Part 4
_v_ _v_ [L′, M′, N′] = ψ (_v_) [L, β(M - ----- Z), β(N + ----- Y)], _c_ _c_
Then by reasoning similar to that followed in §(3), it can be shown that ψ(_v_) = 1.
_v_ _v_ [X′, Y′, Z′] = [X, β(Y - ----- N), β(Z + ------ M)] _c_ _c_
_v_ _v_ [L′, M′, N′] = [L, β(M - ------ Z), β(N + ----- Y)], _c_ _c_
For the interpretation of these equations, we make the following remarks. Let us have a point-mass of electricity which is of magnitude unity in the stationary system K, _i.e._, it exerts a unit force upon a similar quantity placed at a distance of 1 cm. If this quantity of electricity be at rest in the stationary system, then the force acting upon it is equivalent to the vector (X, Y, Z) of electric force. But if the quantity of electricity be at rest relative to the moving system (at least for the moment considered), then the force acting upon it, and measured in the moving system is equivalent to the vector (X′, Y′, Z′). The first three of equations (1), (2), (3), can be expressed in the following way:—
1. If a point-mass of electric unit pole moves in an electro-magnetic field, then besides the electric force, an electromotive force acts upon it, which, neglecting the numbers involving the second and higher powers of _v_/_c_, is equivalent to the vector-product of the velocity vector, and the magnetic force divided by the velocity of light (Old mode of expression).
2. If a point-mass of electric unit pole moves in an electro-magnetic field, then the force acting upon it is equivalent to the electric force existing at the position of the unit pole, which we obtain by the transformation of the field to a co-ordinate system which is at rest relative to the electric unit pole [New mode of expression].
Similar theorems hold with reference to the magnetic force. We see that in the theory developed the electro-magnetic force plays the part of an auxiliary concept, which owes its introduction in theory to the circumstance that the electric and magnetic forces possess no existence independent of the nature of motion of the co-ordinate system.
It is further clear that the asymmetry mentioned in the introduction which occurs when we treat of the current excited by the relative motion of a magnet and a conductor disappears. Also the question about the seat of electromagnetic energy is seen to be without any meaning.
§ 7. Theory of Döppler’s Principle and Aberration.
In the system K, at a great distance from the origin of co-ordinates, let there be a source of electrodynamic waves, which is represented with sufficient approximation in a part of space not containing the origin, by the equations:—
X = X₀ sin Φ Y = Y₀ sin Φ Z = Z₀ sin Φ L = L₀ sin Φ M = M₀ sin Φ N = N₀ sin Φ lx + my + nz Φ = ω(t - ------------ ) c
Here (X₀, Y₀, Z₀) and (L₀, M₀, N₀) are the vectors which determine the amplitudes of the train of waves, (_l_, _m_, _n_) are the direction-cosines of the wave-normal.
Let us now ask ourselves about the composition of these waves, when they are investigated by an observer at rest in a moving medium _k_:—By applying the equations of transformation obtained in §6 for the electric and magnetic forces, and the equations of transformation obtained in § 3 for the co-ordinates, and time, we obtain immediately:—
X′ = X₀ sin Φ′
v Y′ = β(Y₀ - --- N₀) sin Φ′ c
v Z′ = β(Z₀ - --- M₀) sin Φ′ c
L′ = L₀ sin Φ′
v M′ = β(M₀ - --- Z₀) sin Φ′ c
v N′ = β(N₀ - --- Y₀) sin Φ′ c
l′ξ + m′η + n′ζ Φ′ = ω′(t - --------------- ) c
where
$$ \omega' = \omega \beta (1 - \frac {lv}{c}) $$ ,
$$ l' = \frac {l - \frac {v}{c}}{1 - \frac {lv}{c}} $$ ,
$$ m' = \frac {m}{\beta (1 - \frac {lv}{c})} $$ ,
$$ n' = \frac {n}{\beta (1 - \frac {lv}{c})} $$
From the equation for ω′ it follows:—If an observer moves with the velocity _v_ relative to an infinitely distant source of light emitting waves of frequency ν, in such a manner that the line joining the source of light and the observer makes an angle of Φ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency ν′ which is perceived by the observer is represented by the formula
$$ \nu' = \nu \frac {1 - cos \Phi \frac {v}{c}} {\sqrt {1 - \frac {v^2}{c^2}}} $$
This is Döppler’s principle for any velocity. If Φ = 0, then the equation takes the simple form
$$ \nu' = \nu (\frac {1 - \frac {v}{c}}{1 + \frac {v}{c}})^{\frac {1}{2}} $$
We see that—contrary to the usual conception—ν = ∞, for _v_ = -_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 for _l´_ takes the form
$$ \cos \Phi' = \frac {\cos \Phi - \frac {v}{c}} {1 - \frac {v}{c} \cos \Phi} $$
This equation expresses the law of observation in its most general form. If Φ = π/2, the equation takes the simple form
v cos Φ′ = --- c
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
$$ A'^2 = A^2 \frac {(1 - \frac {v}{c} \cos \Phi)^2} {1 - \frac {v^2}{c^2}} $$
If Φ = 0, this reduces to the simple form
$$ A'^2 = A^2 \frac {1 - \frac {v}{c}} {1 + \frac {v}{c}} $$
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.
