Aurora Borealis

Imagine a magnetic charge as shown in the figure of strength \(m\), its magnetic field vector can be expressed as \(\dfrac {kr}{|r|^{2}}\) where \(r\) is the radial position vector of any point from the magnetic charge.

An electric charge has been released in its vicinity at a distance of \(r_{o}\) with a radial velocity component \(v_{o}\) and some arbitary angular velocity \(w_{o}\) about the cone's axis and it now moves in the magnetic field of the magnetic charge.

It moves on the surface of a cone of half angle \(\alpha \) with its vertex at the magnetic charge (Refer to figure above).

If its radial speed \(v\) at any time can be related to other quantities as

\[\displaystyle {(f) v_{o}}^{(e)} - \frac { L^{g}}{(a) m^{(c)} \sin^{(b)}(\alpha )} \left [ \frac {1}{{r_{o}}^{(d)}} - \frac {1}{r^{(d)}} \right ] = {(f) v^{(e)}} \]

For positive integers \(a,b,c,d,e,f,g\), evaluate \( \frac {1}{2} abcdefg\)

Details and Assumptions

\(L\) is the Initial angular momentum about cones axis

\(m\) is the mass of electrical charge

\(r\) is the radial distance

\(v\) is the radial velocity magnitude

\(v_{o}\) is the Initial radial velocity

you may want to try this problem first

\( Details\)

  • A magnetic charge is simply a magnetic analogue of electrical charge, it produces a radial magnetic field and though its existence is uncertain. Assume it exists for this problem.

  • The force upon an electric charge in a magnetic field is given by \(q(v \times B)\)

  • The explicit independence of \(k\) from the relation is not an error.

  • (Fun Fact) The beautiful northern lights or Aurora Borealis is the result of this phenomenon where charged particles funnel in like a cone into the north pole and keep moving in the surface of the cone, Though yes ours is a highly idealised and simplfied situation.


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