A square wire loop of size 2a by 2a lies in the x- y plane with its center at the origin and sides parallel to the x and y axes. A counterclockwise current I runs around the loop.
(a) Find the magnetic field on the z axis. [Answer: Bz = 2μ0Ia2/[(a2 + z2)(2a2 + z2)0.5]]
(b) Show that for z/a >> 1 the field becomes that of a magnetic dipole, and find the magnetic moment.
Field of a magnetic dipole:
where r is the distance to the field point, m is the magnetic moment, and r' is a unit vector pointing towards the field point
The Attempt at a Solution
I have already done part a and got the correct expression for the magnetic field. For part b, I said that since z/a >> 1 the term (a2+z2) could be approximated as z2 and the term (2a2+z2) could be approximated by z2. This gives me z3 in the denominator which is good since it matches the equation for the field of a magnetic dipole but I can't get the rest of the terms to match. Can someone help me? I think I've either made the wrong assumption or I'm not properly evaluating the term in the square brackets in the equation for field of a magnetic dipole.