Page:Die Kaufmannschen Messungen.djvu/4

By this differential equation and by the condition that for φ = φ₀ (x=0) and for φ = φ₁ (x=x₁) y = 0, y is defined as a function of φ.

Between source and diaphragm, according to (1) and

$\xi =\varrho \sin \varphi$,

we have

${\mathfrak {E}}_{m}(-\varphi )={\mathfrak {E}}_{m}(+\varphi )$.

Here, the course of the curve is symmetrical, so that:

$y_{-\varphi }=y_{+\varphi }$ und $\left({\frac {dy}{d\varphi }}\right)_{-\varphi }=-\left({\frac {dy}{d\varphi }}\right)_{+\varphi }$.

The integration of the differential equation (15) yields:

$\left({\frac {dy}{d\varphi }}\right)_{\varphi _{1}}-\left({\frac {dy}{d\varphi }}\right)_{\varphi _{0}}={\frac {\varrho }{q}}\cdot {\frac {1}{\mathfrak {H}}}\int _{\varphi _{0}}^{\varphi _{1}}{\mathfrak {E}}_{m}d\varphi$

or, since φ₀=-φ₁:

$\left({\frac {dy}{d\varphi }}\right)_{\varphi _{1}}={\frac {\varrho }{q{\mathfrak {H}}}}\int _{0}^{\varphi _{1}}{\mathfrak {E}}_{m}d\varphi$.

Between the diaphragm and the plate it is ${\mathfrak {E}}_{m}=0$, so:

${\frac {dy}{d\varphi }}=const=\left({\frac {dy}{d\varphi }}\right)_{1}$,

and by integrating the last equation again:

${\bar {y}}=\left({\frac {dy}{d\varphi }}\right)_{1}\cdot (\varphi _{2}-\varphi _{1})={\frac {\varrho (\varphi _{2}-\varphi _{1})}{q{\mathfrak {H}}}}\int _{0}^{\varphi _{1}}{\mathfrak {E}}_{m}d\varphi$.

16)

The integral occurring here is given, when one considers, that according to (2):

$0<\varphi <\varphi '\qquad {\mathfrak {E}}_{1}=1$

$\varphi '<\varphi <\varphi _{1}\qquad {\mathfrak {E}}_{1}=\varkappa -\lambda \varrho \sin \varphi$,

where

$\varrho \sin \varphi '=\xi '=0,593$.

17)

Then it follows:

$\int _{0}^{\varphi _{1}}{\mathfrak {E}}_{1}d\varphi =\varphi '+\varkappa (\varphi _{1}-\varphi ')-\lambda \varrho (\cos \varphi '-\cos \varphi _{1})$

and by introduction of ${\mathfrak {E}}_{m}$ from (3):

${\bar {y}}={\frac {25\cdot 10^{10}}{0,1242}}\cdot {\frac {\varrho (\varphi _{2}-\varphi _{1})}{q{\mathfrak {H}}}}$ $\cdot \{\varphi '+\varkappa (\varphi _{1}-\varphi ')-\lambda \varrho (\cos \varphi '-\cos \varphi _{1})\}$

18)

§ 5. Various theories.

The relation between the electric deflection ${\bar {y}}$ and the magnetic deflection ${\bar {z}}$ is due to the dependence of momentum p on the velocity q, and this is given by the expression of the kinetic potential H as a function of q, which is different for various theories. I have performed the calculations only for those two theories, which are the most developed today: that of Abraham,^[1] in which the electron has the form of a rigid sphere, and that of Lorentz-Einstein,^[2] in which the "principle of relativity" possesses exact validity. For brevity I shall denote in the following the first theory as "sphere theory", the second as "relative theory". Then, according to the sphere theory, no matter whether volume charge or surface charge will be adopted, since we are concerned only with quasi-stationary motions, the kinetic potential is:

$H=-{\frac {3}{4}}\mu _{0}c^{2}\left({\frac {c^{2}-q^{2}}{2qc}}\log {\frac {c+q}{c-q}}-1\right)$

19)

(μ₁ is the mass of the electron for q = 0). Therefore:

$p={\frac {\partial H}{\partial q}}={\frac {3}{4}}{\frac {\mu _{0}c^{2}}{q}}\left({\frac {c^{2}+q^{2}}{2qc}}\log {\frac {c+q}{c-q}}-1\right)$.

20)

However, according to the relative theory:^[3]

$H=-\mu _{0}c^{2}\left({\sqrt {1-{\frac {q^{2}}{c^{2}}}}}-1\right)$.

21)

Therefore:

$p={\frac {\partial H}{\partial q}}={\frac {\mu _{0}qc}{\sqrt {c^{2}-q^{2}}}}.$

22)

Like Kaufmann, we introduce the two quantities β and u:

$\beta ={\frac {q}{c}}$ und $u={\frac {\mu _{0}c}{p}}$

23)

for the sphere theory it follows:

${\frac {1}{u}}={\frac {3}{4\beta }}\left({\frac {1+\beta ^{2}}{2\beta }}\log {\frac {1+\beta }{1-\beta }}-1\right)$

24)

and in the relative theory:

${\frac {1}{u}}={\frac {\beta }{\sqrt {1-\beta ^{2}}}}$

25)

By introduction of u instead of p, equation (14) for the magnetic deflection becomes:

$u={\frac {\mu _{0}}{\epsilon }}\cdot {\frac {2c\ \sin \varphi _{1}}{x_{1}{\mathfrak {H}}}}$.

26)

§ 6. Numerical values.

The comparison of the observed and the theoretical values was stated by me in a way, so that for each measured magnetic deflection ${\bar {z}}$ for each of the two theories, the corresponding value of the electrical deflection ${\bar {y}}$ was calculated and compared with the observed values. Accordingly, the following table in the first column

1. M. Abraham, Ann. d. Phys. (4) 10, 105, 1903.
2. H. A. Lorentz, Versl. Kon. Akad. v. Wet Amsterdam 1904, S. 809. A. Einstein, Ann. d. Phys. (4) 17, 891, 1905. See also H. Poincaré, C. R. 140, 1504, 1905.
3. for example M. Planck, Verh. D. Phys. Ges. 8, 140, 1906.