Magnetic dipole moment and dipole field

Problem:

A circular loop of wire with radius a = 1 cm and center at the origin is bent, so half lies in the y-z plane and half in the x-y plane.  A current I = 2 A flows in the wire.
(a)  What is the magnetic moment of this loop?
(b)  What is the magnetic field at (x, y, z) = (3, 4, 0) meters from the origin?

Solution:


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We can view the loop as a superposition of loop 1 and loop 2 as shown.  We find m by adding the moments of loop 1 and loop 2, m = m1 + m2.

Problem:

A thin rectangular sheet of composite material has dimensions L x w with L >> w.  The sheet carries a steady surface current that circulates as shown.
A surface current density K = dI/dy = constant flows in the x direction over a width ∆.  At the ends of the rectangle (x = ± L/2) a high conductance “short circuit” completes the current path.
(a)  Find the magnetic moment m.
(b)  Sketch the magnitude of m as a function of the width ∆ of the current.

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Solution:

Problem:

Assume a "Bohr-type model" of the He atom.   Each electron is in circular orbit of radius 3.95*10-11 m around a nucleus with effective charge Zeff = 1.34.
The electrons orbit in a plane perpendicular to the z-axis, with opposite sense of rotation.
(a)  What is the magnitude of the magnetic moment of each electron due to its motion?
(b)  What is the total magnetic moment of the atom?  (Ignore spin.)
(c)  Assume a magnetic field B = B0k is turned on.  If the radius of the electron orbits does not change, but the electron speed changes by ±dv, what is the total magnetic moment of the atom now (ignoring terms higher than first order in dv)?

Solution:

Problem:

A point magnetic dipole m in vacuum (region 1 in the diagram below) is pointing toward (and is normal to) the plane surface of a material with permeability μ (region 2).  The distance between the dipole and the surface is d.
(a)  Use the method of images to find the magnetic field B in both regions, as follows:  Place an image dipole m' = αm a distance d into medium 2 and take the field B1 in region 1 to be due to dipoles m and m' in a medium with μ0.  Take the field B2 in region 2 to be due to a single dipole m" = βm at the location of the real dipole m in a medium with μ.  Solve the boundary value problem at the interface to evaluate B1 and B2.
(b)  Describe physically how each of the image dipoles m' and m" arise and the role they play in determining the fields and the forces on the real dipole and the material of medium 2.
(c)  Evaluate the force acting on the dipole m.

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Solution: