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EMLecture12

# EMLecture12 - Lecture 12 Notes 95.657 Electromagnetic...

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Lecture 12 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. Review of Magnetostatics in Magnetic Materials - Currents give rise to curling magnetic fields: ∇× B = 0 J total or ∇× H = J or ∇× M = J M where J total = J J M - There are no magnetic monopoles: ∇⋅ B = 0 which leads to ∇⋅ H =−∇⋅ M - Defining a vector potential B =∇× A leads to: 2 A =− 0 J total and A = 0 4 J total x ' x x ' d x ' - In a region where the magnetic material is linear and uniform so that B = H we can apply all of the B -field equations to the free current J instead of the total current J total if we replace the permittivity of free space μ 0 with the permittivity of the material μ . For instance: 2 A =− J and A = 4 J x ' x x ' d x ' - The boundary conditions for any type of materials are: B 2 B 1 ⋅ n = 0 and n × H 2 H 1 = K 2. Special Cases in Magnetostatics - If the materials are linear and there is no free current density in the region of space where we want to know the fields ( J = 0), then the equation reduces to: 2 A = 0 - These can be solved in the usual way with appropriate boundary conditions. - An alternate approach is to define a scalar potential B =−∇  M so that the zero-divergence equation becomes: 2 Ψ M = 0 - If there is no current density, J = 0, and if the material is not linear, but instead the magnetization M is known and fixed (such as in permanent magnets), the equations reduces to: 2 A =−μ 0 J M and A = μ 0 4 π J M ( x ' ) x x ' d x ' where J M =∇× M

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- The alternate scalar approach H =−∇  M can also be used in this case. The statement of no magnetic monopoles really means that the divergence of the H field and the M field are equal: ∇⋅ B = 0 ∇⋅ 0 H  0 M = 0 −∇⋅ H =∇⋅ M 2 Φ M =∇⋅ M - where we can now treat the divergence of the magnetization as an effective magnetic charge
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EMLecture12 - Lecture 12 Notes 95.657 Electromagnetic...

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