- #1

quasar_4

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## Homework Statement

A conducting ring with radius a and mass m is placed in a magnetic field at a height H above the origin of the reference frame. The plane of the ring is parallel to the ground, that is, the normal is directed along the z-axis. The electrical resistance per unit length of the ring is R/(2Pia), so that the resistance of the loop is R. The magnetic field has spatial dependence:

[tex]\vec{B} = \frac{B_0}{L}\left(-\frac{\rho}{2} \hat{\rho} + z \hat{z} \right)[/tex]

where [tex] \hat{\rho}, \hat{z} [/tex] are unit vectors in the usual cylindrical coordinate system.

At a certain time, the ring is dropped and falls due to gravity. The plane of the ring remains horizontal as the ring falls. Find the terminal velocity of the ring.

## Homework Equations

We've got Maxwell's equations, F=ma, Lorentz force F=qv X B, etc.

## The Attempt at a Solution

Here's my attempt so far:

[tex] \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \Rightarrow \oint{\vec{E} \cdot d\vec{l}} = -\frac{d}{dt} \int{\vec{B} \cdot \hat{n} \hspace{1mm} da} \Rightarrow E(2 \pi a) = -\frac{d}{dt} \int{B_z \hspace{1mm}da} \Rightarrow E(2 \pi a) = -\frac{d}{dt} \int_0^{2\pi}{d\phi}\int_0^a{B_z \rho\hspace{1mm} d\rho} = - \frac{B_0 a^2 \pi}{L}\frac{dz}{dt} [/tex]

But dz/dt is just the velocity we're looking for!

Here is where I am now stuck. I have a nice expression for the electric field, E, in terms of velocity, dz/dt =v:

[tex] \vec{E} = - \frac{B_0 \pi a v_z}{2L} \hat{\phi} [/tex]

But the electric field points in the circumferential direction, not the z direction. In terms of finding terminal velocity, I want something to the effect of

[tex] \sum F_z = -mg + F_? = ma [/tex], where F? is the net force in the z direction due to my induced electric field. From this point it is easy to set ma =0 and solve for v-terminal. But I can't just say something like F=qE, because E points in the wrong direction.

Furthermore, I haven't at all used the fact that the ring is at a height H or that it has resistance R - are those unnecessary bits of information, or am I approaching this entirely wrong? I almost could solve this using conservation of energy, but it's the

*terminal*velocity I want and not clear to me how to find that in an energy formulation of the problem (might be possible, though). So - can anyone help?