A 2 x 10^(-6) C charge is located at A(4,3,5) in free space, find E_rho, E_phi, E_z, at O(8,12,2). (The problem wants the electric field components in cylindrical components).

Since the points are given in Cartesian, I solved the problem in Cartesian coordinates.

E = [2x10^(-6)] / [4 * pi * 8.854 x 10^12 * sqrt[106]^3] [ 4a_x + 9a_y + 3a_z]

I get:

E = 65.88a_x + 148.24a_y - 49.41a_z

Since in cylindrical components the z component is the same as the z component in Cartesian, I was able to verify that at least the z component is correct. However, I try to determine E_rho and E_phi and cant seem to get the right answer, which is (159.7 and 27.4 respectively).

I do E_rho = 65.88 cos( arctan( 148.24 / 65.88) ) + 148.24 sin( arctan( 148.24 / 65.88) )

and E_phi = -65.88 sin ( arctan( 148.24 / 65.88) ) + 148.24 cos( arctan( 148.24 / 65.88) )

What am I doing wrong?

Since the points are given in Cartesian, I solved the problem in Cartesian coordinates.

E = [2x10^(-6)] / [4 * pi * 8.854 x 10^12 * sqrt[106]^3] [ 4a_x + 9a_y + 3a_z]

I get:

E = 65.88a_x + 148.24a_y - 49.41a_z

Since in cylindrical components the z component is the same as the z component in Cartesian, I was able to verify that at least the z component is correct. However, I try to determine E_rho and E_phi and cant seem to get the right answer, which is (159.7 and 27.4 respectively).

I do E_rho = 65.88 cos( arctan( 148.24 / 65.88) ) + 148.24 sin( arctan( 148.24 / 65.88) )

and E_phi = -65.88 sin ( arctan( 148.24 / 65.88) ) + 148.24 cos( arctan( 148.24 / 65.88) )

What am I doing wrong?

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E_rho = magnitude of E in the r direction as (x^2 + y^2) where x and y are the magnitudes of E in the x and y directions of Cartesian cords.

E_ phi = magnitude of E in the phi direction as arctan y/x where x and y are the magnitudes of E in the x and y directions of Cartesian cords.

I think you may be confusing a bit of spherical coords in here.