THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 5:
 
Lorentz Force, Cyclotrons and Ampère's Law
 
24 Feb. 1999 - finish by 3 Mar. 1999

1.

FRICTION vs. THE LORENTZ FORCE: A 2.0-kg copper rod rests on two horizontal rails 2.0 m apart and carries a current of 100 A from one rail to the other. The coefficient of static friction between the rod and the rails is $\mu_s = 0.50$. What is the smallest magnetic field (not necessarily vertical) that would cause the bar to slide?

2.

CYCLOTRONS: (Neglect any relativistic effects.) Suppose that we want to build a small cyclotron for protons using a magnet with a uniform field over a region 1.0 m in radius such that the protons reach a maximum kinetic energy of 20 MeV at the outer radius of the magnet.

(a) What magnetic field must the magnet produce?

(b) At what frequency must the ``dee'' voltage oscillate?

Now suppose we want to build a cyclotron to accelerate electrons without a magnet, using the Earth's magnetic field (assume $B = 5 \times 10^{-5}$ T) to keep the electrons moving in circles.

(c) What is the radius of the electron orbit at 100 eV?

(d) What is the frequency (in Hz) of the RF electric field we must supply to the cyclotron ``dees?''

3.

HOLLOW CYLINDRICAL CONDUCTOR: A thick-walled hollow conducting cylinder carries a uniformly disrtibuted current I. The (centred) hole in the middle has a radius of R and the outer radius of the conductor is 2R. Derive an expression for the strength of the magnetic field B as a function of radial distance r from the cylinder axis, in the range from r = R to r = 2R; then plot (i.e. sketch, showing axis labels, scales and values at key points) B(r) in the range from r = 0 to r = 4R.