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Homework Help: Expression for induced ε from reducing Loop

  1. Dec 11, 2008 #1
    1. The problem statement, all variables and given/known data

    A circular loop made from a flexible, conducting wire is shrinking. Its radius as a function of time is render?tex=r+%3D+r_0+e%5E%7B-%5Cbeta+t%7D.gif . The loop is perpendicular to a steady, uniform magnetic field B .
    Find an expression for the induced ε in the loop at time t.
    Give your answer in terms of render?tex=r_0.gif , render?tex=%5Cbeta.gif , t, B and appropriate constants.

    2. Relevant equations
    Magnetic Flux (Φm)=A*B
    I(induced) = ε / R
    ε =Absolute value of (dΦ/dt)


    3. The attempt at a solution
    At first get the magnetic flux using the Area * Magnetic field. Area is (pi)r^2 * B. I use this in faradays law with dt but i dont get the write expression. I end up getting

    ε(t)=(B(pi)(r_0*e^(beta*t))^2)/t

    I know im differentiating somewhere wrong but i dont know where.
     
    Last edited by a moderator: Apr 16, 2017
  2. jcsd
  3. Dec 11, 2008 #2

    Redbelly98

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    What's the derivative of

    f(t) = eat
     
  4. Dec 11, 2008 #3
    Wouldnt that just be a'(t)e^at?
     
  5. Dec 11, 2008 #4

    Redbelly98

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    Uh, no. I meant for "a" to be a constant, sorry about the confusion.

    If you can express r2 in the form:

    r2 = ro2eat,​

    and if you can take the derivative of eat correctly, then you'll be able to figure out what dA/dt is.
     
  6. Dec 11, 2008 #5
    wouldn't the derivative of f(t)=e^at be just (1/a)e^at? im not really sure and i dont understand how you were able to square the equation?
     
  7. Dec 11, 2008 #6

    Redbelly98

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    No, the derivative is
    a eat

    You'll have to figure out an expression for A, using A=∏r2, and differentiate it. I mentioned the whole eat thing because the expression for A could be put in that form.

    It sounds like you're having trouble remembering calculus ... has it been a while since you took it?
     
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