Page:Elektrische und Optische Erscheinungen (Lorentz) 045.jpg

It is now necessary to distinguish between the velocity of the considered conductor element and the relative velocity of an ion in the wire. The former shall be called ${\mathfrak {v}}$ and the latter ${\mathfrak {w}}$. From (V_a) it is given

${\mathfrak {E}}=4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p.H}}]+[{\mathfrak {v.H}}]+[{\mathfrak {w.H}}].$

Yet, the velocity ${\mathfrak {w}}$ has the direction of ds; consequently we have $[{\mathfrak {w.H}}]_{s}=0$, and for positive as well as for negative ions

${\mathfrak {E}}_{s}={\mathfrak {E}}'_{s}=4\pi V^{2}{\mathfrak {d}}_{s}+[{\mathfrak {p.H}}]_{s}+[{\mathfrak {v.H}}]_{s}.$

Finally, equation (31) transforms into

$i=c\omega \int \left\{4\pi V^{2}{\bar {{\mathfrak {d}}_{s}}}+[{\mathfrak {p.{\bar {H}}}}]_{s}+[{\mathfrak {v.{\bar {H}}}}]_{s}\right\}\ d\ t{,}$

$c=pe+qe'.$

Let us divide by $c\ \omega$, multiply by ds, and integrate over the whole current-line. If we consider here, that i has everywhere the same value in the current-line, and if we put

$\int {\frac {d\ s}{c\ \omega }}={\frac {1}{C}}{,}$

we shall find

$i=C\int \left\{4\pi V^{2}\int {\bar {{\mathfrak {d}}_{s}}}\ d\ s+\int [{\mathfrak {p.{\bar {H}}}}]_{s}\ d\ s+\int [{\mathfrak {v.{\bar {H}}}}]_{s}\ d\ s\right\}\ d\ t.$

(32)

§ 29. The following discussion is intended to derive the known fundamental law of induction from this formula. Imagine an area σ on which the current-line constantly is located during its motion, and consider the integral

$\int {\overline {{\mathfrak {H}}_{n}}}\ d\ \sigma =P{,}$

(33)

for the part that is cut by the line.

This quantity, which is usually called "the number of magnetic force-lines covered by s", changes over time, namely for two reasons. First, ${\bar {\mathfrak {H}}}$ varies at each point, and second, the area of integration changes.

During time dt, the first cause produces the following increase of P

$d\ t\int {\dot {\overline {{\mathfrak {H}}_{n}}}}\ d\ \sigma .$