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# Homework9 - Homework 9 Answers 95.658 Spring 2011...

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Homework 9 Answers, 95.658 Spring 2011, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell Problem 1 Jackson 10.3 A solid uniform sphere of radius R and conductivity σ acts as a scatterer of a plane-wave beam of unpolarized radiation of frequency ω , with ωR / c << 1. The conductivity is large enough that the skin depth δ is small compared to R . (a) Justify and use a magnetostatic scalar potential to determine the magnetic field around the sphere, assuming the conductivity is infinite. (Remember that ω ≠ 0.) (b) Use the technique of Section 8.1 to determine the absorption cross section of the sphere. Show that it varies as ( ω ) 1/2 provided σ is independent of frequency. SOLUTION: (a) The statement ωR / c << 1 is equivalent to R << λ. This is the long-wavelength region. In this region, we can safely assume that the fields are constant across the object, so that the problem reduces to a electrostatics/magnetostatics problem. In this case, we want to find the absorption, which is caused by induced currents meeting resistance. So we first need to find the currents. So we choose to approach this problem as a magnetostatics problem, because static magnetic fields are linked to currents. A perfectly conducting sphere is placed in an originally uniform magnetic field. The region of interest, outside the sphere, is free space and contains no charges or currents, so that we immediately know: ∇⋅ B = 0 , ∇× B = 0 (Because B 0 H and J = 0) The curl being zero (the second equation) lets us define the magnetic field in terms of the gradient of a scalar potential: B =−∇ Ψ M Plugging this into the first equation leads to: 2 Ψ M = 0 This is just the familiar Laplace equation. If we place the sphere at the origin, and set the original field pointing along the z-axis, the problem is azimuthally symmetric. The solution to the Laplace equation

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Homework9 - Homework 9 Answers 95.658 Spring 2011...

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