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Electrostatic force problem

  1. Feb 22, 2017 #1
    1. The problem statement, all variables and given/known data
    Two point charges q and λq located at the points, x=a & x=μa respectively. If the sum of the two charges is constant,what is the value of λ for which the magnitude of the electrostatic force is maximum?


    2. Relevant equations



    3. The attempt at a solution
    For force to be maximum dF/dq =0 and d^F/dq^2 <0. When I tried to calculate dF/dq =0 I got λ=0.
     
  2. jcsd
  3. Feb 22, 2017 #2

    BvU

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    Well, how did you do that? Can you post your steps in detail ?

    By the way: why calculate ##dF\over dq## ?
     
  4. Feb 22, 2017 #3
    I already wrote that in The attempt at a solution section.

    F= λq2/4πε0 (μa-a)2
    ⇒ dF/dq = 2λq/4πε0(μa-a)2 =0
    ⇒ 2λq = 0
    q≠0 ⇒λ=0
     
  5. Feb 22, 2017 #4

    BvU

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    If you want to find the maximum of a function of x you take the derivative wrt x.
    Here, do you want to consider the force as a function of q ?

    Note also that you forgot to make use of the given that
     
  6. Feb 22, 2017 #5
    If I used q+λq = C ⇒ λq=C-q
    Then 2λq=0 ⇒ 2(C-q) = 0 ⇒q=C

    Let F is function of distance

    F = λq2/4πε0a2(μ-1)2
    ⇒ -2λq2/4πε0a3(μ-1)2 = 0
    ⇒ -2λq2 = 0
    ⇒ λq= 0 or λq=C
     
  7. Feb 22, 2017 #6

    BvU

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    Are you saying ##\lambda = 0 \Rightarrow F = 0 ## too ? That is easily proven wrong !
     
  8. Feb 22, 2017 #7
    You are right. F shouldn't be zero but I don't find any mistake in my calculation.
     
  9. Feb 22, 2017 #8

    BvU

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    I did and I tried to point it out. You want to express F in terms of ##\lambda## and differentiate wrt ##\lambda##. Make a start ...
     
  10. Feb 22, 2017 #9
    You mean I have to differentiate F wrt λ and distance between charges and q is constant.
     
  11. Feb 22, 2017 #10

    BvU

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    Yes and No. In that order:
     
  12. Feb 22, 2017 #11
    λ+qλ = C
    ⇒dq = -(q+1)dλ/λ ............... (1)

    dF/da = [(μa-a)2 { 2λq dq + q2 dλ} - 2q2λa (μ-1)2 ]/ 4πε0 (μa-a)2 = 0

    After putting value of eq (1) I got final eq
    -q(q+1) dλ = 2λa
     
    Last edited: Feb 22, 2017
  13. Feb 23, 2017 #12

    BvU

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    Can you explain why you are now differentiating wrt a ? I thought we agreed to seek a value for ##\lambda## that gives the maximum ##F## ?

    And: do you think we can leave out the constants ##\displaystyle {1\over 4\pi\varepsilon_0 (\mu a - a)^2 } ## ?
     
  14. Mar 2, 2017 #13
    I got λ=1 if we leave the constants.

    But I don't understand why are we differentiating wrt λ?
     
  15. Mar 2, 2017 #14

    gneill

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    Because the problem asks, "what is the value of λ for which the magnitude of the electrostatic force is maximum?"
     
  16. Mar 2, 2017 #15

    BvU

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    Can you explain how you found ##\lambda = 1 ## ?
    And can you justify leaving out the constants ? Why is that allowed ?
     
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