Page:BatemanElectrodynamical.djvu/34

These equations are of the form

$\left.{\begin{array}{lll}{\mathsf {curl}}\ H={\frac {\partial D}{\partial t}}+s,&&{\mathsf {div}}\ D=\rho \\\\{\mathsf {curl}}\ E=-{\frac {\partial B}{\partial t}},&&{\mathsf {div}}\ B=0\end{array}}\right\}$

(I)

where the vectors E, H, D, B denote the electric force, magnetic force, electric displacement, and magnetic induction respectively, s denotes the current and ρ the volume density of electricity. The equations differ from those used by Lorentz by the fact that the vector H - [Pw] occurring in Lorentz's equations is replaced here by the vector H.^[1] It should be remarked that Frank^[2] has obtained Minkowski's equations by a process of averaging in the case of non-magnetic bodies.

We shall suppose that the electric polarisation P and the magnetic polarisation Q are connected with D, E, B, and H by the formulae

$P=D-E,\ Q=B-H.$

The electrodynamical equations (I) can be replaced by the two integral equations

$\int \int \left(B_{x}dy\ dz+B_{y}dz\ dx+B_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\right)=0,$

(II)

${\begin{array}{l}\int \int \left(D_{x}dy\ dz+D_{y}dz\ dx+D_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\right),\\\qquad =-\int \int \int \left(s_{x}dy\ dz\ dt+s_{y}dz\ dx\ dt+s_{z}dx\ dy\ dt-\rho dx\ dy\ dz\right).\end{array}}$

(III)

These equations are unaltered in form by a transformation from (x, y, z, t) to (x', y', z', t') if

${\begin{array}{l}B_{x}dy\ dz+B_{y}dz\ dx+B_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\=\theta \left[B'_{x}dy'dz'+B'_{y}dz'dx'+B'_{z}dx'dy'+E'_{x}dx'dt'+E'_{y}dy'dt'+E'_{z}dz'dt'\right],\end{array}}$

(IV)

${\begin{array}{l}D_{x}dy\ dz+D_{y}dz\ dx+D_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\\=\phi \left[D'_{x}dy'dz'+D'_{y}dz'dx'+D'_{z}dx'dy'-H'_{x}dx'dt'-H'_{y}dy'dt'-H'_{z}dz'dt'\right],\end{array}}$

(V)

${\begin{array}{l}s_{x}dy\ dz\ dt+s_{y}dz\ dx\ dt+s_{z}dx\ dy\ dt-\rho dx\ dy\ dz\\\qquad =\phi \left[s'_{x}dy'dz'dt'+s'_{y}dz'dx'dt'+s'_{z}dx'dy'dt-\rho 'dx'dy'dz'\right],\end{array}}$

(VI)

where θ and φ are constants.

1. This simply means that a different definition is adopted for H, the object being to retain the symmetry of the equations.
2. Ann. d. Phys., Bd. 27, p. 1059 (1908).