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Homework12

# Homework12 - Homework 12 Answers 95.657 Fall 2011...

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Homework 12 Answers, 95.657 Fall 2011, Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Problem 1 A permanent magnet is in the shape of a right circular cylinder, contains a uniform permanent magnetization along its axis pointing in the positive z direction, and is centered at the origin. A solid sphere of uniform paramagnetic material, previously unmagnetized, is kept on the z axis above the cylinder and is brought close enough that it feels the fields of the cylinder. (a) Sketch the system and draw all the effective magnetic charge. (b) Sketch all the currents. (c) On three separate diagrams, sketch the B , H , and M field lines in all regions. SOLUTION: There are three regions to this problem: inside the cylinder, inside the sphere, and outside both. There is no applied current in this problem, so J = 0 in all regions and on all boundaries. In each region, the material is uniform, so that all effective magnetic charges and magnetization currents exist on the boundary between regions, i.e. on the surface of the objects. The external region is free space, so we immediately know B = μ 0 H , M = 0 in this region. (a) The effective magnetic surface-charge density is given by ρ M =−∇⋅ M . This means that negative effective magnetic charge creates M field lines and positive magnetic charges destroy field lines. The permanent magnet also induces a magnetization in the sphere which roughly follows the pattern the of field lines in this region before the sphere was introduced. Because the sphere is paramagnetic, it sucks in field lines and is attracted to the permanent magnet. This means that is has induced effective magnetic charge. (b) The magnetization current density is given by J M =∇× M . This means that effective current loops give rise to a curling magnetization in the same way free currents give rise to curling B fields.

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