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Current due to varying magnetic field.

  1. May 2, 2010 #1
    1. The problem statement, all variables and given/known data
    In this problem I have to find the current through the section PQ of two loops of wire sharing one side.
    The loop thing is in a uniform magnetic field pointing into the page which varies in based on the equation B=zt where t is in seconds and B is in teslas. z is a constant.
    The wires in this problem have a resistance per length of y ohms/meter. y is a constant.
    The circuit is composed of parts of length a. a is a constant.

    |xxxxxxxxxxx|xxxxxx| *height is also a


    In the above diagram, | and - represent wires. xxxxx represents the uniform magnetic field. .... is for spacing. the right loop is a square with side lengh a. The left loop is a rectange with one side length a and the other side length 2a.

    2. Relevant equations
    area of a rectangle: side1 * side2
    flux through a closed loop = magnetic field * area
    emf = change in flux / change in time
    kirchoff's rules

    3. The attempt at a solution
    To start, I found the area of each loop.
    area of loop 1 (larger loop) A_1 = 2*a*a =2a^2
    area of loop 2 (smaller loop) A_2 = a*a = a^2
    I then found the rate of change of flux which gives the emf:
    (d\dt)(zt) * A so
    for loop 1 the change in flux is zA_1 = z2a^2
    for loop 2 the changfe in flux is zA_2 =za^2
    I then determined the resistance in a segment of wire with length a:
    R= a*y
    which I can use to find the resistance in the sections of the loops that aren't shared.
    This is 5R for loop 1 and 3R for loop 2.Also the resistance in the shared section is R.
    I then used kirchoff's rules to come up with the equations and currents:
    current 1 = I_1 which runs through loop 1.
    current 2 = I_2 which runs through loop 2.
    current 3 = I_3 which runs through the joined side of loop 1 and 2.
    I1-I2 = I3 since the currents are flowing in opposite directions through PQ.
    emf1 (change in flux for loop 1) - 5RI_1 - RI_3 = 0
    emf2 (change in flux for loop 2) - 3RI_2 + RI_3 = 0 (+RI_3 because the current is flowing against the emf)
    I then expanded, simplified etc which eventually got me
    (3emf_1 -5emf_2)/7R = I_3 which is what I need to solve for except that my answer is wrong. I checked through my manipulations of the equations and I don't see the error but I can't rule out the possibility. So help with regards to where I went wrong in my approach/execution of this problem would be appreciated.
  2. jcsd
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