Page:CunninghamExtension.djvu/6

current density in the two systems are,

$\delta j={\frac {\sum e\left(w'-w\right)}{\delta a}},$ $\delta J={\frac {\sum e\left(W'-W\right)}{\delta A}},$

these being vector equations.

Using equations (3) and (5) these give

${\begin{array}{ll}\delta J_{x}&={\frac {\sum e\left(w'_{x}-w_{x}\right)}{\delta a\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)}}={\frac {\delta j_{x}}{\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)}}=\beta \left(1+{\frac {vW_{x}}{c^{2}}}\right)\delta j_{x},\\\\\delta J_{y}&={\frac {\sum e}{\delta a}}\left\{\left(w'_{y}-w_{y}\right)-{\frac {vw_{y}\left(w'_{x}-w_{x}\right)}{c^{2}\left(1-{\frac {vw_{x}}{c^{2}}}\right)}}\right\}\\\\&=\delta j_{y}-{\frac {\beta v}{c^{2}}}W_{y}\delta j_{x},\\\\\delta J_{z}&=\delta j_{z}-{\frac {\beta v}{c^{2}}}W_{z}\delta j_{x}.\end{array}}$

Summing for all electrons, we obtain Minkowski's equation,

$\left.{\begin{array}{l}J_{x}=\beta j_{x}\left(1+{\frac {vW_{x}}{c^{2}}}\right)\\\\J_{y}=j_{y}-{\frac {\beta vW_{y}}{c^{2}}}j_{x}\\\\J_{z}=j_{z}-{\frac {\beta vW_{z}}{c^{2}}}j_{x}\end{array}}\right\}.$

(7)

The Polarization p and Magnetization m.

So far no distinction has been made between various types of motion of the electrons; but now they must be differentiated.

Consider first the so-called polarization electrons which move with the body, but have an electric moment. Taking δa, δA to be the volumes occupied by such a polarized element, its contributions to the polarization in the two systems are related as follows:

$\delta P_{x}={\frac {\sum eX}{\delta A}},$

where X is measured relative to the centre of the element.