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I Electric field equation

  1. Mar 19, 2017 #1
    I was reading a book on Electromagnetism and it's said on deriving the electric field that $$\nabla \frac{1}{|x-x'|} = - \frac{x-x'}{|x-x'|^3}$$ where ##|x-x'|## is the magnitude of the distance between two point charges. I've tried to derive this result and I found that $$\nabla |x-x'| = \frac{x-x'}{|x-x'|}$$ must be true for the first identity to be valid. Is this right?
     
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  3. Mar 19, 2017 #2

    Orodruin

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    Yes, it essentially boils down to ##\nabla r = \vec e_r## in spherical coordinates.
     
  4. Mar 19, 2017 #3
    Thanks. Can you point out where I'm wrong?

    I'm assuming a Minkowskian metric, so vectors- and covectors- components are equal. So we have ##|x-x'| = \sum_i(x^i - x'^{\ i})^2##. I guess we need some formalism to express ##x-x'##, because it's a vector. Let's say an arbitrary vector can be written as ##V = V^i \partial_i##. Then ##x - x' = (x^i - x'^{\ i})\partial_i ##. We have then $$ \nabla |x-x'| = \frac{x-x'}{|x-x'|} = \sum_i\partial_i[(x^i-x'^{\ i})^2] = \sum_i\frac{(x^i - x'^{\ i})\partial_i}{(x^i - x'^{\ i})^2}$$
    If we apply the derivative on the far right-hand-side on ##|x-x'|##, we get a bad result, namely ##(x^i - x'^{\ i}) = 1##. I guess this can't be right..
     
  5. Mar 19, 2017 #4

    Orodruin

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    Why are you using ##|\vec x| = x^i x^i##?? This does not even make sense dimensionally. You are missing some square roots ...
     
  6. Mar 19, 2017 #5
    Oh yea. I got the correct result now. Thank you.
     
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