Assignment 5

Due: Wednesday, 11 February 2004

23. The magnetic dipole moment m for an atom immersed in a magnetic field B changes in the sense that the change in m opposes the imposed B. That is, m changes to try and cancel the imposed B. A model for this effect can be developed as follows:

• Imagine a nucleus with charge +e, about which an electron with mass m_e and charge -e moves in a circular orbit of radius R. Find the orbital frequency in terms of R, m_e, and e. If the orbital period is very short compared to any relevant time scale, one can define a current due to this motion. In this case, find the dipole moment m for the atom.
  
• A magnetic field of strength B is applied parallel to the orbital angular momentum vector of the electron. If R does not change, find the change in the orbital frequency in terms of B, m_e, e, and R, under the assumption that the change in frequency is small. Under what condition is the change in frequency small?

24. A solenoid has length L, radius R, and N turns per unit length where each turn carries current I. Find the field in the vicinity of the axis of the solenoid. You may use the result for the finite solenoid from an earlier homework assignment.

25. A uniformly charged sphere (radius R and total charge Q) spins with frequency f. Find the dipole moment for the sphere and find the vector potential for r >> R.

26. An electron and a proton are separated by a distance D. Assume that the spins of the electron and the proton lie in the same plane. Find the force and torque between the electron and the proton

• if the proton's spin and electron's spin are perpendicular to the line connecting the electron and proton and both spins point upward. Is this configuration stable?
  
• if the proton's spin is perpendicular to the line connecting the electron and proton and again points upward, and the electron's spin points toward the proton along the line connecting the electron and proton. Is this configuration stable?