q1) What are the boundary conditions for the electric field on the infinitely/perfectly conducting plates P1 and P2?
    We have to define a measure of approximation for Problem2(image charges). In other words, as an equivalent problem, how close is Problem 2 to Problem 1(Parallel plates)? If we use a finite number of image sources,we know that Problem 2will not be exactly equivalent to Problem 1. Nevertheless, we need to
    understand how much error we are making by using a finite number of image sources.For this
    purpose, let us define a measure of error as the ratio of the magnitude of the tangential
    component of the electric field to the normal component of the electric field at the point E:

    magnitude of the tangential component of the electric field (At the point on P1 plate)
    e= -----------------------------------------------------------
    electric field to the normal component of the electric field (At the point on P1 plate)

    Q2) Let us consider a case where we would like to have a
    maximum of 1 error. In other words, we would like to have e<0.01.
    .How many image sources do we need to satisfy this constraint?


    2. Relevant equations
    E2t = E1t
    D1n - D2n = ρ
    E= -∇Φ
    Φ = q/r


    3. The attempt at a solution

    For Q1 i know that E1t = E2t and as back side of the plate Et1=0=Et2 and n(normal) . D1 = ρ

    For Q2 i know for not to make any error i need infinite number of image sources.

    the equation Φ = qƩ(n = 0 to ∞) { (-1)n+1 / [(x-nd)2 + y2 + z2]1/2 + (-1)n+1 / [ (x + nd)2 + y2 + z2]1/2}
    The attempts that i made maybe totally incorrect.
     
    Last edited: Jan 4, 2012
  2. jcsd
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