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MIT OpenCourseWare http://ocw.mit.edu 6.642 Continuum Electromechanics �� Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.642 Continuum Electromechanics Problem Set #4 Fall Term 2008 Issued: 9/24/08 Due: 10/03/08 Problem 1 A sphere of radius R and infinite magnetic permeability is placed within a uniform magnetic field at infinity, H (r f, T ) H 0 iz H 0 (ir cosT iT sin T ) . The medium outside the sphere is free space. a) Find the magnetic flux density B(r, T ) for r ! R . b) Find the equation of the magnetic field lines dr rdT Br BT c) Find the vector potential A(r, T ) . d) For r ! R , the governing equation for the vector potential is 2 A 0 . Show that the solution of part (c) satisfies 2 A 0 . e) For the separation magnetic field line that passes through the point (r R, T S / 2) find its distance D from x 0 at (r f, T S ) . f) Draw the magnetic field lines similar to those shown above. Problem 2 The Kelvin force density for charged dielectric media is ( P<) E U f E F where P D H 0 E is the polarization field, E is the electric field , D is the displacement field, and U f < D is the free charge density. Do not assume that the dielectric has a linear permittivity. Find the stress tensor Ti j for this force density in the form Fi wTi j wx j 1 of 2 Problem 3 The Kelvin force density for current carrying magnetic media with magnetization M is F where M P0 (M <)H J u P0 H B H is the magnetization field, B is the magnetic flux density, H is the magnetic P0 field intensity, and J u H is the current density. Do not assume that the magnetic media has a linear magnetic permeability. Find the stress tensor Ti j for this force density in the form Fi wTi j wx j 2 of 2

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