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How to get maximum electrostatic force?

  1. Oct 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Let's take two spheres that are somewhat nearby. On one sphere, sphere A, there is Q units of charge. Now, we take some little bit of charge q off of sphere A, and put it on sphere B. What should the ratio q/Q be if we want the electrostatic force between the spheres to be a maximum?

    2. Relevant equations
    F = K Q q / r2

    3. The attempt at a solution

    For this question, I just took plotted numbers and found that the charge q has to be half of Q in order to have the greatest electrostatic force between the spheres. So the ratio of q/Q = 1/2

    How would one do this problem without plotting numbers?
     
  2. jcsd
  3. Oct 26, 2015 #2

    BvU

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    Hi catch,

    Do you know what the criteria are for the maximum of a function of one variable ? If so, all you need to do is write F as a function of one variable !
     
  4. Oct 26, 2015 #3
    unfortunately, I don't. do you mean like global maximum/minimums of a function?
     
  5. Oct 26, 2015 #4

    BvU

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    I mean like if f(x) has a maximum in x0, then what about df/dx at x = x0
     
  6. Oct 26, 2015 #5
    oh, so if I picture a graph, the highest point is where the slope is 0.
    and to get the slope, we need the derivative.
     
  7. Oct 26, 2015 #6

    BvU

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    Yessss! Now, what derivative ? f is easy: F. But what is a candidate for the variable x ?
     
  8. Oct 26, 2015 #7
    q or Q?
     
  9. Oct 26, 2015 #8

    BvU

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    Well, that's a no-brainer: Q is a number, so it must be q ! And now it's time to realize that the Q in the problem statement is not the Q in the relevant equation ! But you knew that, right ?
     
  10. Oct 26, 2015 #9
    whoops, for a second I thought Q was the charge of the sphere that was taking in q.
     
  11. Oct 26, 2015 #10

    BvU

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    So what is now the one-variable function you are going to differentiate with respect to ##q## ?
     
  12. Oct 26, 2015 #11
    F = k q Q / r^2
     
  13. Oct 26, 2015 #12
    whoops, should be F = k (Q-q)(q) / r^2
     
  14. Oct 26, 2015 #13

    BvU

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    Agreed. Does it work out OK now ?
     
  15. Oct 26, 2015 #14
    hmm, my q turned into a 1.
    dF / dq = K(Q-1) / r^2 = 0

    Q = 1

    doesn't seem right.?:)
     
  16. Oct 26, 2015 #15

    BvU

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    Ah ! So not the same as when looking at the plotted figure ?
    Or perhaps a second try differentiating F = some constant times ( Qq - q2 ) ?
     
  17. Oct 26, 2015 #16
    where did ( Qq - q2 ) come from?
     
  18. Oct 26, 2015 #17

    BvU

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    !
     
  19. Oct 26, 2015 #18
    oh, we shouldn't have factored out the q?

    anyways, k (Q-2q)/ r^2 = 0

    Q - 2q = 0

    2q = Q

    q/Q = 1/2
     
  20. Oct 26, 2015 #19

    BvU

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    Bingo ! Well done.
     
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