# Superconducting Frame in Magnetic Field (Moscow Phys-Tech)

1. Nov 8, 2009

### tresgog

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.