§ 8. Transformation of the Energy of the Rays of Light. Theory of the Radiation-pressure on a perfect mirror.
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 and _k_ being equal. Yet this is not the case. If _l_, _m_, _n_ are 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 system _k_, _i.e._, the energy of the light-complex relative to the system _k_.
Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time τ = 0, the equation:—
$$ (\beta \xi - l \beta \frac {v}{c} \xi)^2 + (\eta - m \beta \frac {v}{c} \xi)^2 + (\zeta - n \beta \frac {v}{c} \xi)^2 = R^2 $$
If S = volume of the sphere, S′ = volume of this ellipsoid, then a simple calculation shows that:
$$ \frac {S'}{S} = \frac {\beta}{\sqrt{1 - \frac {v}{c} \cos \Phi}} $$
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
$$ \frac {E'}{E} = \frac {\frac {A'^2}{8\pi} S'}{\frac {A^2}{8\pi}S} = \frac {1 - \frac {v}{c} \cos \Phi}{\sqrt{1 - \frac {v^2}{c^2}}} $$
If Φ = 0, we have the simple formula:—
$$ \frac {E'}{E} = (\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}})^{\frac{1}{2}} $$
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 from _k_, we have the corresponding magnitudes:
$$ A' = A \frac{1 - \frac{v}{c} \cos \Phi}{\sqrt{1 - \frac{v^2}{c^2}}} $$
$$ \cos \Phi' = \frac{\cos \Phi - \frac{v}{c}}{1 - \frac{v}{c} \cos \Phi} $$
$$ \nu' = \nu \frac{1 - \frac{v}{c} \cos \Phi}{\sqrt{1 - \frac{v^2}{c^2}}} $$
For the reflected light we obtain, when the process is referred to the system _k_:—
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:—
$$ A''' = A'' \frac{1 + \frac{v}{c}\cos \Phi''}{\sqrt{1 - \frac{v^2}{c^2}}} = A \frac{1 - 2\frac{v}{c} \cos \Phi + \frac{v^2}{c^2}}{1 - \frac{v^2}{c^2}} $$ ,
$$ \cos \Phi''' = \frac{\cos \Phi'' + \frac{v}{c}}{1 + \frac{v}{c}\cos \Phi''} = - \frac{(1 + \frac{v^2}{c^2}) \cos \Phi - 2 \frac{v}{c}} {1 - 2 \frac{v}{c} \cos \Phi + \frac {v^2}{c^2}} $$ ,
$$ \nu''' = \nu'' \frac{1 + \frac{v}{c} \cos \Phi''}{\sqrt{1 - \frac{v^2}{c^2}}} = \nu \frac{1 - 2 \frac{v}{c} \cos \Phi + \frac{v^2}{c^2}} {(1 - \frac{v}{c})^2} $$
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
$$ P = 2 \frac{A^2}{8\pi} \frac{(\cos \Phi - \frac{v}{c})^2} {1 - (\frac{v}{c})^2} $$
As a first approximation, we obtain
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.
§ 9. Transformation of the Maxwell-Hertz Equations.
Let us start from the equations:—
$$ \frac{1}{c}(\rho u_{x} + \frac{\partial X}{\partial t}) = \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} $$
$$ \frac{1}{c}(\rho u_{y} + \frac{\partial Y}{\partial t}) = \frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} $$
$$ \frac{1}{c}(\rho u_{z} + \frac{\partial Z}{\partial t}) = \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} $$
$$ \frac{1}{c} \frac{\partial L}{\partial t} = \frac{\partial Y}{\partial z} - \frac{\partial Z}{\partial y} $$
$$ \frac{1}{c} \frac{\partial M}{\partial t} = \frac{\partial Z}{\partial x} - \frac{\partial X}{\partial z} $$
$$ \frac{1}{c} \frac{\partial N}{\partial t} = \frac{\partial X}{\partial y} - \frac{\partial Y}{\partial x} $$
where
$$ \rho = \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} + \frac{\partial Z}{\partial z} $$
denotes 4π times the density of electricity, and (_u__{_x_}, _u__{_y_}, _u__{_z_}) 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 system _k_ with the aid of the transformation-equations given in § 3 and § 6, then we obtain the equations:—
$$ \frac{1}{c} (\rho' u_{\xi} + \frac{\partial X'}{\partial \tau}) = \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} $$ ,
$$ \frac{\partial L'}{\partial \tau} = \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} $$ ,
$$ \frac{1}{c} (\rho' u_{\eta} + \frac{\partial Y'}{\partial \tau}) = \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} $$ ,
$$ \frac{\partial M'}{\partial \tau} = \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} $$ ,
$$ \frac{1}{c} (\rho' u_{\zeta} + \frac{\partial Z'}{\partial \tau}) = \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} $$ ,
$$ \frac{\partial N'}{\partial \tau} = \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} $$ ,
where
$$ \frac{u_{x} - v}{1 - \frac{u_{x}v}{c}} = u_{\xi} $$ ,
$$ \frac{u_{y}}{\beta(1 - \frac{vu_{x}}{c^2})} = u_{\eta} $$ ,
$$ \rho' = \frac{\partial X'}{\partial \xi} + \frac{\partial Y'}{\partial \eta} + \frac{\partial Z'}{\partial \xi} = \beta(1 - \frac{vu_{x}}{c^2}) \rho $$ ,
$$ \frac{u_{x}}{\beta(1 - \frac{vu_{x}}{c^2})} = u_{\zeta} $$ ,
Since the vector (_u__{ξ}, _u__{η}, _u__{ζ}) is nothing but the velocity of the electrical mass measured in the system _k_, 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.
