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

  1. 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?
     
  2. jcsd
  3. Tom Mattson

    Tom Mattson 5,538
    Staff Emeritus
    Science Advisor
    Gold Member

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

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

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

    Vector potential [tex]\vec{A}[/tex] is

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

    Magnetic field [tex]\vec{B}[/tex] is

    [tex]\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})[/tex]

    To get coordinate-free form, you just need to express [tex]\vec{m}[/tex] 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 [tex]+\hat{z}[/tex] direction, then in spherical it would be [tex]\vec{m} = m \cos \theta \hat{r} - m \sin \theta \hat{\theta}[/tex]. Now use dot products of [tex]\vec{m}[/tex] with the necessary spherical unit vectors in order to eliminate those pesky sine and cosine functions.
     
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