# Cross product in spherical coordinates.

1. Apr 30, 2012

### mancini0

1. The problem statement, all variables and given/known data

i am trying to solve for the magnetic torque a circular loop of radius R exerts on a square loop of side length b a distance r away. The circular loop has a normal vector towards the positive z axis, the square loop has a normal towards the +y axis. The current is I in both loops.(Griffiths question 6.1)

2. Relevant equations
I used a dipole approximation and found the magnetic vector potential of the circular loop. I then took the curl of this in spherical coordinates and found an expression for B1, the magnetic field due to the loop. I know the torque exerted on the square loop is m2 X B1, where m2 is the magnetic dipole moment of the square (m2 = Ib^2 y)

3. The attempt at a solution
B1 = curl(A) =( u_0(m1) /(4 *pi* r^3) ) {2cosθ r + sinθ vartheta }
m1 = I(pi)R^2 z.

In order to take the curl, I switched B1 to cartesian coordinates. This is where I believed i made my mistake... I just used the following relationships:
x = Rcos$\phi$sin$\theta$
(for r I used the r component of B1, that is, R = u_0(m1) /(4 *pi* r^3) ) *2cosθ )
$\phi$ is 0 here because there is no $\phi$ component of B1.

y = 0 since sin($\phi$ is 0.

z = R cos $\vartheta$

Now i just cross m2 = <0,Ib^2,0> with B1 using its components found above. However, my answer is incorrect. (The book uses the coordinate free form of B_dip in its solution, but I am trying to understand how to convert my expression for B1 from spherical to Cartesian coordinates.