Alfvén Speed

The Alfvén speed is how fast a disturbance runs along a magnetic field line through a conducting gas. The picture to keep in mind is a stretched string. The field line carries tension, the gas supplies the mass, and a pluck travels along it at a speed set by the ratio of the two, just as a string's wave speed is the square root of tension over mass per length. In $v_A = B/\sqrt{\mu_0\,\rho}$ the field strength $B$ plays the part of the tension, the mass density $\rho$ of the gas plays the part of the mass, and the permeability $\mu_0$ is the constant that ties them together.

Reading the answer comes down to two proportions. A stronger field is a tighter string and the wave runs faster; a denser gas is a heavier string and it runs slower, though only as the square root of the density. The density here is mass of gas per volume, not a count of particles, so a particle count has to be turned into mass first: 5 protons per cm³ is about 8.4 × 10⁻²¹ kg/m³. In the solar wind at the Earth, a field of about 5 nT in gas like that gives an Alfvén speed of about 50 km/s. That is a tenth of the wind's own speed, so the wind is super-Alfvénic: it moves faster than a disturbance can run along its field, and it piles up in a bow shock ahead of the Earth.

The page's example runs the relation backwards for the density. With $B$ = 5 nT and $v_A$ = 48.77 km/s, the rearranged form $\rho = B^{2}/(\mu_0\,v_A^{2})$ gives 8.363 × 10⁻²¹ kg/m³, which is the 5 protons per cm³ of the solar wind. Going forward from that density and field returns 48.75 km/s, and this agrees with the NRL Plasma Formulary (Huba, 2019), which writes the speed in Gaussian units as $2.18\times10^{11}\,\mu^{-1/2}\,n_i^{-1/2}\,B$ cm/s, with $B$ in gauss and $n_i$ in cm⁻³. For 5 nT, 5 protons per cm³ and $\mu = 1$ that is 48 746.3 m/s. The formulary's 2.18 is rounded to three figures, so the library's 48 773 m/s sits 0.055 % higher; the gap comes from that rounding.

The relation is Alfvén's, from his 1942 paper in Nature on electromagnetic-hydrodynamic waves, and the formulary quotes it in practical form. It can be solved for any of the three quantities: the field from the speed and density, or the density from the field and speed, as above.