# Page:A Dynamical Theory of the Electromagnetic Field.pdf/43

501
PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

we should call the electric fluid, and select either vitreous or resinous electricity as the representative of that fluid, then we might have normal vibrations propagated with a velocity depending on this density. We have, however, no evidence as to the density of electricity, as we do not even know whether to consider vitreous electricity as a substance or as the absence of a substance.

Hence electromagnetic science leads to exactly the same conclusions as optical science with respect to the direction of the disturbances which can be propagated through the field; both affirm the propagation of transverse vibrations, and both give the same velocity of propagation. On the other hand, both sciences are at a loss when called on to affirm or deny the existence of normal vibrations.

Relation between the Index of Refraction and the Electromagnetic Character of the substance.

(101) The velocity of light in a medium, according to the Undulatory Theory, is

${\displaystyle {\frac {1}{i}}V_{0}}$

where ${\displaystyle i}$ is the index of refraction and ${\displaystyle V_{0}}$ is the velocity in vacuum. The velocity, according to the Electromagnetic Theory, is

${\displaystyle {\sqrt {\frac {k}{4\pi \mu }}}}$

where, by equations (49) and (71), ${\displaystyle k={\tfrac {1}{D}}k_{0}}$, and ${\displaystyle k_{0}=4\pi V_{0}^{2}}$.

Hence

 ${\displaystyle D={\frac {i^{2}}{\mu }}}$ (80)

or the Specific Inductive Capacity is equal to the square of the index of refraction divided by the coefficient of magnetic induction.

Propagation of Electromagnetic Disturbances in a Crystallized Medium.

(102) Let us now calculate the conditions of propagation of a plane wave in a medium for which the values of ${\displaystyle k}$ and ${\displaystyle \mu }$ are different in different directions. As we do not propose to give a complete investigation of the question in the present imperfect state of the theory as extended to disturbances of short period, we shall assume that the axes of magnetic induction coincide in direction with those of electric elasticity.

(103) Let the values of the magnetic coefficient for the three axes be ${\displaystyle \lambda ,\mu ,\nu }$, then the equations of magnetic force (B) become

 ${\displaystyle \left.{\begin{array}{l}\lambda \alpha ={\frac {dH}{dy}}-{\frac {dG}{dz}},\\\\\mu \beta ={\frac {dF}{dz}}-{\frac {dH}{dx}},\\\\\nu \gamma ={\frac {dG}{dx}}-{\frac {dF}{dy}}.\end{array}}\right\}}$ (81)