Physics · Electricity and magnetism · Grade 8

Aurora and the Lorentz force

Why does the aurora glow only around the poles?

The situation

On a winter night in Norway or northern Canada, green curtains “dance” across the sky — this is the aurora. It is produced by charged particles from the Sun. But why do they not fall over the whole Earth, only in a ring around the poles? The answer lies in the Lorentz force: a magnetic field does not stop a particle, it only turns it, making it travel in a spiral along the field lines. When does a particle stay in the magnetic “trap”, and when does it plunge into the atmosphere?

Topic

Motion of a charge in a magnetic field. Aurora and a particle accelerator

Key ideas

Lorentz force
Larmor radius
Magnetic mirror

Formulas

\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})

Lorentz force. Its magnetic part is always perpendicular to the velocity — it does no work, and the energy of the particle does not change.

r_{L} = \frac{m v_{⟂}}{| q | B}

Larmor (gyration) radius. For fast particles, use $γ m$ instead of $m$ .

f_{c} = \frac{| q | B}{2 π m}

Cyclotron frequency: independent of speed (in the non-relativistic case).

{\vec{v}}_{d} = \frac{\vec{E} \times \vec{B}}{B^{2}}

E×B drift: independent of the sign of the charge and of the mass.

\sin^{2} α > \frac{B_{m i n}}{B_{m a x}} = \frac{1}{R}

Condition for trapping in a magnetic mirror; $α$ is the pitch angle between the velocity and $\vec{B}$ .

Theory

Why a circle and a spiral?

The Lorentz force is perpendicular to the velocity, so it only turns the direction: from $| q | v B = m v^{2} / r$ we get $r_{L} = m v / (| q | B)$ . The force does not act on the velocity component along the field, $v_{∥}$ — so the circle becomes a spiral. The simulation solves the equation of motion $d \vec{p} / d t = q (\vec{E} + \vec{v} \times \vec{B})$ , $\vec{p} = γ m \vec{v}$ by the Boris method: at each step the velocity vector is rotated by an exact angle and its length does not change — so energy is conserved without error. If the radius and speed are measured, the specific charge can be found: $| q | / m = γ v_{⟂} / (B r)$ — in the results table for “Uniform B” the electron gives ≈ 1.76·10¹¹ C/kg (laboratory methods are in the “Electric current in vacuum and gases” experiment).

The field of a moving charge itself

A particle does not just “feel” the field — it also creates one. For a slow charge ( $v ≪ c$ ) the magnetic field is $\vec{B} = \frac{μ_{0}}{4 π} \frac{q \vec{v} \times \vec{r}}{r^{3}}$ ; for a uniformly moving charge at any speed, $\vec{B} = \vec{v} \times \vec{E} / c^{2}$ . The magnetic field of a current in a wire is the sum of the fields of millions of such drifting electrons (the “Magnetic field” experiment). The field of a single particle is far smaller than the external $B$ , so it is ignored in the simulation. The electric and magnetic parts of the Lorentz force depend on the frame of reference: an observer moving with the particle sees an electric field ${\vec{E}}^{'} \approx \vec{v} \times \vec{B}$ ( $v ≪ c$ ) (the “Special relativity” experiment).

E×B drift

The electric field speeds the particle up over one half of the orbit (the radius grows) and slows it down over the other half (the radius shrinks). As a result the orbit shifts sideways: $v_{d} = E / B$ , perpendicular to both $\vec{E}$ and $\vec{B}$ . Electron and proton both drift the same way. If $E = v B$ , the particle flies in a straight line — this is a velocity filter (Wien filter), used in mass spectrometers.

Magnetic bottle

If the field changes slowly, the magnetic moment $μ = m v_{⟂}^{2} / (2 B)$ is conserved. When the particle enters a stronger field, $v_{⟂}$ increases, and since energy is conserved, $v_{∥}$ decreases; at the point $B = B_{m i n} / \sin^{2} α$ the particle is reflected. If $\sin^{2} α < 1 / R$ , it escapes through the “loss cone”. This is the first adiabatic invariant. For the oscillation between the mirrors, the second one, $J = \oint p_{∥} d s$ , is conserved, and for the slow drift around the Earth, the third (the magnetic flux enclosed by the drift shell). Gyration, bouncing and drift — three periodic motions — hold particles in the trap for a long time: this is how the Earth's radiation belts form. The field of two coils is calculated exactly using elliptic integrals.

Aurora

The Earth's field is approximately a dipole: $B = B_{0} {(R_{\oplus︎} / r)}^{3} \sqrt{1 + 3 \sin^{2} λ}$ , with field lines $r = L R_{\oplus︎} \cos^{2} λ$ . Near the Earth the field is strong — the Earth itself is a giant magnetic bottle. Particles in the loss cone reach the foot of the field line, at latitude $\cos^{2} λ \approx 1 / L$ : for $L = 6 \dots 8$ this is 65°–70° — the auroral oval. Simplifications: a pure dipole (no compression by the solar wind and no tail), no collisions and no electric fields; in “Earth” mode electrons are calculated by the guiding-centre method (they gyrate millions of times per second, with a radius of ~1 km), and the particles of the flow are fed directly into the loss cone.

The Sun, the solar wind and the ionosphere

The visible surface of the Sun is the photosphere (≈ 5800 K); above it lie the thin chromosphere and the very tenuous, million-degree corona. A continuous stream of plasma flows from the corona — the solar wind — at 300–800 km/s. Solar activity (spots, flares, coronal mass ejections) varies with a cycle of about 11 years; a strong eruption reaches the Earth in a few days and causes a magnetic storm and an aurora that can be seen even farther south than usual. The wind compresses the Earth's field and forms the boundary of the magnetosphere (on the day side ≈ 10 $R_{\oplus︎}$ — the “Solar wind” button). The altitudes of 100–300 km where the aurora glows lie in the ionosphere: a layer of air (roughly 60–1000 km) ionised by the Sun's ultraviolet rays and by incoming particles, which reflects short radio waves. For more on cosmic plasma, see the “Plasma — the fourth state” experiment.

Did you know?

The most common green colour of the aurora is the emission of oxygen atoms at a wavelength of 557.7 nm; it forms at an altitude of roughly 100–200 km. Above that, oxygen emits red light (630.0 nm).
The radiation belts around the Earth were discovered in 1958 with instruments by James Van Allen on the satellites Explorer 1 and Explorer 3 — which is why they are called the Van Allen belts.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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