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Physics 322 Midterm 2 Nov 30, 2015 name: Box your final answer. 1 (15 pt) 2 (50 pt) 3 (20 pt) 4 (15 pt) total (100 pt) 1 ~ = −M ẑ. Compute 1. (15 pt) An infinitely long cylinder of radius R whose axis is parallel to the ẑ axis has a uniform magnetization M ~ H everywhere. 2 2. Suppose for t < 0, the capacitor is charged to V0 in the circuit below. The switch is closed at t = 0. a) (10 pt) The current behaves as a function of time as I(t) = (envelope function of time) × cos ( f t + δ ) where f > 0 for (R/L)2 < 4/(LC). What is f ? 3 b) (15 pt) Suppose the inductance L is due to the self-inductance of a toroidal coil with rectangular cross section (innder radius a, outer radius b, and height h as shown in the figure below) carrying a total of N turns. Express L in terms of N, h, a, and b. c) (10 pt) Suppose the toroidal coil described in part b) has a magnetic field ~B(s, φ , z) = f (s)φ̂ (where s is a cylindrical coordinate variable: e.g. the inner radius is described as s = a). In this magnetic field, suppose a pointlike magnetic dipole with ~m = mẑ is placed at rest at s = u ∈ (a, b) inside the torus. What are the forces and torques on the dipole due to the magnetic field? 4 d) (15 pt) Suppose the capacitor is made of parallel circular plates each with radius R with vacuum in between. Suppose the charge density on one of the plates for t > 0 is σ (t) and the uniform electric field region between the capaciator plates is then characterized by ~E ≈ σ (t) ẑ. ε0 Neglecting the edge effects as usual and assuming rotational symmetry about the axis passing through the center of the plates, find the resulting Bφ and the Poynting vector at s = R, somewhere between the parallel plates in the uniform field region. 5 3. Suppose you are given that the Maxwell equations were replaced by ~∇ · ~Ẽ = 0 ~ ~ ~∇ × ~Ẽ = − ∂ B̃ − L̃ ∂ Ẽ ∂t ∂t ~ ~∇ ×~B̃ = Ñ ∂ Ẽ − L̃~∇ × ~Ẽ ∂t ~∇ ·~B̃ = 0 where L̃ and Ñ are complex constants and the tilde indicates that these fields are complex. a) (10 pt) What is the wave equation governing ~Ẽ? b) (10 pt) Find a plane wave solution ~Ẽ to this Maxwell equation system proportional to e−iωt , traveling in the y direction, having a single wavelength, and the boundary condition ~Ẽ(t = 0,~x = 0) = ~C̃. Be sure to specify the condition on ~C̃ coming from the Maxwell equation system. 6 4. (15 pt) A plane wave is incident normal to a dielectric interface as shown: where the index of refraction of medium i ∈ {1, 2} is ni and µ1 = µ2 . The incident wave is ~Ẽ (t, z) = Ẽ exp [i (k z − ωt)] x̂ I 0I 1 and the electric field in the rest of the regions can be written as ~Ẽ (t, z) = Ẽ exp [i (−k z − ωt)] x̂ R 0R 1 ~Ẽ (t, z) = Ẽ exp [i (k z − ωt)] x̂. T 0T 2 Recall that one can derive from one of the Maxwell equations the magnetic field for the reflected wave being ω µ1 ε1 B̃~R (t, z) = − Ẽ0R ei(−k1 z−ωt) ŷ k1 ~ k continuity where the minus sign in the amplitude comes from the minus sign in~k = −k1 ẑ for the reflected wave. Use ~Ek continuity and H to find ~Ẽ just in terms of {~Ẽ , n , n }. 0T 0I 1 2 7 (4 continued) (extra space ) 8