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

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Unformatted text preview: 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 , 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 , where is the unit vector in the direction. SetCoordinates @ Cylindrical @ r, , z DD Cylindrical @ r, , z D In cylindrical coordinates the vector 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, , DD Spherical @ r, , D scbv = Transpose @ JacobianMatrix @D . Inverse @ DiagonalMatrix @ ScaleFactors @DDDD 88 Cos @ D Sin @ D , Sin @ D Sin @ D , Cos @ D< , 8 Cos @ D Cos @ D , Cos @ D Sin @ D ,- Sin @ D< , 8- Sin @ D , Cos @ D , 0 << zhat = 8 0, 0, 1 < 8 0, 0, 1 < Compute dot products : scbv @@ 1 DD .zhat Cos @ D scbv @@ 2 DD .zhat- Sin @ D scbv @@ 3 DD .zhat so zhat = Cos[ ] rhat - hat Sin[ ] 2. Compute S vdS , 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, 3 x y z ^ 2, a y z < 9 x 3 , 3 x y z 2 , a y z = SetCoordinates @ Cartesian @ x, y, z DD Cartesian @ x, y, z D div = Div @ vcart D 3 x 2 + a y + 3 x z 2 Thread @8 x, y, z <-> 8 r Cos @ D Sin @ D , r Sin @ D Sin @ D , r Cos @ D<D 8 x fi r Cos @ D Sin @ D , y fi r Sin @ D Sin @ D , z fi r Cos @ D< divsp = div . Thread @8 x, y, z <-> 8 r Cos @ D Sin @ D , r Sin @ D Sin @ D , r Cos @ D<D Simplify r Sin @ D I 3 r 2 Cos @ D 2 Cos @ D + 3 r Cos @ D 2 Sin @ D + a Sin @ DM Integrate @ divsp r ^ 2 Sin @ D , 8 , 0, 2 < , 8 , 0, < , 8 r, 0, 1 <D 4 5 3. A plane wave polarized in the x direction and propagating in the z direction has an electric field E1={ex, 0, 0} 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} 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 <- t LL 9 H km z- t L ex, 0, 0 = SetCoordinates @ Cartesian @ x, y, z DD Cartesian @ x, y, z D Curl @ Ef D 9 0, H km z- t L ex km, 0 = Bf = 8 bx, by, bz < E ^ H I H8 0, 0, km < . 8 x, y, z <- t L L 9 bx H km z- t L , by H km z- t L , bz H km z- t L = 2 212HomeWork6sol.nb Thread @ Curl @ Ef D- D @ Bf, t DD 9 bx H km z- t L , H km z- t L ex km by H...
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212HomeWork6sol - Physics 212A Homework #6 Name: 1 a) In...

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