1 Problem set 10 - Magnetic materials due: Monday June 9th, by 5pm. 1. A solenoid with N windings, of length L, and volume V is filled with a material with magnetic permeability μ = μ 0 (1 + χ m ). Recall that ∇× B = μ j free . (a) What is the inductance of the solenoid? (b) When a current I is flowing through the solenoid, what is the magnetization in the magnetic material filling the solenoid? (c) Consider the solenoid carrying a current I , with its two sides chort-circuited by an ideal conductor. Also neglect any resistance in the circuit. The solenoid is then cooled, which primarily results in the magnetic permeability decreasing with time: μ(t)= μ + νt. (1) What is the current in the circuit as a function of time? 2. A uniformly magnetized sphere has radius R, and magnetization M pointing along the z axis. Using the analogy with our discussion of a uniformly polarized sphere, find: (a) The dipole magnetic field, B out outside the sphere. (b) The constant magnetic field inside the sphere B in . (c) The surface current density (magnitude and direction) at the perimeter of the sphere. 3. Following the discussion in class of the origin of diamagnetic susceptibility (mirroring section 11.5 of Purcell), and using the formulas derived, what is the magnetic susceptibility in a response to a field in the z-direction of a material with each atom having electronic current loops of radius 0.5 ˚ A lying in the x − y plane, and an atom density of ρ = 10 30 m -3 ?

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Problem set 10 - Magnetic materials

due: Monday June 9th, by 5pm.

1. A solenoid with N windings, of length L, and volume V is filled with a material with magnetic permeability

= 0(1 + m). Recall that ~B = ~jfree.

(a) What is the inductance of the solenoid?

(b) When a current I is flowing through the solenoid, what is the magnetization in the magnetic material fillingthe solenoid?

(c) Consider the solenoid carrying a current I, with its two sides chort-circuited by an ideal conductor. Alsoneglect any resistance in the circuit. The solenoid is then cooled, which primarily results in the magneticpermeability decreasing with time:

(t) = + t. (1)

What is the current in the circuit as a function of time?

2. A uniformly magnetized sphere has radius R, and magnetization ~M pointing along the z axis. Using the analogywith our discussion of a uniformly polarized sphere, find:

(a) The dipole magnetic field, ~Bout outside the sphere.

(b) The constant magnetic field inside the sphere ~Bin.

(c) The surface current density (magnitude and direction) at the perimeter of the sphere.

3. Following the discussion in class of the origin of diamagnetic susceptibility (mirroring section 11.5 of Purcell),and using the formulas derived, what is the magnetic susceptibility in a response to a field in the z-direction ofa material with each atom having electronic current loops of radius 0.5A lying in the x y plane, and an atomdensity of = 1030m3?