March 28, 2017

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The equation describing the (r, \theta ) coordinates of points along a single field line of a magnetic dipole is r=R_0 \sin ^2(\theta ) where \theta =0 is in the direction of the dipole moment and R_0 is a constant which is different for each field line.

The question requires only an understanding of this field line equation. To increase your understanding of the properties of magnetic fields, we give the full derivation of this result. From the equations of the magnetic field of a dipole,
\displaystyle B_ r= \displaystyle \frac{\mu _0}{4\pi }\, \frac{2\, m\, \cos \theta }{r^3}
\displaystyle B_{\theta }= \displaystyle \frac{\mu _0}{4\pi }\, \frac{m\, \sin \theta }{r^3}

we have
\displaystyle \frac{B_{\theta }}{B_{r}}=\frac{\tan \theta }{2}.

In order to find the direction of the field lines, we equate with the ratio of the infinitesimal length element in the r and \theta direction. You can convince yourself that the infinitesimal length in the radial direction is dr and in the \theta direction is rd\theta.
(a) Fig. 1

(b) Fig. 2

At any arbitrary point, an infinitesimal vector along the field line itself has r and \theta components of:
\displaystyle r d\theta = \displaystyle \frac{B_{\theta }}{B_{r}}dr
\displaystyle r d\theta = \displaystyle \frac{\tan \theta }{2}dr

This leads to the following differential equation:
\displaystyle 2\frac{d\theta }{\tan \theta }=\frac{dr}{r}

which has r=R_0\, \sin ^2\theta as a solution.

The solar wind ram pressure causes the magnetic field of the earth to terminate at about 10\, R_ E on the sunward side of the earth (see figure).

The auroral zone is defined by the last field line from the Earth that returns to the Earth. If the field of the earth extends no further than 10\, R_ E in the sunward direction, at what angle (in degrees) in the picture above does the last field line that returns to the earth on the sunward side leave the polar regions of the earth at 1\, R_ E, assuming that the fields are always described by r = R_0\, \sin ^2(\theta )? In the drawing, this is equivalent to asking what is the angle with respect to the vertical of a line from the center of the earth to the point that the red curve intersects the light blue circle.

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