Equation for magnetic field line of dipole

In summary, the conversation discusses the equation for a dipole magnetic field in spherical coordinates and how to show that the equation for a magnetic field line is r = R sin^2 θ, where R is the radius of the magnetic field at the equator (θ = π/2). The conversation also mentions using the gradient of B and the vector potential for a dipole magnetic field in spherical coordinates. Finally, it is mentioned that the equation for the field lines is derived by considering the direction of the field at each point along the line.
  • #1
erogard
62
0
Hi, given the equation for a dipole magnetic field in spherical coordinates:

[itex]
\vec{B} = \frac{\mu_0 M}{4 \pi} \frac{1}{r^3} \left[ \hat{r} 2 \cos \theta + \bf{\hat{\theta}} \sin \theta \right]
[/itex]

I need to show that the equation for a magnetic field line is [itex] r = R \sin^2 \theta [/itex]
where R is the radius of the magnetic field at the equator (theta = pi/2)

Not sure where to start. I know that the gradient of B would give me a vector that is perpendicular to a given field line...

I also know that a vector potential for a dipole magnetic field in spherical coordinate is given by

[itex]
A_\theta = \frac{\mu_0 M}{4 \pi} \frac{sin\theta}{r^2}
[/itex]
 
Last edited:
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  • #2
The equation for a field line is [tex]\frac{dr}{d\theta}=B_r/B_\theta[/tex].
I don't think this gives [tex]sin^2\theta[/tex].
 
  • #3
Clem has omitted an r.

Clem meant: [itex]\frac{B_r}{B_\theta} = \frac{dr}{r d\theta}[/itex].

dr is the radial increment in the line corresponding to a tangential increment r d[itex]\theta[/itex].

The resulting DE is solved by separating variables, and yield logs on each side. You use the condition that r = R when [itex]\theta[/itex] = [itex]\pi[/itex]/2 to re-express the arbitrary constant. You do get just what you said.
 
  • #4
But why is the equation for the field lines:

[itex]\frac{B_r}{B_\theta} = \frac {dr}{rd\theta}[/itex] ??

I can see how solve this to give the equation:

[itex] r = R sin^2 \theta [/itex]

where R is r when θ is ∏/2. Any help would be greatly appreciated.
 
  • #5
You need to recall what is meant by a field line: a line whose direction at every point along it is the direction of the field at that point. So the ratio of radial to tangential field components must be the same as the ratio of tangential to radial components of the line increment.
 
  • #6
Thanks that is great I get it now.
 

1. What is the equation for the magnetic field line of a dipole?

The equation for the magnetic field line of a dipole is B = μ0/4π x (3(m⃗ .r̂)r̂ - m⃗ ), where B is the magnetic field, μ0 is the permeability of free space, π is the mathematical constant pi, m⃗ is the magnetic moment of the dipole, and r̂ is the unit vector in the direction of the point where the magnetic field is being calculated.

2. How is the magnetic field line of a dipole different from that of a single pole?

Unlike a single pole, which has only one magnetic field line extending out from it, a dipole has two equal and opposite magnetic poles, resulting in a more complex magnetic field line that curves and loops between the poles.

3. What is the significance of the magnetic moment in the equation for the magnetic field line of a dipole?

The magnetic moment, denoted as m⃗, represents the strength and orientation of the dipole. It is a vector quantity that points from the negative pole to the positive pole and is directly proportional to the strength of the magnetic field produced by the dipole.

4. How does the distance from a dipole impact the strength of the magnetic field?

According to the equation for the magnetic field line of a dipole, the strength of the magnetic field is inversely proportional to the cube of the distance from the dipole. This means that the further away from the dipole, the weaker the magnetic field will be.

5. Can the equation for the magnetic field line of a dipole be used to calculate the magnetic field at any point?

Yes, the equation can be used to calculate the magnetic field at any point in space. However, it is most accurate for points that are far away from the dipole compared to its size.

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