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EMF in a loop; non-constant magnetic field

  1. Apr 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Refer to Figure attached.

    The current in the long straight wire is

    i = (4.5A/s2)t2-(10A/s)t

    Find the EMF in the square loop at t=3.0s.


    2. Relevant equations

    [itex]\xi = -\frac{d \Phi}{dt}[/itex]

    And Biot-Savart law for straight wires of infinite length:

    [itex]B = \frac{\mu_0 i}{2 \pi R}[/itex]


    3. The attempt at a solution

    Solution: 600 nV

    I cannot recreate this result.

    I consider two loops, one below the wire in the picture and one above.
    I calculate their EMFS separately as:

    [itex]\xi = -\frac{A \mu_0}{2 \pi R} \frac{di}{dt}[/itex]

    and find the difference between them.
    This doesn't reveal 600 nV.

    I try integrating with respect to R, and get mathematical gibberish (ln[0]).

    I must be missing something fundamental in my setup - any suggestions?
     

    Attached Files:

    Last edited: Apr 6, 2013
  2. jcsd
  3. Apr 6, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Think about the flux through the shaded areas in the figure.
    [EDIT: This can help avoid dealing with r = 0. But I don't get 600 nV either. You will need to integrate.]
    [Edit 2: OK, it does come out 600 nV.]
     

    Attached Files:

    Last edited: Apr 6, 2013
  4. Apr 6, 2013 #3
    The change in flux for those areas cancel.

    So, I consider the third region.

    4cm away from the source.
    8cm long, 16cm wide (not shown in figure)

    [itex]\xi = \frac{\mu_0 * 8cm * 16 cm * 17A/s}{2 \pi} \int^{12cm}_{4cm} \frac{1}{R}[/itex] = 47 nV.

    If I don't integrate, and instead just find the difference, I get closer, but it doesn't make sense to do that. ( = 725 nV)
     
  5. Apr 6, 2013 #4
    AH!

    You were right, it does come out to 600 nV.
    Infinitesimal lengths :).

    Thanks
     
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