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

498 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),

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

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

{{MathForm2|(67)|$$\left.\begin{array}{l} P=-\frac{dF}{dt}-\frac{d\Psi}{dx},\\ \\Q=-\frac{dG}{dt}-\frac{d\Psi}{dy},\\ \\R=-\frac{dH}{dt}-\frac{d\Psi}{dz}.\end{array}\right\} $$}}

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

{{MathForm2|(68)|$$\left.\begin{array}{l} k\left(\frac{dJ}{dx}-\nabla^{2}F\right)+4\pi\mu\left(\frac{d^{2}F}{dt^{2}}+\frac{d^{2}\Psi}{dxdt}\right)=0,\\ \\k\left(\frac{dJ}{dy}-\nabla^{2}G\right)+4\pi\mu\left(\frac{d^{2}G}{dt^{2}}+\frac{d^{2}\Psi}{dydt}\right)=0,\\ \\k\left(\frac{dJ}{dy}-\nabla^{2}H\right)+4\pi\mu\left(\frac{d^{2}H}{dt^{2}}+\frac{d^{2}\Psi}{dzdt}\right)=0.\end{array}\right\} $$}}

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

{{MathForm2|(69)|$$\left.\begin{array}{l} k\nabla^{2}\mu\alpha=4\pi\mu\frac{d^{2}}{dt^{2}}\mu\alpha,\\ \\k\nabla^{2}\mu\beta=4\pi\mu\frac{d^{2}}{dt^{2}}\mu\beta,\\ \\k\nabla^{2}\mu\gamma=4\pi\mu\frac{d^{2}}{d^{2}t}\mu\gamma.\end{array}\right\} $$}}

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

or

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