Page:Elektrische und Optische Erscheinungen (Lorentz) 106.jpg

For the determination of ${\mathfrak {d}}$ and ${\mathfrak {H}}$, the formulas (IX) (§ 56) and ($VI_{b}$) (§ 20) can serve, which we may replace by

$4\pi V^{2}{\mathfrak {d}}=4\pi V^{2}{\mathfrak {d}}'-[{\mathfrak {p.H}}']$

and

${\mathfrak {H}}={\mathfrak {H}}'+4\pi [{\mathfrak {p.d}}']$

If follows

${\mathfrak {d}}_{x}=a\left\{f-{\frac {{\mathfrak {p}}_{y}}{V}}(lg-mf)+{\frac {{\mathfrak {p}}_{z}}{V}}(nf-lh)\right\}\cos \psi '$, etc.

(100)

${\mathfrak {H}}_{x}=4\pi a\left\{V(mh-ng)+({\mathfrak {p}}_{y}h-{\mathfrak {p}}_{z}g)\right\}\cos \psi '$, etc.

(101)

or, if we put by (99)

${\frac {{\mathfrak {p}}_{x}}{V}}=l'\left(1+{\frac {{\mathfrak {p}}_{s}}{V}}\right)-l$, etc.

and if we consider (95).

${\mathfrak {d}}_{x}=a\left(1+{\frac {{\mathfrak {p}}_{s}}{V}}\right)\left\{-m'(lg-mf)+n'(nf-lh)\right\}\cos \psi '$, etc.

(102)

${\mathfrak {H}}_{x}=4\pi aV\left(1+{\frac {{\mathfrak {p}}_{s}}{V}}\right)(m'h-n'g)\cos \psi '$, etc.

By that we see, that ${\mathfrak {d}}$ and ${\mathfrak {H}}$ are both perpendicular to the wave normal, as it was expected. Additionally, both vectors are mutually perpendicular, which can be seen most easily, when by replace (100) by

${\mathfrak {d}}_{x}=a\left\{f-{\frac {{\mathfrak {p}}_{y}}{V}}(l'g-m'f)+{\frac {{\mathfrak {p}}_{z}}{V}}(n'f-l'h)\right\}\cos \psi '$, etc.

Furthermore, we can conclude, that the vector $[{\mathfrak {d.H}}]$ which is present in Poynting's theorem, falls into the wave normal. We can easily convince ourselves, that it has the direction, in which the waves are propagating, and we find for its magnitude

$4\pi a^{2}(V+2{\mathfrak {p}}_{s})\cos ^{2}\psi '$.

The energy flux through a plane which is parallel to the waves, thus amounts for the unit of area and time

$4\pi a^{2}V^{2}(V+2{\mathfrak {p}}_{s})\cos ^{2}\psi '$

(103)

§ 76. From a light bundle as the one considered above, others of the same kind can arise by refraction and mirroring at plane limiting surfaces.