Cross product in spherical coordinates.

[mancini0]

Homework Statement

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)

Homework 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)

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.

 

[mark726]

Hello! Your approach seems to be on the right track. However, there are a few things that may have caused your incorrect answer.

First, when you converted B1 to Cartesian coordinates, you used the r component of B1, but you should have used the theta component since you are taking the cross product with m2 which has a y component. This would give you a different expression for B1 in Cartesian coordinates.

Second, when taking the cross product, make sure you are using the correct order of the vectors. In this case, you should be taking the cross product of m2 with B1, not the other way around.

Finally, make sure you are using the correct units for all your quantities. The magnetic dipole moment m2 should have units of A*m^2, and the magnetic field B1 should have units of T. Double check your units to make sure they all match up correctly.

I hope this helps! Keep up the good work.