Chapter 5

Formula.

This is Döppler’s principle for any velocity. If Φ = 0, then the equation takes the simple form

Formula.

We see that—contrary to the usual conception—ν = ∞, forv= -c.

If Φ′ = angle between the wave-normal (direction of the ray) in the moving system, and the line of motion of the observer, the equation forl´takes the form

Formula.

This equation expresses the law of observation in its most general form. If Φ = π/2, the equation takes the simple form

vcos Φ′ = ---c

cos Φ′ = ---

We have still to investigate the amplitude of the waves, which occur in these equations. If A and A′ be the amplitudes in the stationary and the moving systems (either electrical or magnetic), we have

Formula.

If Φ = 0, this reduces to the simple form

Formula.

From these equations, it appears that for an observer, which moves with the velocity c towards the source of light, the source should appear infinitely intense.

Since A²/8π is equal to the energy of light per unit volume, we have to regard A²/8π as the energy of light in the moving system. A′²/A² would therefore denote the ratio between the energies of a definite light-complex “measured when moving” and “measured when stationary,” the volumes of the light-complex measured in K andkbeing equal. Yet this is not the case. Ifl,m,nare the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface

(x-clt)² + (y-cmt)² + (z-cnt)² = R²,

which expands with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses, when regarded from the systemk,i.e., the energy of the light-complex relative to the systemk.

Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time τ = 0, the equation:—

Formula.

If S = volume of the sphere, S′ = volume of this ellipsoid, then a simple calculation shows that:

Formula.

If E denotes the quantity of light energy measured in the stationary system, E′ the quantity measured in the moving system, which are enclosed by the surfaces mentioned above, then

Formula.

If Φ = 0, we have the simple formula:—

Formula.

It is to be noticed that the energy and the frequency of a light-complex vary according to the same law with the state of motion of the observer.

Let there be a perfectly reflecting mirror at the co-ordinate-plane ξ = 0, from which the plane-wave considered in the last paragraph is reflected. Let us now ask ourselves about the light-pressure exerted on the reflecting surface and the direction, frequency, intensity of the light after reflexion.

Let the incident light be defined by the magnitudes A cos Φ,v(referred to the system K). Regarded fromk, we have the corresponding magnitudes:

Formula.

For the reflected light we obtain, when the process is referred to the systemk:—

A″ = A′, cos Φ″ = -cos Φ″, ν″ = ν′

By means of a back-transformation to the stationary system, we obtain K, for the reflected light:—

Formula.

The amount or energy falling upon the unit surface of the mirror per unit of time (measured in the stationary system) is A²/(8π (c cos Φ -v)). The amount of energy going away from unit surface of the mirror per unit of time is A‴²/(8π (-c cos Φ″ +v)). The difference of these two expressions is, according to the Energy principle, the amount of work exerted, by the pressure of light per unit of time. If we put this equal to P.v, where P = pressure of light, we have

Formula.

As a first approximation, we obtain

A²P = 2 ---- cos² Φ8π

A²

P = 2 ---- cos² Φ

8π

which is in accordance with facts, and with other theories.

All problems of optics of moving bodies can be solved after the method used here. The essential point is, that the electric and magnetic forces of light, which are influenced by a moving body, should be transformed to a system of co-ordinates which is stationary relative to the body. In this way, every problem of the optics of moving bodies would be reduced to a series of problems of the optics of stationary bodies.

Let us start from the equations:—

Formula.

where

Formula.

denotes 4π times the density of electricity, and (ux,uy,uz) are the velocity-components of electricity. If we now suppose that the electrical-masses are bound unchangeably to small, rigid bodies (Ions, electrons), then these equations form the electromagnetic basis of Lorentz’s electrodynamics and optics for moving bodies.

If these equations which hold in the system K, are transformed to the systemkwith the aid of the transformation-equations given in§ 3and§ 6, then we obtain the equations:—

Formula.

where

Formula.

Since the vector (uξ,uη,uζ) is nothing but the velocity of the electrical mass measured in the systemk, as can be easily seen from the addition-theorem of velocities in§ 4—so it is hereby shown, that by taking our kinematical principle as the basis, the electromagnetic basis of Lorentz’s theory of electrodynamics of moving bodies correspond to the relativity-postulate. It can be briefly remarked here that the following important law follows easily from the equations developed in the present section:—if an electrically charged body moves in any manner in space, and if its charge does not change thereby, when regarded from a system moving along with it, then the charge remains constant even when it is regarded from the stationary system K.

Let us suppose that a point-shaped particle, having the electrical chargee(to be called henceforth the electron) moves in the electromagnetic field; we assume the following about its law of motion.

If the electron be at rest at any definite epoch, then in the next “particle of time,” the motion takes place according to the equations

d²xd²yd²zm----- =eX,m----- =eY,m----- =eZdt²dt²dt²

d²xd²yd²z

m----- =eX,m----- =eY,m----- =eZ

dt²dt²dt²

Where (x,y,z) are the co-ordinates of the electron, andmis its mass.

