# 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 χ_{m}. 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.