56.

In the surface of separation of the two media $x = 0$ .

According to the general electromagnetic boundary conditions the components of the field-strengths in the surface of separation, i.e., the four quantities $⋿_{y}$ , $⋿_{z}$ , $𝖧_{y}$ , $𝖧_{z}$ must be equal to each other on the two sides of the surface of separation for this value of $x$ .

In the conductor the electric field-strength $⋿$ is infinitely small in accordance with the assumption made above.

Hence $⋿_{y}$ and $⋿_{z}$ must vanish also in the vacuum for $x = 0$ .

This condition cannot be satisfied unless we assume in the vacuum, besides the incident, also a reflected wave superposed on the former in such a way that the components of the electric field of the two waves in the $y$ and $z$ direction just cancel at every instant and at every point in the surface of separation.

By this assumption and the condition that the reflected wave is a plane wave returning into the interior of the vacuum, the other four components of the reflected wave are also completely determined. They are all functions of the single argument $t - \frac{- x \cos θ + y \sin θ}{c} . \Label [e q n] (57) \upshape (57)$

The actual calculation yields as components of the total electromagnetic field produced in the vacuum by the superposition of the two waves, the following expressions valid for points of the surface of separation $x = 0$ , $\mathsf{E}{x} &= -\sin\theta · f - \sin\theta · f = -2\sin\theta · f\ \mathsf{E}} &= \phantom{-}\cos\theta · f - \cos\theta · f = 0\ \mathsf{E{z} &= g - g = 0 \ \mathsf{H}} &= \phantom{-}\sin\theta · g - \sin\theta · g = 0\ \mathsf{H{y} &= -\cos\theta · g - \cos\theta · g = -2\cos\theta · g\ \mathsf{H} &= f + f = 2f.

\Label[eqn]{(58)}\tag*{\upshape (58)}$ In these equations the argument of the functions $f$ and $g$ is, according to [eqn:(56)] (56) and [eqn:(57)] (57), $t - \frac{y \sin θ}{c} .$ From these values the electric and magnetic field-strength within the conductor in the immediate neighborhood of the separating surface $x = 0$ is obtained: $\mathsf{E}{x} &=0 \qquad & \mathsf{H}} &= 0\ \mathsf{E{y} &=0 & \mathsf{H}} &= -2\cos\theta · g\ \mathsf{E{z} &=0 & \mathsf{H} &= 2f

\Label[eqn]{(59)}\tag*{\upshape (59)}$ where again the argument $t - \frac{y \sin θ}{c}$ is to be substituted in the functions $f$ and $g$ . For the components of $⋿$ all vanish in an absolute conductor and the components $𝖧_{x}$ , $𝖧_{y}$ , $𝖧_{z}$ are all continuous at the separating surface, the two latter since they are tangential components of the field-strength, the former since it is the normal component of the magnetic induction $𝖡$ ([sect:55.] Sec. 55), which likewise remains continuous on passing through any surface of separation.

On the other hand, the normal component of the electric field-strength $⋿_{x}$ is seen to be discontinuous; the discontinuity shows the existence of an electric charge on the surface, the surface density of which is given in magnitude and sign as follows: $\frac{1}{4 π} 2 \sin θ \cdot f = \frac{1}{2 π} \sin θ \cdot f . \Label [e q n] (60) \upshape (60)$ In the interior of the conductor at a finite distance from the bounding surface, i.e., for $x > 0$ , all six field components are infinitely small. Hence, on increasing $x$ , the values of $𝖧_{y}$ and $𝖧_{z}$ , which are finite for $x = 0$ , approach the value $0$ at an infinitely rapid rate.