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

by eyenkay
Tags: curl, dipole, magnetic field, vector potential
 P: 7 1. The problem statement, all variables and given/known data 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] 2. Relevant equations Adip(r)= (μ_o/(4πr^2))(m*sin(theta)) Bdip= curl A = (μ_o*m/(4πr^3))(2cos(theta)(r-direction)+sin(theta)(theta-direction) 3. 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 dont 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?
 Emeritus Sci Advisor PF Gold P: 5,540 "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.
 P: 2 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.

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