Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/421

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790.]

PLANE WAVES.

389

counted for by errors of observation, and shews that our theories of the structure of bodies must be much improved before we can deduce their optical from their electrical properties. At the same time, I think that the agreement of the numbers is such that if no greater discrepancy were found between the numbers derived from the optical and the electrical properties of a considerable number of substances, we should be warranted in concluding that the square root of $K$ , though it may not be the complete expression for the index of refraction, is at least the most important term in it.

Plane Waves.

790.] Let us now confine our attention to plane waves, the front of which we shall suppose normal to the axis of $z$ . All the quantities, the variation of which constitutes such waves, are functions of $z$ and $t$ only, and are independent of $x$ and $y$ . Hence the equations of magnetic induction, (A), Art. 591, are reduced to

a=-{\frac {dG}{dz}},\;b={\frac {dF}{dz}},\;c=0,

(13)

or the magnetic disturbance is in the plane of the wave. This agrees with what we know of that disturbance which constitutes light.

Putting $\mu \alpha$ , $\mu \beta$ and $\mu \gamma$ for $a$ , $b$ and $c$ respectively, the equations of electric currents, Art. 607, become

\left.{\begin{aligned}4\pi \mu u&=-{\frac {db}{dz}}&=&-{\frac {d^{2}F}{dz^{2}}},\\4\pi \mu v&=\;\;{\frac {da}{dz}}&=&-{\frac {d^{2}G}{dz^{2}}},\\4\pi \mu w&=0.\end{aligned}}\right\}

(14)

Hence the electric disturbance is also in the plane of the wave, and if the magnetic disturbance is confined to one direction, say that of $x$ , the electric disturbance is confined to the perpendicular direction, or that of $y$ .