- #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]
[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]
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