Homework12

# Homework12 - Homework 12 Answers 95.657 Electromagnetic...

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Homework 12 Answers, 95.657 Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Jackson 5.19 A magnetically “hard” material is in the shape of a right circular cylinder of length L and radius a . The cylinder has a permanent magnetization M 0 , uniform through-out its volume and parallel to its axis. (a) Determine the magnetic field H and magnetic induction B at all points on the axis of the cylinder, both inside and outside. SOLUTION Let us place the center of the cylinder at the origin and its axis aligned with the z axis. Then it is natural to use cylindrical coordinates. Let us state everything we know about this problem outright. There are no free currents present, so there cannot be any curl to the H field. ∇× H = 0 This is valid inside and out. The curl of the gradient of any function is always zero, so we can define a magnetic scalar potential H =−∇  M Outside the object, there is no material, so that the magnetization outside must be zero and the total field B depends only on the H field. B = 0 H and M = 0 outside We should also note that there are no applied external fields, only the fields created by the shape. This means that all fields should die down to zero infinitely away from the shape. Inside, the magnetization is a known: M = M 0 z inside Generally speaking, the equation that syas there are no magnetic monopoles can be cast as a relationship between M and H: ∇⋅ B = 0 ∇⋅ 0 H  0 M = 0 ∇⋅ H =−∇⋅ M This is a totally general results, applicable everywhere, no matter what material is present. It means that whatever divergence that may exist in the magnetization must be perfectly canceled by the divergence in the H field in order to keep the actual, real total field B divergenceless. We now substitute in the definition of H in terms of the magnetic scalar potential and recognize an entity that acts like a

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## This note was uploaded on 03/21/2012 for the course PHY 2030 taught by Professor Avery during the Spring '11 term at University of Florida.

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Homework12 - Homework 12 Answers 95.657 Electromagnetic...

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