Class_27new - PHYS809 Class 27 Notes Boundary value...

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1 PHYS809Class27Notes Boundary value problems in magnetostatics A variety of techniques are available for solving boundary value problems in magnetostatics. For linear media with piecewise uniform magnetic permeabilities, the vector potential in the Coulomb gauge satisfies a Poisson equation in each piece 2 . μ = A J (27.1) The piecewise solutions must be matched together by using the boundary conditions at the interfaces. In regions where J = 0 , the magnetic field H is given in terms of a scalar potential . m = -∇Φ H (27.2) For linear media, 0 ∇⋅ = B gives ( ) 0. m μ ∇⋅ ∇Φ = (27.3) If μ is piecewise uniform, the potential satisfies the Laplace equation in each piece 2 0. m ∇ Φ = (27.4) Again the solutions in each piece are connected through the boundary conditions. For situations in which the magnetic field is due to regions of fixed magnetization (as in so called hard ferromagnets), we can again use a scalar potential. Since ( ) 0 , μ = + B H M (27.5) we have 2 0 0 0 0 0. m μ μ μ μ ∇⋅ = ∇⋅ + ∇⋅ = - ∇ Φ + ∇⋅ = B H M M (27.6) The magnetic scalar potential satisfies a Poisson equation 2 , m m ρ ∇ Φ = - (27.7) where the magnetic ‘charge’ density is . m ρ = -∇⋅ M (27.8) If there are no bounding surfaces, the solution for the potential is ( ) ( ) 3 1 . 4 m d x π ∇ ⋅ Φ = - - M x x x x (27.9) Integrating by parts and using
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2 1 1 , = -∇ - - x x x x (27.10) we get ( ) ( ) 3 1 . 4 m d x π Φ = -∇⋅ - M x x x x (27.11) Note that this expression for the potential avoids consideration of how to treat any discontinuities in M , which would need to be considered if equation (27.9) were used. However equation (27.9) is useful if M is uniform in a volume V and drops discontinuously to zero outside V . Applying the divergence theorem to a Gaussian pillbox straddling the surface, equation (27.8) shows that there is an effective surface
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