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Electric Potential and bounds of integration

  1. Mar 27, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the potential difference between V(P) - V(R) and V(c) - V(a)


    2. Relevant equations

    Electric potential:

    V(a) - V(b) = [itex]\int^{b}_{a}[/itex]E*dr

    V(b) - v(a) = -[itex]\int^{a}_{b}[/itex]E*dr

    Fundamental Theorem of Calculus:

    F(b) - F(a) = [itex]\int^{b}_{a}[/itex]f(x)dx


    3. The attempt at a solution

    My question is about the solutions attached. In the first example, we have V(P)-V(R), which makes the integral [itex]\int^{R}_{P}[/itex]E*dr. However, when evaluating the integral, the solution takes V(P)-V(R) which seems to disagree with the fundamental theorem of calculus. As per the bounds, I think it should be V(R)-V(P).

    Note: while I think it is wrong, the online homework said that evaluting it V(R)-V(P) is incorrect.

    Likewise, to add to the confusion, the second picture is the solution for V(c)-V(a), which makes an integral [itex]\int^{a}_{c}[/itex]E*dr. In this case, though, the solution is found by taking v(a)-v(c). (technically v(a) - v(b) because v(c)-v(b) is a constant). This result seems to follow the fundamental theorem of calculus and is the result I expected.

    Finally, I thought maybe it had to do with the geometry so I attached a picture of the scenario. Hopefully some one can shed some light on the confusing integral bounds. Thanks
     

    Attached Files:

  2. jcsd
  3. Mar 27, 2013 #2

    rude man

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    .

    No. V(b) - V(a) = -∫E*dr with lower limit of r(a) and upper limit of r(b), with * denoting the vector dot-product. If a unit test charge is moved from a to b where r(a) > r(b) then this integral is positive and represents both the gain in potential and the work done in moving a unit test charge from a to b.

    OK, yor formula is not wrong quantitatively but you should think of going from a to b as integrating from a to b, and that means V = -Edr.
     
    Last edited: Mar 27, 2013
  4. Mar 27, 2013 #3

    rude man

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    This is a contradiction in terms!
     
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