# 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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