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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 scbv @@ 3 DD .zhat 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, 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 @ Φ D Sin @ Θ D , r Sin @ Θ D Sin @ Φ D , r Cos @ Θ D<D 8 x fi r Cos @ Φ D Sin @ Θ D , y fi r Sin @ Θ D Sin @ Φ D , z fi r Cos @ Θ D< divsp = div . Thread @8 x, y, z <-> 8 r Cos @ Φ D Sin @ Θ D , r Sin @ Θ D Sin @ Φ D , r Cos @ Θ D<D Simplify r Sin @ Θ D I 3 r 2 Cos @ Θ D 2 Cos @ Φ D + 3 r Cos @ Φ D 2 Sin @ Θ D + a Sin @ Φ DM Integrate @ divsp r ^ 2 Sin @ 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-Ω 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-Ω 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 km z- t Ω L ex, 0, 0 = SetCoordinates @ Cartesian @ x, y, z DD Cartesian @ x, y, z D Curl @ Ef D 9 0, ª H km z- t Ω L ex km, 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 km z- t Ω L , by ª H km z- t Ω L , bz ª H km z- t Ω L = 2 212HomeWork6sol.nb Thread @ Curl @ Ef D- D @ Bf, t DD 9 bx ª H km z- t Ω L Ω , ª H km z- t Ω L ex km by ª H...
View Full Document

This note was uploaded on 12/07/2010 for the course PHYS PHYSICS 21 taught by Professor Alicea,j. during the Fall '10 term at UC Irvine.

Page1 / 15

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online