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Superconducting Frame in Magnetic Field (Moscow Phys-Tech)

  1. Nov 8, 2009 #1
    Hello everybody,

    I've been trying to understand fully the problem I found in a famous Physics book (A guide to physics Problems Part 1). The maths behind the problem are understandable. However, even if I found the partial solution to the problem at the end of the book, several points are not clear for me.

    1. The problem statement, all variables and given/known data

    A superconducting square rigid frame of mass m inductance L, and side a
    is cooled down (in a magnetic field) to a temperature below the critical
    temperature. The frame is kept horizontal (parallel to the x-y plane)
    and constrained to move in the z direction in a nonuniform but constant magnetic field described by a vector potential A=(-Bo*alpha,alpha*x*z,0) and a
    uniform gravitational field given by the acceleration g .
    The thickness of the frame is much smaller than a . Initially, the frame
    is at rest, with its center coinciding with the origin. Find the equations of
    motion of the frame and solve for the position of the frame as a function of
    time.


    2. Relevant equations



    3. The attempt at a solution
    I found the magnetic field from the vector potential: B=(-alpha*x,0,alpha*z+Bo)

    I found the flux throught the plate due to this magnetic field:

    Phi[e]=B.S=(Bo+alpha*z)*a^2

    What I don't understand is why the flux from the current inside the frame is (according to the solution) Phi=L*I, I being the current inside de plate.

    Conseptually, I don't get the idea of an inductance L for a non coil-shaped inductor.

    Then The total flux throught the plate is: Phi=Phi[e]+Phi

    Before the establishement of the current: Phi[0]=Bo*a^2

    By stating that the flux is constant in time, we have Phi=Phi[0]. So we can deduce the current inside the plate.

    I=-(alpha*z*a^2)/L

    To calculate the force, we remember the lapalce force:

    F=int[I*dlXB]

    the solution states: "due to physical constraint, we only need the component in the z direction", why?

    For me when we take the cross product of dl and B we have: (ds,dy,0)X(Bx,0,Bz)=(-Bx*dy,-By*dx,-Bx*dy)

    Is the solution the integrale is calculated as follow:

    Fz=2*I*int[(-dy)*(-alpha*(a/2) ),-a/2,a/2]

    for me Bx=-alpha*x

    why are the taking x=a/2 ?

    once we have the force the solution to the differential equation is for me straightfoward.

    to sum up my questions:

    1) How to conceptualize the inductance L of a plate ?
    2) why is the internal flux Phi=L*I ?
    3)why the total flux equals to Phi[0]=Bo*a^2 at t=o and not (Bo+alpha*z)*a^2
    4)Causality speaking, we calculate the current I from the flux which is calcutalted from the current I. Is it normal?
    5)Why is their only a laplace force in z?
    6) why taking x=a/2 for calculating Fz

    I hope my querries belong to the scope of this forum. I spent time on this problem. I could have just accepted the solution. But I prefer to be picky and understand the problem fully.

    Thx for reading.
     
  2. jcsd
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