Approximating magnetic field as the field of magnetic dipole

[phys-student]

Homework Statement

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.

Homework Equations

Field of a magnetic dipole:
B=(μ₀/4πr³)*[3(m⋅r')r'-m]

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.

 

[nrqed]

Homework Statement

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.

Homework Equations

Field of a magnetic dipole:
B=(μ₀/4πr³)*[3(m⋅r')r'-m]

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.

For a dipole located at the origin with its axis along z, what would be the vector [itex] \vec{m} [/itex]? For a point on the z axis, what would be the unit vector [itex] \vec{r}' [/itex] ?