Reference
1 Most notably, by Hans Christian Ørsted who discovered the effect of electric currents on magnetic needles, and André-Marie Ampère who has extended this work by finding the magnetic interaction between two currents.
2 For details, see appendix CA: Selected Physical Constants. In the Gaussian units, the coefficient ( mu_{0} / 4 pi) is replaced with ( 1 / c^{2}).
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3 An important case when the electroneutrality may not hold is the motion of electrons in vacuum. (However, in this case the electron speed is often comparable with the speed of light, so that the magnetic forces may be comparable in strength with electrostatic forces, and hence important.) In some semiconductor devices, local violations of electroneutrality also play an important role – see, e.g., SM Chapter 6.
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4 The first detailed book on this subject, De Magnete by William Gilbert (a.k.a. Gilberd), was published as early as 1600.
5 In particular, until very recently (2018), Eq. (7) was used for the legal definition of the SI unit of current, one ampere (A), via the SI unit of force (the newton, N), with the coefficient ( mu_{0}) considered exactly fixed.
6 The SI unit of the magnetic field is called tesla (T) – after Nikola Tesla, a pioneer of electrical engineering. In the Gaussian units, the already discussed constant ( 1 / c^{2}) in Eq. (1) is equally divided between Eqs. (8) and (9), so that in them both, the constant before the integral is ( 1 / c). The resulting Gaussian unit of the field ( mathbf{B}) is called gauss (G); taking into account the difference of units of electric charge and length, and hence of the current density, 1 G equals exactly ( 10^{-4} mathrm{~T}). Note also that in some textbooks, especially old ones, ( mathbf{B}) is called either the magnetic induction or the magnetic flux density, while the term “magnetic field” is reserved for the field ( mathbf{H}) that will be introduced in Sec. 5 below.
7 Named after Jean-Baptiste Biot and Félix Savart who made several key contributions to the theory of magnetic interactions – in the same notorious 1820.
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8 Named after Hendrik Antoon Lorentz, famous mostly for his numerous contributions to the development of special relativity – see Chapter 9 below. To be fair, the magnetic part of the Lorentz force was implicitly described in a much earlier (1865) paper by J. C. Maxwell, and then spelled out by Oliver Heaviside (another genius of electrical engineering – and mathematics!) in 1889, i.e. also before the 1895 work by H. Lorentz.
9 From the magnetic part of Eq. (10), Eq. (8) may be derived by the elementary summation of all forces acting on ( n >> 1) particles in a unit volume, with ( mathbf{j}=q n mathbf{v}) – see the footnote on Eq. (4.13a). On the other hand, the reciprocal derivation of Eq. (10) from Eq. (8) with ( mathbf{j}=q mathbf{v} deltaleft(mathbf{r}-mathbf{r}_{0}right)), where ( mathbf{r}_{0}) is the current particle’s position (so that ( d mathbf{r}_{0} / d t=mathbf{v})), requires care and will be performed in Chapter 9.
10 See, e.g., MA Eq. (7.5).
11 Just as in electrostatics, one needs to exercise due caution transforming these expressions for the limit of discrete classical particles, and extended wavefunctions in quantum mechanics, to avoid the (non-existing) magnetic interaction of a charged particle with itself.
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