Midterm Test 1
Spring Semester 2009
A small magnet is falling along the axis of a vertical pipe made of non-magnetic conducting material. Find the terminal velocity of the magnet.
The mass of the magnet is m, the magnetic moment is μ; assume that the magnetic moment is oriented along the axis of the pipe. The conductivity of the pipe's material is σ, the relative magnetic permeability is 1. Assume that the radius of the pipe, a, is much larger than the thickness of the pipe's walls, d, and the size of the magnet.
Suggested solution steps:
- Google for "magnet falling through pipe". The search will bring you several articles published since 1990. You'll find the required answer (which you still need to derive yourself).
- Using reasonable physical approximations, calculate (a) magnetic flux through the pipe's circular crossection, (b) emf and electric field indiced along a pipe's circular crossection, (c) current density, and (d) total power dissipated in the pipe. You may 'recycle' the result of Problem 7.20b (HW2) for step (a). Eq. 8.6 of the textbook will help you in step (d); you may use a computer algebra systems to evaluate the integral required in step (d).
- Determine the drag force acting on the magnet by equating the dissipated power and the power of the drag force.
- Write the equation of motion of the magnet and determine the terminal speed.
- Estimate the value of the terminal speed. Of course, you have no idea what is the the magnetic moment of the magnet. However, you can easily measure the force between two such magnets when they are separated by distance a (the radius of the pipe). Let us assume that this force is 100 times the weight of the magnet. You do not know what is the material of the pipe. Let us assume that you managed to measure the resistance of the cube of the material of the size d (the thickness of the pipe's walls): R = 10-6 Ω.