Rutgers Physics 385 Electromagnetism I (Fall15/Gershtein)
Homework 12 - Due TUESDAY December 8, 2015 in class
1) Verify that for uniform magnetic field the potential is
That is, check that divergence of the potential is zero and curl is B.
2) A uniform surface current flows along the xy plane. Find the magnetic potential.
3) Show that the magnetic field of a dipole can be written in coordinate-free form:
4) Find the exact magnetic field a distance z above the center of a square loop of side w carrying current I.
Verify that it reduces to a dipole field for z>>w, with the appropriate dipole moment (what is it?).
5) An infinitely long cyllinder carries a uniform magnetization M parallel to its axis. Find the magnetic field due to M
inside and outside the cyllinder.
6) An iron rod of length L and square cross section (side a) is given a uniform longitudinal magnetization M, and then bent into a a circle
with a narrow gap between (width w). Find the magnetic filed at the center of the gap. Assume thar w << a.
Hint: note the similarity between this problem and problem 4) from HW02 .
7) A coaxial cable consists of two very long cylindrical tubes (radii a and b),
separated by linear material of magnetic susceptability χm. A current I
flows down the inner conductor and returns along the outer one; in each case the current uniformly distributes itself over the surface of the tubes.
a) Find the magnetic field in the region between the tubes
b) As a check, calculate the bound currents, and confirm that they, together with the free currents, generate correct field.