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Homework Help: Electric Field Due to Infinite Line Charge

  1. Jan 28, 2010 #1
    1. The problem statement, all variables and given/known data

    "Compute the electric field due to an infinite line charge by integrating the expression obtained from the inverse square law."

    2. Relevant equations

    I think that the equation required is:

    [itex]E(\bold{r}) = \frac{1}{4 \pi \epsilon_0} \int_{-\infty}^{+\infty} \frac{(\bold{r} - \bold{r}') \sigma}{|\bold{r} - \bold{r}'|^3} dx[/itex]

    3. The attempt at a solution

    Well, I don't know what to put in the above integral for r and r', or even if the above integral is even right...

    Thanks for any help :) .
  2. jcsd
  3. Jan 28, 2010 #2
    Your equation looks correct.

    Build a coordinate system. Include the wire in some convenient place, like along the x-axis. The wire will extend to plus and minus infinite x. As presented, the integral already assumes the wire lies parallel to the x-axis.

    r is a displacement vector that varies with x, and locates points within the wire.

    r' is a displacement vector of the point for which we wish to calculate the electric field. It is a constant with respect to the variable of integration, x.
  4. Jan 28, 2010 #3


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    im working on the same problem. it appears that if infinite limits of integration are used the expression blows up, is it better to set the limits from 0-> L and then take the limit as L goes to infinity?
  5. Jan 29, 2010 #4
    Mmm, I'm getting an integral of something similar to 1/x² between -infinity and infinity, that doesn't look good...
  6. Jan 29, 2010 #5


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    [tex]\int_{-\infty}^{+\infty} \frac{dx}{(x^2+y^2)^{3/2}}= \left( \frac{x}{y^2 \sqrt{y^2+x^2}} \right) |_{-\infty}^{+\infty}[/tex]

    The main observation you need to make to solve this is:

    [tex]\lim_{x \rightarrow \infty} \frac{x}{y^2 \sqrt{y^2+x^2}} = \frac{1}{y^2}[/tex]

    Since x is a first order variable on both the numerator and denominator.
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