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I Weird thinking of electric field inside a hollow cylinder.

  1. Sep 27, 2016 #1
    While I was studying with electric field about cylinder, I learned that for a very long cylinder, the electric field in the hollow of cylinder will be zero.

    http://physics.stackexchange.com/questions/156789/electric-field-of-hollow-cylinder

    However, I couldn't accept this intuitively, and thought up this weird idea.

    We can express electric field E of charged line like

    ##E=\frac \lambda {2\pi\epsilon_0 r}##

    Thus, we knows that (+) charge between two parallel lines with same charge density will always move to their center, right?

    Then, suppose we have a (+) charge in a cylinder other than on its axis, and let's see that cylinder above from it.

    1.png

    And this is what really confuses me.

    2.png

    Draw a line that passes charge, then it'll meet with circle(cylinder) at two points(lines). Since a red dot(line) is always closer than a blue dot(line), sum of all forces will head to the left(?).

    3.png

    But this weird calculation conflicts with the fact that E=0 in the hollow of the cylinder.

    What is a critical mistake of this logic(?). Will it be possible to explain why this image is wrong without using exact calculation?
     
  2. jcsd
  3. Sep 27, 2016 #2

    kuruman

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    It is good that you worry about this. The critical mistake in the logic is this. Imagine two intersecting lines crossing at your off-center point. They define a blue arc dsblue and a red arc dsred. We make the ds arcs very small, not like in your figures, so that the contributions to the E-field from each arc are antiparallel and can be treated as contributions from lines of charge . The charge on each arc is proportional to ds, so that the magnitude of its contribution to the E-field is $$ dE \sim \frac{ds}{r} = \frac{r d \theta}{r} = d \theta $$ Since the subtended angle by the two arcs is the same, the fields cancel. This argument is similar to the 3d argument for the electric field inside a uniformly charged shell, except there one uses solid angles.
     
  4. Sep 27, 2016 #3
    Thanks for cool explanation! They are canceling out each other so clearly.... awesome!
     
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