212HomeWork6sol - Physics 212A Homework#6 Name 1 a In...

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Physics 212A Homework #6 Name: 1 a) In Spherical coordinates, calculate 2 H r n L , and 2 I r 2 Y 2 1 H Θ , Φ LM (see SphericalHarmonicY in Help). b) In Spherical coordinates, calculate · I r n r M , where r is the unit vector in the r direction. Div @8 r^n,0,0 <D H 2 + n L r - 1 + n c) In Cylindrical coordinates, calculate × Φ , where Φ is the unit vector in the Φ direction. SetCoordinates @ Cylindrical @ r, Φ ,z DD Cylindrical @ r, Φ ,z D In cylindrical coordinates the vector Φ is represented as {0,1,0}. Despite the fact that it is a constant in the canonical cylindrical basis, it has a non-zero Curl: Curl @8 0,1,0 <D : 0,0, 1 r > d) Express z in terms of spherical coordinate basis vectors. The spherical coordinate basis vectors are : SetCoordinates @ Spherical @ r, Θ , Φ DD Spherical @ r, Θ , Φ D scbv = Transpose @ JacobianMatrix @D .Inverse @ DiagonalMatrix @ ScaleFactors @DDDD 88 Cos @ Φ D Sin @ Θ D ,Sin @ Θ D Sin @ Φ D ,Cos @ Θ D< , 8 Cos @ Θ D Cos @ Φ D ,Cos @ Θ D Sin @ Φ D , - Sin @ Θ D< , 8 - Sin @ Φ D ,Cos @ Φ D ,0 << zhat = 8 0,0,1 < 8 0,0,1 < Compute dot products : scbv @@ 1 DD .zhat Cos @ Θ D scbv @@ 2 DD .zhat - Sin @ Θ D
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scbv @@ 3 DD .zhat 0 so zhat = Cos[ Θ ] rhat - Θ hat Sin[ Θ ] 2. Compute S v·dS , where S is the surface of the unit sphere and v= x 3 x + 3 x y z 2 y + a y z z . Convert this to a volume integral and do the integration in spherical coordinates. vcart = 8 x^3,3xyz^2, ayz < 9 x 3 ,3xyz 2 ,ayz = SetCoordinates @ Cartesian @ x,y,z DD Cartesian @ x,y,z D div = Div @ vcart D 3x 2 + ay + 3xz 2 Thread @8 x,y,z < -> 8 rCos @ Φ D Sin @ Θ D ,rSin @ Θ D Sin @ Φ D ,rCos @ Θ D<D 8 x fi rCos @ Φ D Sin @ Θ D ,y fi rSin @ Θ D Sin @ Φ D ,z fi rCos @ Θ D< divsp = div .Thread @8 x,y,z < -> 8 rCos @ Φ D Sin @ Θ D ,rSin @ Θ D Sin @ Φ D ,rCos @ Θ D<D Simplify rSin @ Θ D I 3r 2 Cos @ Θ D 2 Cos @ Φ D + 3rCos @ Φ D 2 Sin @ Θ D + aSin @ Φ DM Integrate @ divspr^2Sin @ Θ D , 8 Φ ,0,2 Π < , 8 Θ ,0, Π < , 8 r,0,1 <D 4 Π 5 3. A plane wave polarized in the x direction and propagating in the z direction has an electric field E1={ex, 0, 0} ª H k . x t L , where ex is the magnitude of the field and k is a constant vector k={0, 0, km} . According to Maxwell's equations, this wave has a magnetic field associated with it which obeys the equation ×E= - B t . a) Assume B={bx, by, bz} ª H k . x t L , and determine bx, by, and bz. Ef = 8 ex,0,0 < E^ H I H8 0,0,km < . 8 x,y,z < - Ω t LL 9 ª H kmz - t Ω L ex,0,0 = SetCoordinates @ Cartesian @ x,y,z DD Cartesian @ x,y,z D Curl @ Ef D 9 0, ª H kmz - t Ω L exkm,0 = Bf = 8 bx, by, bz < E^ H I H8 0,0,km < . 8 x,y,z < - Ω t L L 9 bx ª H kmz - t Ω L ,by ª H kmz - t Ω L ,bz ª H kmz - t Ω L = 2 212HomeWork6sol.nb
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Thread @ Curl @ Ef D - D @ Bf,t DD 9 0 bx ª H kmz - t Ω L Ω , ª H kmz - t Ω L exkm by ª H kmz - t Ω L Ω ,0 bz ª H kmz - t Ω L Ω = Solve @ Thread @ Curl @ Ef D - D @ Bf,t DD , 8 bx, by, bz <D Flatten : bx fi 0,by fi exkm Ω ,bz fi 0 > Bfsol = Bf . % : 0, ª H kmz - t Ω L exkm Ω ,0 > b) Compute the Poynting vector S= E * · B Μ 0 .Verify that it is proportional to the magnitude of E1 squared.
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