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Questions based on derivation of electrical potential energy

  1. Nov 13, 2015 #1
    Consider a system of two charges ## q_1## and ##q_2## separated by distance ##r_1##.This configuration is associated with a potential energy ##U_1##.When the separation is increased to ##r_2##.Potential energy becomes ##U_2##

    diag.png



    ##dW_E##=##\vec{F}##.##\vec{dr}##

    ##dW_E##=##Fdrcos0##=##\frac{1}{4πε0}\frac{q_1q_2}{r^2}##dr

    ⇒##W_E##=##\int_{r_1}^{r_2}####\frac{1}{4πε0}####\frac{q_1q_2}{r^2}##dr

    ##W_E##=##\frac{-q_1q_2}{4πε0}####[\frac{1}{r_2}##-##\frac{1}{r_1}]##

    By definition of potential energy ,

    ##U_2##-##U_1##=##-W_E##

    ⇒##U_2##-##U_1##=##\frac{q_1q_2}{4πε0}####[\frac{1}{r_2}##-##\frac{1}{r_1}]##

    Taking infinity as reference i.e ##r_1##=∞ and ##U_1##=0

    ⇒##U_2##-0=##\frac{q_1q_2}{4πε0}####[\frac{1}{r_2}##-##\frac{1}{∞}]##

    ⇒##U_2##=##\frac{1}{4πε0}####\frac{q_1q_2}{r^2}##

    Taking ##U_2##=##U## and ##r_2##=##r_1##

    ##U##=##\frac{1}{4πε0}####\frac{q_1q_2}{r}##

    I want to ask as we can see ##r_2 ##is >##r_1##

    and then problem assumes ##r_1## to be ∞ my question is what is ##r2## then?how can ##r_2## be greater than ##r_1##?How can any number be greater than infinity?
     
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  3. Nov 13, 2015 #2

    jbriggs444

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    The only place where I see an assumption that r2 > r1 is in the limits of integration in ##W_E =\int_{r_1}^{r_2} \frac{1}{4πε0}\frac{q_1q_2}{r^2}dr##

    But that is not an assumption that r2 > r1. You can evaluate an integral from a larger endpoint to a smaller. The result is the negative of the same integral evaluated from smaller to larger.
     
  4. Nov 13, 2015 #3
    This is what we see looking at the figure, yes.

    Yes, now they want you to imagine increasing ##r_1## so that it's not only bigger than ##r_2## but bigger than any value of ##r## you can imagine.

    They replaced ##r_2## with ##r##, meaning that instead of thinking of it as a fixed value for the separation distance, it's now thought of as a variable.

    It can't. Very poorly authored example, if you ask me.
     
  5. Nov 14, 2015 #4
    But they don't want to increase ##r_1## such that distance between them becomes greater than ##r_2## instead they want to increase ##r_1## such that distance between them becomes ##r_2##
     
  6. Nov 14, 2015 #5

    haruspex

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    Unless you have mistyped the final equation, it contains r, not r1. So it's a misprint: they mean replacing r2 with r.
     
  7. Nov 14, 2015 #6
    Yes.
     
  8. Nov 14, 2015 #7

    haruspex

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    It isn't. When you write an integral from a to b, there is no requirement for b to be greater than a:
    ##\int_a^bf(x)dx = -\int_b^af(x)dx##
     
  9. Nov 14, 2015 #8
    Yes,you have told me once.Upper and lower limit just indicate final and initial positions but this line confused me
     
  10. Nov 14, 2015 #9

    haruspex

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    Ok, but the equation is valid either way. With hindsight, the author might have preferred to write "changed to r2".
     
  11. Nov 14, 2015 #10
    I can also refer to this http://aakashtestguru.com/document/askExpert/08-10-2015-18:26:171608102015DP.pdf [Broken]
    for derivation of potential energy between the two charges ,right?
     
    Last edited by a moderator: May 7, 2017
  12. Nov 14, 2015 #11
    Is this another typo then?

    I suggest you present the entire example, as it was written by the author, along with a reference to the text you are quoting from.
     
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