Question

Points P1 and P2 have Cartesian coordinate:s (2.2,0) and (1,0,1), respectively (a) Find the spherical coordinates of P1. (b) Express erl , ^e01,and" eφ1, the spherical unit vectors at P1,intermsofthe Cartesian unit vectors

Answer 1

P1 has the co-ordiantes (2,2,0)

In spherical terms we are to represent (x1,y1,z1) as (r1, theta 1, phi 1 )
Let theta = t and let phi = p
r= sqrt( x² + y²+ z²)
r= sqrt ( 2² + 2²) = sqrt ( 4+4) = sqrt(8) = 2 sqrt(2)

cos p = z/ r = 0/2sqrt(2) = 0
cos p = 0 so
p = cos ^-1 0 = pi/2
Hence p = phi = pi/2

sin t = y / r sin p = 2/ (2sqrt(2) sin pi/2) = 2/2sqrt(2) = 1/sqrt(2)
t= sin^-1 ( 1/sqrt(2) ) = pi/4;
Thus, in spherical form, the co-ordinates are (2sqrt(2) , pi/2 , pi/4)

b)
The spherical units vectors and cylindrical unit vectors are related by:
[ r = [ sint cosp sint sinp cos t [ x
t cost cosp cost sinp -sint y
p ] -sin p cos p 0 ] z ]

[ r = [ 0 1/sqrt(2) 1/sqrt(2) [ x
t 0 1/sqrt(2) -1/sqrt(2) y
p ] -1 0 0 ] z ]
r= (y+z) / sqrt(2)
t= (y-z) / sqrt(2)
p = -x

Note: In the 'b' part, x,y,z, r,t,p, represent unit vectors