# Loop falling in a magnetic field

1. Aug 31, 2014

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

2. Sep 3, 2014

Nothing?

3. Sep 3, 2014

### vanhees71

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

4. Sep 3, 2014

### ShayanJ

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

5. Sep 10, 2014

### 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?