226 Week 7: Sources of the Magnetic Field
Advanced Problem 9.
A semi-infinite thin solenoid aligned with (say) the negativez-axis so that the “+” end is at the
origin creates a magnetic field thatlookslike that of a point magnetic chargeqmat the origin:
B~=kmqmˆr
r^2at points “near” the end and outside of the solenoid itself. Note thatkm=μ 0 / 4 π= 10−^7 N-m/A^2 is
the magnetic field constant, analogous tokefor the electric field, and thatμ 0 is called themagnetic
permeability, none of which matters more than algebraically for this problem but which is important
next week!
Suppose you take a small bar magnet and place it at~r=rˆrso its magnetic moment~mis aligned
withˆr. Find the force acting on it (if any).
What would you expect its motion to be if you placed it at the same pointso that its moment
wasnotinitially aligned with the magnetic field?
Advanced Problem 10.
For a presumed e.g. proton in a magnetic field, evaluate:dL~
dt=μpL~×B~ (484)in Cartesian components, assumingvB=B 0 zˆand arbitraryL~=Lxˆx+Lyyˆ+Lzˆz. Identify and
solve the resulting system of equations to prove that the angular momentum does indeed precess
around the applied magnetic field with the constant angular velocityωp=μpB 0 independent of~L’s
magnitude or direction.