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Field on a plane from line charge

  1. Nov 20, 2013 #1
    1. The problem statement, all variables and given/known data
    A wire at extends for -L to L on the z-axis with charge ##\lambda##. Find the field at points on the xy-plane


    2. Relevant equations

    ##E(r)=k\int\frac{\rho}{r^2}dq##
    ##k=\frac{1}{4\pi ε_o}##
    3. The attempt at a solution

    First time I've looked for field on a plane so I wasn't sure if I'm doing this correctly.

    ##dq=\lambda dz=2L\lambda##
    I made a right triangle with one side L and the other two x and y.
    ##r^2=L^2+x^2+y^2## where only x and y change so I have dxdy.

    So, the final integral

    ##k(2L\lambda)\int\int\frac{1}{L^2+x^2+y^2}dxdy## with limits 0→∞.

    I expect at large x,y that the field should be very small and small x,y it should be large. This integral satisfies both of those conditions.

    The method I saw our professor do for another, somewhat similar, problem was exceedingly more complicated than this.
     
  2. jcsd
  3. Nov 20, 2013 #2

    CompuChip

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    Science Advisor
    Homework Helper

    Hmm, the question is to find the field at some point (x, y, 0). So you should fix x and y, and add up all the contributions from the charges on the z-axis.
    Therefore I would expect something like
    $$E(x, y, 0) = k \int_{-L}^L \frac{\rho}{r^2} \, dz$$
    (where did your ##\rho## go? Should there not be something with ##\lambda## in there)?

    Try drawing such a point, and calculating ##r## and ##\rho \, dz## from the geometry of the problem rather than by trying to copy your notes.
     
  4. Nov 20, 2013 #3
    ##\rho## is the charge density ##\lambda##
    I was trying to use ##\rho## in the general sense but I sort of got ahead of myself putting it in there.

    ##E(r)=k\int\frac{1}{r^2}dq##

    ##dq=\lambda dz##

    so, ##E(r)=k\int\frac{\lambda}{r^2}dz##


    As soon as I walked away from the computer I knew I was wrong. My whole way of thinking was stuck in previous problems I've done.

    X and Y are my points of interest and Z is my variable (for the line charge not on the plane). So if I want to figure out what the field is at X and Y I need to count up all the little pieces of charge at distance 'r' from them to the line.

    I have ##\int_{-L}^{L} \frac{\lambda}{z^2+x^2+y^2}dz##
     
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