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Motional emf with gravity as the pulling force

  1. Feb 21, 2010 #1
    1. The problem statement, all variables and given/known data
    A square loop of mass m resistance R and side l] is halfway inside a magnetic field pointing into the page. It is then released from rest and gravity pulls it out of the magnetic field.
    a) Calculate the induced emf and the current as functions of the velocity.
    b) Calculate the magnitude and direction of the magnetic force.
    c) At some time the magnetic force will be equal to the gravitational force, calculate the velocity of the loop at this time.
    d) Calculate the velocity of the loop as a function of time.

    2. Relevant equations
    induced emf = -d(phi)/dt
    phi = integral(B * da)
    V= emf = IR
    magnetic force = qv X B (I think there are two components so this one is in the direction opposite gravity?)

    3. The attempt at a solution
    I'm pretty confident I figured out the emf and current that are induced.
    da = l*v (with v = -dx/dt)
    d(phi)/dt = -Blv and emf = Blv
    I = Blv / R

    for the magnetic force the component opposing the graviational pull is in the positive x direction and is equal to the force of gravity in magnitude so it is equal to:
    and then there is a horizontal component that is allowing the charges to move in the positive y direction (or collectively counter-clockwise) with magnitude:
    =vb per unit charge q
    so finally:
    magnetic force = vb - mg

    I'm not so sure how to compute the direction unless I just described it like I have just now
    I tried using the fact that the components form a triangle and saying:
    tan(theta) = -vb/mg but I'm sure this isn't the way to go about it.

    I set the gravitational pull equal to the magnetic force so:
    vb - mg = mg; vb = 2mg so:
    v = 2mg / B

    I tried using f= ma and saying:
    gravitational force - magnetic force= mg - VB + mg = ma
    a = 2g - vB/m
    but if I tried to find the velocity and integrated I would get and answer that doesn't depend on time and that doesn't even involve gravity which puzzles me.

    Any help at all would be greatly appreciated
  2. jcsd
  3. Feb 22, 2010 #2


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    Science Advisor
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    Gold Member

    The v in the equation that you have is the velocity of a charge carrier in a magnetic field. You don't want that. You need to use FM = I l x B where I is the induced current (your expression for I is correct.)
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