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

In summary: So finally, the coordinate-free form of the magnetic field for a dipole is \vec{B}_{dip}(\vec{r}) = \frac{\mu_0}{4 \pi r^3}[3(\vec{m} \cdot \hat{r})\hat{r} - \vec{m}]
  • #1
eyenkay
7
0

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


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)

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?
 
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  • #2
"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.
 
  • #3
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.
 

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

1. What is a dipole?

A dipole is a pair of equal and opposite electric charges that are separated by a small distance. It can also refer to any object or system that has two poles, such as a magnet or a molecule.

2. What is a magnetic field?

A magnetic field is a region in space where a magnetic force can be detected. It is created by moving electric charges, such as electrons, and is characterized by both strength and direction.

3. How is the magnetic field of a dipole written in coordinate-free form?

The magnetic field of a dipole can be written in coordinate-free form as B = μ0/4π * (3 * (m * r̂) * r̂ - m), where B is the magnetic field vector, μ0 is the permeability of free space, m is the dipole moment vector, and r̂ is the unit vector pointing in the direction of r, the position vector from the dipole to the point where the field is being measured.

4. What does the dipole moment vector represent?

The dipole moment vector represents the strength and direction of the dipole. It is a vector that points from the negative to the positive charge and has a magnitude equal to the product of the charge and the distance between them.

5. Can the magnetic field of a dipole be simplified in certain situations?

Yes, in certain situations such as when the distance from the dipole is much greater than the distance between the charges, the magnetic field can be approximated as B = μ0/4π * (m * r̂) / r^3, where r is the distance from the dipole to the point where the field is being measured.

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