Write the Magnetic Field of a dipole in coordinate-free form?

[eyenkay]

Homework Statement

Show that the magnetic field of a dipole can be written in coordinate-free form: B_dip (r)=(μ_o/(4πr^3 ))[3(m*r ̂ ) r ̂-m]

Homework Equations

A_dip(r)= (μ_o/(4πr^2))(m*sin(theta))

B_dip= curl A = (μ_o*m/(4πr^3))(2cos(theta)(r-direction)+sin(theta)(theta-direction)

The Attempt at a Solution

I figure this must have something to do with the above equations for the vector potential dipole and magnetic field dipole, I just don't have any idea what it means to write in 'coordinate-free form', or how to go about that..
Can anybody point me in the right direction?

 

[quantumdude]

"Coordinate free" simply means "in terms of dot products". See how the expression they want you to derive has dot products in it? That's what they want.

 

[drnairb]

You can \TeX{}-ify your posts. It helps a *lot.* Here's the coordinate-free form:

\vec{B}_{dip}(\vec{r}) = \frac{\mu_0}{4 \pi r^3}[3(\vec{m} \cdot \hat{r})\hat{r} - \vec{m}]

where \vec{m} is the dipole moment, right? (I've always used \vec{p}.)

Vector potential \vec{A} is

\vec{A}_{dip}(\vec{r}) = \frac{\mu_0}{4 \pi r^2} (m \sin \theta)

Magnetic field \vec{B} is

\vec{B}_{dip}(\vec{r}) = \vec{\nabla} \times \vec{A} = \frac{\mu_0 m}{4 \pi r^3} (2 \cos \theta \hat{r} + \sin \theta \hat{\theta})

To get coordinate-free form, you just need to express \vec{m} in spherical coordinates and manipulate the properties of dot products in that coordinate system. If you assume your dipole is at the origin and points in the +\hat{z} direction, then in spherical it would be \vec{m} = m \cos \theta \hat{r} - m \sin \theta \hat{\theta}. Now use dot products of \vec{m} with the necessary spherical unit vectors in order to eliminate those pesky sine and cosine functions.