Magnetic Monopole on U-shaped Track with Wire Loop · USAPhO 2012 #B2
Physics · electromagnetism
Problem
For this problem, assume the existence of a hypothetical particle known as a magnetic monopole. Such a particle would have a "magnetic charge" qm, and in analogy to an electrically charged particle would produce a radially directed magnetic field of magnitude
B = μ₀qm/(4πr²)
and be subject to a force (in the absence of electric fields)
F = qmB
A magnetic monopole of mass m and magnetic charge qm is constrained to move on a vertical, nonmagnetic, insulating, frictionless U-shaped track. At the bottom of the track is a wire loop whose radius b is much smaller than the width of the "U" of the track. The section of track near the loop can thus be approximated as a long straight line. The wire that makes up the loop has radius a ≪ b and resistivity ρ. The monopole is released from rest a height H above the bottom of the track.
Ignore the self-inductance of the loop, and assume that the monopole passes through the loop many times before coming to a rest.
a. Suppose the monopole is a distance x from the center of the loop. What is the magnetic flux φB through the loop?
b. Suppose in addition that the monopole is traveling at a velocity v. What is the emf E in the loop?
c. Find the change in speed Δv of the monopole on one trip through the loop.
d. How many times does the monopole pass through the loop before coming to a rest?
e. Alternate Approach: You may, instead, opt to find the above answers to within a dimensionless multiplicative constant (like 2 or π²/3). If you only do this approach, you will be able to earn up to 60% of the possible score for each part of this question.
You might want to make use of the integral
∫_{-∞}^∞ 1/(1+u²)³ du = 3π/8
or the integral
∫₀^π sin⁴θ dθ = 3π/8