Magnetic Dipoles and Gilbert Model · USAPhO 2015 #B2

Physics · electromagnetism

Problem

The nature of magnetic dipoles. a. A "Gilbert" dipole consists of a pair of magnetic monopoles each with a magnitude

q_m

but opposite magnetic charges separated by a distance d, where d is small. In this case, assume that

-q_m

is located at z = 0 and

+q_m

is located at z = d. [Figure: Diagram showing two magnetic monopoles on z-axis:

-q_m

at z = 0 and

+q_m

at z = d] Assume that magnetic monopoles behave like electric monopoles according to a coulomb-like force

F = \frac{\mu_0}{4\pi} \frac{q_{m1}q_{m2}}{r^2}

and the magnetic field obeys

B = \frac{F}{q_m}.

i. What are the dimensions of the quantity

q_m

? ii. Write an exact expression for the magnetic field strength B(z) along the z axis as a function of z for z > d. Write your answer in terms of

q_m

, d, z, and any necessary fundamental constants. iii. Evaluate this expression in the limit as d → 0, assuming that the product

q_m d = p_m

is kept constant, keeping only the