Page:A Dynamical Theory of the Electromagnetic Field.pdf/40

If the medium in the field is a perfect dielectric there is no true conduction, and the currents $p',q',r'$ are only variations in the electric displacement, or, by the equations of Total Currents (A),

$p'={\frac {df}{dt}},\ q'={\frac {dg}{dt}},\ r'={\frac {dh}{dt}}$

(65)

But these electric displacements are caused by electromotive forces, and by the equations of Electric Elasticity (E),

$P=kf,\ Q=kg,\ R=kh$

(66)

These electromotive forces are due to the variations either of the electromagnetic or the electrostatic functions, as there is no motion of conductors in the field; so that the equations of electromotive force (D) are

$\left.{\begin{array}{l}P=-{\frac {dF}{dt}}-{\frac {d\Psi }{dx}},\\\\Q=-{\frac {dG}{dt}}-{\frac {d\Psi }{dy}},\\\\R=-{\frac {dH}{dt}}-{\frac {d\Psi }{dz}}.\end{array}}\right\}$

(67)

(94) Combining these equations, we obtain the following: –

$\left.{\begin{array}{l}k\left({\frac {dJ}{dx}}-\nabla ^{2}F\right)+4\pi \mu \left({\frac {d^{2}F}{dt^{2}}}+{\frac {d^{2}\Psi }{dxdt}}\right)=0,\\\\k\left({\frac {dJ}{dy}}-\nabla ^{2}G\right)+4\pi \mu \left({\frac {d^{2}G}{dt^{2}}}+{\frac {d^{2}\Psi }{dydt}}\right)=0,\\\\k\left({\frac {dJ}{dy}}-\nabla ^{2}H\right)+4\pi \mu \left({\frac {d^{2}H}{dt^{2}}}+{\frac {d^{2}\Psi }{dzdt}}\right)=0.\end{array}}\right\}$

(68)

If we differentiate the third of these equations with respect to $y$, and the second with respect to $z$, and subtract, J and $\Psi$ disappear, and by remembering the equations (B) of magnetic force, the results may be written

$\left.{\begin{array}{l}k\nabla ^{2}\mu \alpha =4\pi \mu {\frac {d^{2}}{dt^{2}}}\mu \alpha ,\\\\k\nabla ^{2}\mu \beta =4\pi \mu {\frac {d^{2}}{dt^{2}}}\mu \beta ,\\\\k\nabla ^{2}\mu \gamma =4\pi \mu {\frac {d^{2}}{d^{2}t}}\mu \gamma .\end{array}}\right\}$

(69)

(95) If we assume that $\alpha ,\beta ,\gamma$ are functions of $lx+my+nz-Vt=w$, the first equation become

$k\mu {\frac {d^{2}\alpha }{dw^{2}}}=4\pi \mu ^{2}V^{2}{\frac {d^{2}\alpha }{dw^{2}}}$

(70)

or

$V=\pm {\sqrt {\frac {k}{4\pi \mu }}}$

(71)

The other equations give the same value for V, so that the wave is propagated in either direction with a velocity V.