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Consider the 2D system shown in Figure 1, where multiple parallel wires are surrounded bya perfectly conducting duct. The wires are located at the mid-plane y = b. The wires arethin enough to be modeled as a surface current density \mathrm{K}=K_{o} \sin (x) \hat{\mathrm{e}}_{z} The wires are driven such that the surface current is sinusoidally distributed in 0 < x < π.Due to the planar symmetry, we know that the tangential flux density right above and rightbelow the mid-plane have the same magnitude but opposite direction, that is, \left.B_{x}^{(1)}\right|_{y=b}=-\left.B_{x}^{(2)}\right|_{y=b} . Answer the following questions only for the region (2) - below the mid-plane. Assign zeromagnetic vector potential (A₂ = 0) to the perfectly conducting duct.

Figure 1: Multiple wires surrounded by a perfectly conducting duct. (a) Find the tangential flux density Byt right below the mid-plane. Note that this isa function of x. (Hint: apply the general boundary condition K = nx [B/μ|]). (b) Find the magnetic vector potential A₂ (x, y). Note that A. (x, y) should satisfy the zero-potential conditions on x = 0, x = π, and y = 0, as well as the tangential-flux-densitycondition at y = b obtained in Part (b).1 (c) Find the surface current densities induced on the three sides of the perfect conductor,that is, at x = 0, x = 7, and y = 0. Show that the net currents induced on the threesides sum to -I/2, where I_{o}=\int_{0}^{\pi} \mathbf{K}(x) \mathrm{d} x .

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