Let the electron possess the velocityvat a certain epoch of time. Let us now investigate the laws according to which the electron will move in the ‘particle of time’ immediately following this epoch.

Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering, the electron is at the origin of co-ordinates, and moves with the velocityvalong the X-axis of the system. It is clear that at this moment (t= 0) the electron is at rest relative to the systemk, which moves parallel to the X-axis with the constant velocityv.

From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the systemk, the electron moves according to the equations

d²ξd²ηd²ζm----- =eX′,m----- =eY′,m----- =eZ′ ,dτ²dτ²dτ²

d²ξd²ηd²ζ

m----- =eX′,m----- =eY′,m----- =eZ′ ,

dτ²dτ²dτ²

in the time immediately following the moment, where the symbols (ξ, η, ζ, τ, X’, Y’, Z’) refer to the systemk. If we now fix, that fort=v=y=z= 0, τ = ξ = η = ζ = 0, then the equations of transformation given in§ 3(and§ 6) hold, and we have:

vτ = β(t- ----x), ξ = β(x-vt), η =y, ζ =z,c²vvX′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M)cc

vτ = β(t- ----x), ξ = β(x-vt), η =y, ζ =z,c²

τ = β(t- ----x), ξ = β(x-vt), η =y, ζ =z,

c²

vvX′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M)cc

X′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M)

With the aid of these equations, we can transform the above equations of motion from the systemkto the system K, and obtain:—

Formula.

Let us now consider, following the usual method of treatment, the longitudinal and transversal mass of a moving electron. We write the equations (A) in the form

d²xmβ² ----- =eX =eX′dt²d²yvmβ² ----- =eβ (Y - --- N) =eY′dt²cd²zvmβ² ----- =eβ (Z - --- M) =eZ′dt²c

d²xmβ² ----- =eX =eX′dt²

d²x

mβ² ----- =eX =eX′

dt²

d²yvmβ² ----- =eβ (Y - --- N) =eY′dt²c

d²yv

mβ² ----- =eβ (Y - --- N) =eY′

dt²c

d²zvmβ² ----- =eβ (Z - --- M) =eZ′dt²c

d²zv

mβ² ----- =eβ (Z - --- M) =eZ′

dt²c

and let us first remark, thateX′,eY′,eZ′ are the components of the ponderomotive force acting upon the electron, and are considered in a moving system which, at this moment, moves with a velocity which is equal to that of the electron. This force can, for example, be measured by means of a spring-balance which is at rest in this last system. If we briefly call this force as “the force acting upon the electron,” and maintain the equation:—

Mass-number × acceleration-number = force-number, and if we further fix that the accelerations are measured in the stationary system K, then from the above equations, we obtain:—

Longitudinal mass:

Formula.

Transversal mass:

Formula.

Naturally, when other definitions are given of the force and the acceleration, other numbers are obtained for the mass; hence we see that we must proceed very carefully in comparing the different theories of the motion of the electron.

We remark that this result about the mass hold also for ponderable material mass; for in our sense, a ponderable material point may be made into an electron by the addition of an electrical charge which may be as small as possible.

Let us now determine the kinetic energy of the electron. If the electron moves from the origin of co-ordinates of the system K with the initial velocity 0 steadily along the X-axis under the action of an electromotive force X, then it is clear that the energy drawn from the electrostatic field has the value ∫eXdx. Since the electron is only slowly accelerated, and in consequence, no energy is given out in the form of radiation, therefore the energy drawn from the electro-static field may be put equal to the energy W of motion. Considering the whole process of motion in questions, the first of equationsA)holds, we obtain:—

Formula.

Forv=c, W is infinitely great. As our former result shows, velocities exceeding that of light can have no possibility of existence.

In consequence of the arguments mentioned above, this expression for kinetic energy must also hold for ponderable masses.

We can now enumerate the characteristics of the motion of the electrons available for experimental verification, which follow from equationsA).

1. From the second of equationsA), it follows that an electrical force Y, and a magnetic force N produce equal deflexions of an electron moving with the velocityv, when Y = Nv/c. Therefore we see that according to our theory, it is possible to obtain the velocity of an electron from the ratio of the magnetic deflexion Am, and the electric deflexion Ae, by applying the law:—

Formula.

This relation can be tested by means of experiments because the velocity of the electron can be directly measured by means of rapidly oscillating electric and magnetic fields.

2. From the value which is deduced for the kinetic energy of the electron, it follows that when the electron falls through a potential difference of P, the velocityvwhich is acquired is given by the following relation:—

Formula.

3. We calculate the radius of curvature R of the path, where the only deflecting force is a magnetic force N acting perpendicular to the velocity of projection. From the second of equationsA)we obtain:

Formula.

mvβcR = --------eN

mvβc

R = --------

These three relations are complete expressions for the law of motion of the electron according to the above theory.

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