Loop falling in a magnetic field

[ShayanJ]

There is a conducting circular loop with resistance R falling in the magnetic field $\vec{B}=B_\rho(\rho,z)\hat\rho+B_z(\rho,z)\hat z$ and gravitational field $\vec g=-g\hat z$. How does z and the current in the loop change in time?(assume the loop remains horizontal!)
The flux through the loop is $\int_0^a\int_0^{2\pi} B_z \rho d\varphi d\rho$, Its time derivative is $\int_0^a \int_0^{2\pi} \frac{\partial B_z}{\partial z} \frac{dz}{dt} \rho d\varphi d\rho$ and so the induced current is $I=\frac{1}{R} \int_0^a \int_0^{2\pi} \frac{\partial B_z}{\partial z} \frac{dz}{dt} \rho d\varphi d\rho$. Now we can write the z component of the magnetic force as $-\frac{2\pi a B_\rho}{R} \int_0^a \int_0^{2\pi} \frac{\partial B_z}{\partial z} \frac{dz}{dt} \rho d\varphi d\rho$. So we have:
$<br /> \ddot z=-g-\frac{dz}{dt}\frac{4\pi^2 a B_\rho}{m R} \int_0^a \frac{\partial B_z}{\partial z} \rho d\rho<br />$
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

 

[ShayanJ]

Nothing?

 

[vanhees71]

Looks good to me. I've not checked the details.

 

[ShayanJ]

My main question is the third. What should I do if the loop is superconducting?

 

[ShayanJ]

Is it right to say that the superconducting loop excludes the magnetic field and so there will be no induced current and so no magnetic force is applied to it?