Page:AbrahamMinkowski1.djvu/12

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From them it follows, that when \dot{\epsilon} and \dot{\mu} are set equal to zero:

\begin{array}{l}
\mathfrak{E'\dot{D}-D\dot{E}'=\dot{q}[E'H']+q[E'\dot{H}']+q[\dot{E}'H']},\\
\mathfrak{H'\dot{B}-B\dot{H}'=\dot{q}[E'H']+q[\dot{E}'H']+q[E'\dot{H}']},\end{array}

Now, since relation (18) requires

2\mathfrak{\dot{q}}c\mathfrak{g=E'\dot{D}-D\dot{E}'+H'\dot{B}-B\dot{H}}'

one thus places Cohn's theory in our system, by setting

(27) c\mathfrak{g=[E'H']=}\frac{\mathfrak{S}'}{c}

In Cohn's electrodynamics, the momentum density has to be set equal to the relative ray divided by c^{2}.

That relation (18a) is satisfied by (26) and (27), can easily be confirmed by considering, that the identity

\mathfrak{\left[q[E'H']\right]=\left[E'[qH']\right]-\left[H'[qE']\right]}

exists. From (19) it follows now for the electromagnetic energy density.

(28) \psi=\frac{1}{2}\mathfrak{E'D}+\frac{1}{2}\mathfrak{H'B}+\mathfrak{q[E'H']},

an expression, which according to (26) can also be written

(28a) \psi=\frac{1}{2}\epsilon\mathfrak{E}'^{2}+\frac{1}{2}\mu\mathfrak{H}'^{2}+2\mathfrak{q[E'H']};

it is in agreement with E. Cohns approach.

I later return to the calculation of the ponderomotive force.


§ 8. Theory of H. A. Lorentz.


When we modify the connecting equations of the theory of H. A. Lorentz, so that symmetry exists between the electric and magnetic vectors, then we arrive at the approach:

(29) \begin{cases}
\mathfrak{D}=\epsilon\mathfrak{E}'-[\mathfrak{qH}],\\
\mathfrak{B}=\mu\mathfrak{H}'+[\mathfrak{qE}];\end{cases}
(30) \begin{cases}
\mathfrak{E'}=\mathfrak{E}+[\mathfrak{qH}],\\
\mathfrak{H'}=\mathfrak{H}-[\mathfrak{qE}].\end{cases}

Besides four vectors contained in the main equations, two new vectors \mathfrak{E,H} occur here. This circumstance makes Lorentz's theory more complicated than Cohn's one. The latter directly connects the components of \mathfrak{D,B} with those of \mathfrak{E',H'} by equations, which are linear in the velocity components; at this one, however, the connecting equations (§ 10, eq. 37b) given by elimination of \mathfrak{EH}, are not linear in the velocity components any more.