Question: The very large magnetic field B needed for the cyclotron at the NSCL in Michigan is generated by passing a high current through a solenoid, or coil, containing thousands of loops of superconducting wire. The magnetic field contributed by each loop adds to create a large total field. But the electrons that flow through the solenoid wire to produce B feel a Lorentz force from the field they have generated. At high fields and currents, this force can be strong enough to destroy the coil.
a. You know how to find the force on a single charge q moving with velocity v in a magnetic field B. A current I is made up of many such moving charges. Find an expression for I in terms of the number of charges per unit volume, and then derive an expression for the force on a long straight wire of length L carrying a current I in a direction perpendicular to a uniform magnetic field B.
b. Consider a single loop of the solenoid, which for simplicity take to be square-shaped with side length 1m, carrying a current of 100 A in the sense shown in Fig. What is the direction of the field within the loop? What is the direction of the force on each of the four sides of the loop?
c. If B = 5 T, as in the NSCL K500 magnet, find the magnitude of the force on each side of the loop. Assume that the field is uniform. Compare this force to your weight. In magnets made with coils of superconducting wire great care must be used to assure that there are no small movements of coil wires because these can generate enough heat to turn off the superconductivity with the result that the energy stored in the magnetic field explosively dissipates in the coils. Designing a stable solenoid is not an easy task!
Fig. - Geometry of the current loop