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Force acting on a dipole in non-uniform electric field.

  1. May 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Calculate the force acting on a dipole of dipole moment P due to a line charge of density λ
    at a distance r from it??

    2. Relevant equations
    field due to a line charge= λ/2εr


    3. The attempt at a solution
    tried caculating force on each individual charge but i dont see how dipole moment should come in play here?
     
  2. jcsd
  3. May 27, 2013 #2

    TSny

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    This approach should lead to the answer. If you can show some of the details of your calculation, maybe we can see how the dipole moment will come in.

    A better approach is to use energy concepts. Are you familiar with the expression for the potential energy of a dipole in an electric field? Do you know how to relate potential energy to force?
     
  4. Jun 2, 2015 #3
    Hello TSny ,

    What is the orientation of the dipole with respect to line charge in this problem ?

    Is it perpendicular to the line charge such that the center of dipole is at a distance 'r' ? Am I interpreting it correctly ?
     
    Last edited: Jun 2, 2015
  5. Jun 2, 2015 #4
    Or is it that the end closer to the line charge is at a distance 'r' ?
     
    Last edited: Jun 2, 2015
  6. Jun 2, 2015 #5

    TSny

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    The OP was not clear on the orientation. So I guess that P is oriented parallel to the E field. I think r can be taken as the distance to the center of the dipole.
     
  7. Jun 2, 2015 #6
    Sir,

    If that is the case and if +q is closer to line charge ,then

    Net force on dipole = ##\frac{2kλ}{(r-a)}q - \frac{2kλ}{(r+a)}q## = ##-\frac{2kλ\vec{p}}{(r^2-a^2)}##

    If I consider r>>a ,then net force = ##-\frac{2kλ\vec{p}}{r^2}##

    Have I done it correctly ?
     
  8. Jun 2, 2015 #7

    TSny

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    Looks very good. I believe that's correct.
     
  9. Jun 2, 2015 #8
    Here I have a doubt . First I will show the work .

    Potential Energy of a dipole in Electric field is ##U =-\vec{p} \cdot \vec{E}## . Since ##\vec{p}## and ## \vec{E}## are oppositely aligned , U = pE .

    ##U =\frac{2kλ\vec{p}}{r}##

    ##F=-\frac{dU}{dr}##

    ##F=-\frac{2kλp}{r^2}##

    Have I done it correctly ?

    If you think I have done it correctly , my doubt is that even though electric field across the length of dipole is non uniform , still expression for potential energy of dipole remains ##U =-\vec{p} \cdot \vec{E}## .

    But this expression for U was for uniform electric field . Can you explain it ?

    Thanks .
     
    Last edited: Jun 2, 2015
  10. Jun 2, 2015 #9

    TSny

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    I had in mind an "ideal dipole" where the length of the dipole is infinitesimally small. Then you can use ##U = -\vec{p} \cdot \vec{E}##.

    For a finite length you can still use potential energy. The potential energy of a point charge ##q## in the field of the line charge is ##U = -2 k \lambda q \ln \frac{r}{r_0}## where ##r## is the distance of ##q## from the line charge and ##r_0## is an arbitrarily chosen distance from the line charge for defining zero potiential.

    So, the potential energy of the dipole (for the case where the +q is farther away) is ##U = -2 k \lambda q \ln \frac{r+a}{r-a}##. Here, ##r## is the location of the center of the dipole.

    You can then get the force from ##F = -\frac {dU}{dr}##.
     
  11. Jun 2, 2015 #10
    Thank you Sir .
     
    Last edited: Jun 3, 2015
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