Find the exact magnetic field a distance z above the center of a square loop of side w, carrying a current I. Verify that it reduces to the field of a dipole, with the appropriate dipole moment, when z >> w.
(1) dB = μ0I/4πr2 dl × rhat
(2) r = √((½w)2+z2)
The Attempt at a Solution
I treated the loop like four current-carrying wires of finite length and used Biot-Savart. I think due to symmetry, the magnetic field in x-y should cancel at the center so all the magnetic field is in z. It seems intuitive to me that dB should just be 4*(1).
But what's confusing me is the dl × rhat term; I'm not sure my approach to it is sound. If rhat is the unit vector pointing along the distance r from a point on the loop to the point on the z-axis, it must be at some angle to x-y plane.
I can define an angle φ such that sinφ = z/r and write r in terms of z and w. Can I say the contribution from dl × r to the magnetic field dB is dl sinφ and write:
dB = μ0Iz/4πr3 dl
for each segment of the loop, where is r is as in (2) and dl = dw? Then my limits of integration would be 0 to w and I would multiply the result by 4. Does that make sense? Thank you.