Why auroras happen at the poles

Classical Mechanics Level 4

The charged particles from solar eruptions hit Earth mainly around the North and South poles (and cause auroras). This is because our Earth is similar to a big magnet. The Earth generates a magnetic field and this magnetic field funnels the charged particles towards the poles. In order to see an example of this funneling, we can think of the following problem:

Consider each magnetic pole separately (so there's only one pole in the problem). The magnetic field near a single pole is \( \vec{B} = k\vec{r}/r^3=k\hat{r}/r^2 \) where \(\vec{r}\) is the radial vector point of the pole and the point of interest and \(\hat{r}=\vec{r}/r\) is the radial unit vector. The sign of \(k\) is opposite for the North and South poles. If there's an electric charge moving in that magnetic field, its trajectory is on a surface of a cone, i.e. a big funnel. Find the vertex angle (the angle between the axis and a line on the surface of the cone) in degrees with these given initial conditions: the distance between the charge and the pole is \( r = 1~\mbox{m} \) and the velocity vector of the charge is \( v = 2~\mbox{m/s}\) perpendicular to the line connecting the pole and the charge. We will consider a north pole and so let \( k = 3~\mbox{T}\cdot\mbox{m}^2 \). The charge of the particle is \( q = 4~\mbox{C} \) and the mass is \( m = 5~\mbox{kg}\).

Details and assumptions

  • Hint: For anyone who has not taken electromagnetism yet, the force on the charge particle is given by the Lorentz force law: \(\vec{F}=q\vec{v} \times \vec {B}\).

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