Page:Die Kaufmannschen Messungen.djvu/3



Then, because of the continuity of $$\mathfrak{E}_{1}$$:

$$\varkappa-\lambda\xi'=1$$ und $$\varkappa-\lambda\frac{x_{1}}{2}=$$0.

The value of the constant ξ' I assumed to be as large, so that that the value of the "electric field integral" is the same as 's. Its value is:

$$(x_{1}-x_{1})\cdot\left\{ \int_{x_{0}}^{x_{1}}\mathfrak{E}_{1}dx-\frac{1}{x_{1}-x_{0}}\int_{x_{0}}^{x_{1}}dx\int_{x_{0}}^{x}\mathfrak{E}_{1}dx\right\} =1,565$$.

Substituting the above values, it follows:

$$\frac{1}{2}(x_{2}-x_{1})\cdot\left(\frac{x_{1}}{2}+\xi'\right)=1,565$$

and from this:

ξ = 0,593 ϰ = 2,468 λ = 2,475.

To reduce the electric field strength to the absolute electrostatic unit: $$\mathfrak{E}$$, or to the absolute electromagnetic unit: $$\mathfrak{E}_{m}$$, one has:

Whether the simplistic assumptions made in relation to the electric and magnetic field are really sufficient for the relevant calculations, will be shown below.

§ 3. Magnetic deflection.
If we introduce into the equations of motion (§ 1) the momentum vector (quantity of motion)

and also the electromagnetic unit of the electric field strength ($$\mathfrak{E}_{m}$$) and for the electrical elementary quantum (ε), then they are:

Because $$\mathfrak{H}$$ is constant, (5) and (7) can be integrated with respect to time t. Dividing the two resulting equations, t, p and q are entirely eliminated, and a second integration yields the equation of the trajectory of projection on the xz-plane, a circle, which goes through the points x = 0, z = 0, x = x1, z = 0 and x = x2, $$z = \bar{z}$$ and is determined by it. The current coordinates x, z of the points of this circle can be represented as functions of one variable parameter: the angle φ which is the tangent of the circle in the direction of motion on the x-axis, and it is positive when the motion is to the side of the positive z-axis:

Where

is the radius of the circle and φ1 the value of φ for x = x1. In these equations it is already expressed that for x = 0 and x = x1, z = 0. If it is considered that x = x2, $$z=\bar{z}$$, then we have the values:

and

and also $$\varrho$$ from (9).

By inserting (8) into (5) or (7) we have:

Now it is:

$$q^{2}=\varrho^{2}\dot{\varphi}^{2}+\dot{y}^{2}$$,

for which we can put with sufficient approximation:

Therefore:

The momentum p of each electron is independent of time and, without entering into a special theory, can be calculated from the magnetic deflection $$\bar{z}$$.

Since p is independent of time t, the same follows for the velocity q and by (12) for the angular velocity $$\dot{\varphi}$$. So angle φ is linearly dependent on time t.

§ 4. Electrical deflection.
From (6) it follows:

$$\frac{p}{q}\frac{d^{2}y}{d\varphi^{2}}\cdot\dot{\varphi}^{2}=\epsilon\mathfrak{E}_{m}$$

and from this according to (12) and (13)