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Total electric potential energy

  1. Nov 24, 2015 #1
    1. The problem statement, all variables and given/known data
    My text book states that for point charges ##q_1, q_2, ... ,## where distance between ##q_i## and ##q_j## is ##r_{ij},## the total potential energy U is the sum ## U = \dfrac{1}{4\pi\epsilon_0} \sum_{i<j} \dfrac{q_iq_j}{r_{ij}} ## and specifically mentions not to count them twice as (i,j) and (j,i), but I don't see why.

    2. Relevant equations
    Potential energy of a point charge ##q_i## is ##U_i = \dfrac{q_i}{4\pi\epsilon_0}\sum_{i \neq j}\dfrac{q_j}{r_j} ##

    3. The attempt at a solution
    Using the above equation, I think the total potential energy should be ## U = \dfrac{1}{4\pi\epsilon_0} \sum \dfrac{q_iq_j}{r_{ij}} ##, counting both cases (i,j) and (j,i). For instance, when there are two point charges ##q_1## and ##q_2##, the potential energies of individuals is ##U_1 = \dfrac{q_1}{4\pi\epsilon_0}\dfrac{q_2}{r_{12}}##, ##U_2 = \dfrac{q_2}{4\pi\epsilon_0}\dfrac{q_1}{r_{21}}## so that the total potential energy is the sum ## U = U_1 + U_2 = \dfrac{1}{4\pi\epsilon_0} \sum_{1\leqslant i,j\leqslant2} \dfrac{q_iq_j}{r_{ij}} ## which counts both the cases (1,2) and (2,1).
    Where am I wrong?
     
  2. jcsd
  3. Nov 24, 2015 #2
    This condition should add to the last equation: ##i\neq j##
     
  4. Nov 25, 2015 #3
    Suppose you have only 2 charges.
    The work done in moving charge 1 into place is zero.
    Then you can move charge 2 into place by doing work that depends on R12.
    So you don't need R21, since you would be counting the work twice.
    Further, if you add an additional charge then the additional work done
    depends on R13 and R23 and you don't need to count R31 and R32.
     
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