Loop falling in a magnetic field

There is a conducting circular loop with resistance R falling in the magnetic field [itex] \vec{B}=B_\rho(\rho,z)\hat\rho+B_z(\rho,z)\hat z [/itex] and gravitational field [itex]\vec g=-g\hat z[/itex]. How does z and the current in the loop change in time?(assume the loop remains horizontal!)

The flux through the loop is [itex] \int_0^a\int_0^{2\pi} B_z \rho d\varphi d\rho[/itex], Its time derivative is [itex] \int_0^a \int_0^{2\pi} \frac{\partial B_z}{\partial z} \frac{dz}{dt} \rho d\varphi d\rho [/itex] and so the induced current is [itex] I=\frac{1}{R} \int_0^a \int_0^{2\pi} \frac{\partial B_z}{\partial z} \frac{dz}{dt} \rho d\varphi d\rho [/itex]. Now we can write the z component of the magnetic force as [itex]-\frac{2\pi a B_\rho}{R} \int_0^a \int_0^{2\pi} \frac{\partial B_z}{\partial z} \frac{dz}{dt} \rho d\varphi d\rho [/itex]. So we have:

[itex]



\ddot z=-g-\frac{dz}{dt}\frac{4\pi^2 a B_\rho}{m R} \int_0^a \frac{\partial B_z}{\partial z} \rho d\rho



[/itex]

Which gives us z as a function of time and then I can be calculated easily.



1- Is everything OK?

2-Any hints or suggestions or further explanations?

3-How does this change if the loop is superconducting?



Thanks