I am trying to read through classical fields by Landau and Lifshitz but I am struggling to understand their phrasing of the magnetic field intensity.
They state the equation of motion as:
$$ \frac{dp}{dt} = – \frac{e}{c} \frac{\partial A}{\partial t} – e \nabla\phi + \frac{e}{c} v \times (\nabla \times A) $$
They then go on to say the first two terms per unit charge are called the electric field intensity. So set $e =1$, I'm good with that. My confusion is when they follow with "The factor of $\frac{v}{c}$ in the force of the second type, per unit charge, is called the magnetic field intensity. We denote it by H. So by definition $H = \nabla \times A$."
I am confused because it is worded as though $H = \frac{v}{c}$ to me, I have taken electrodynamics so I know the formula for H they have given is indeed correct. What am I missing?
For reference: Section 17 in Landau and Lifshitz, Volume 2: The Classical Theory of Fields.
Best Answer
Must be a language thing. It must be intended such that "the factor of $\frac{v}{c}$" is the thing being multiplied with $\frac{v}{c}$ and not $\frac{v}{c}$ itself. I share your confusion.
In the German edition it reads: "Der im zweiten Kraftanteil bei $\frac{v}{c}$ stehende Faktor, wieder auf die Ladungs- einheit bezogen, wird magnetische Feldstärke genannt und durch $H$ bezeichnet: [...]" (My translation of that: "The factor standing with/next to $\frac{v}{c}$ in the second force term ...") Here it is clearer that indeed the term next to $\frac{v}{c}$ is the magnetic field intensity, still the formulation feels a bit awkward, maybe old-fashioned.