§ 10. Dynamics of the Electron (slowly accelerated).
Let us suppose that a point-shaped particle, having the electrical charge _e_ (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
_d²x_ _d²y_ _d²z_ _m_ ----- = _e_X, _m_ ----- = _e_Y, _m_ ----- = _e_Z _dt²_ _dt²_ _dt²_
Where (_x_, _y_, _z_) are the co-ordinates of the electron, and _m_ is its mass.
Let the electron possess the velocity _v_ at 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 velocity _v_ along the X-axis of the system. It is clear that at this moment (_t_ = 0) the electron is at rest relative to the system _k_, which moves parallel to the X-axis with the constant velocity _v_.
From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the system _k_, the electron moves according to the equations
_d²_ξ _d²_η _d²_ζ _m_ ----- = _e_X′, _m_ ----- = _e_Y′, _m_ ----- = _e_Z′ , _d_τ² _d_τ² _d_τ²
in the time immediately following the moment, where the symbols (ξ, η, ζ, τ, X’, Y’, Z’) refer to the system _k_. If we now fix, that for _t_ = _v_ = _y_ = _z_ = 0, τ = ξ = η = ζ = 0, then the equations of transformation given in § 3 (and § 6) hold, and we have:
_v_ τ = β(_t_ - ---- _x_), ξ = β(_x_ - _vt_), η = _y_, ζ = _z_, _c²_
_v_ _v_ X′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M) _c_ _c_
With the aid of these equations, we can transform the above equations of motion from the system _k_ to the system K, and obtain:—
(A)
$$ \frac{d^2 x}{dt^2} = \frac{e}{m} \frac{1}{\beta} X $$ ,
$$ \frac{d^2 y}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Y - \frac{v}{c} N) $$ ,
$$ \frac{d^2 z}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Z + \frac{v}{c} M) $$
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
_d²x_ _m_β² ----- = _e_X = _e_X′ _dt²_
_d²y_ _v_ _m_β² ----- = _e_β (Y - --- N) = _e_Y′ _dt²_ _c_
_d²z_ _v_ _m_β² ----- = _e_β (Z - --- M) = _e_Z′ _dt²_ _c_
and let us first remark, that _e_X′, _e_Y′, _e_Z′ 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:
$$ \frac{m}{(\sqrt{1 - \frac{v^2}{c^2}})^{\frac{3}{2}}} $$
Transversal mass:
$$ \frac{m}{\sqrt{1 - \frac{v^2}{c^2}}} $$
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 ∫_e_X_dx_. 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 equations A) holds, we obtain:—
$$ W = \int eXdx = \int_0^v m\beta^3 vdv = mc^2 (\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1) $$
For _v_ = _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 equations A).
1. From the second of equations A), it follows that an electrical force Y, and a magnetic force N produce equal deflexions of an electron moving with the velocity _v_, when Y = N_v_/_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 A_{_m_}, and the electric deflexion A_{_e_}, by applying the law:—
$$ \frac{A_{m}}{A_{e}} = \frac{v}{c} $$
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 velocity _v_ which is acquired is given by the following relation:—
$$ P = \int Xdx = \frac{m}{e}c^2 (\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1) $$
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 equations A) we obtain:
$$ - \frac{d^2y}{dt^2} = \frac{v^2}{R} = \frac{e}{m} \frac{v}{c} N \sqrt{1 - \frac{v^2}{c^2}} $$
or
_mv_β_c_ R = ---------- _e_N
These three relations are complete expressions for the law of motion of the electron according to the above theory.
Footnote 6:
_Vide_ Note 9.
Footnote 7:
_Vide_ Note 9.
Footnote 8:
_Vide_ Note 12.
ALBRECHT EINSTEIN [_A short biographical note._]
The name of Prof. Albrecht Einstein has now spread far beyond the narrow pale of scientific investigators owing to the brilliant confirmation of his predicted deflection of light-rays by the gravitational field of the sun during the total solar eclipse of May 29, 1919. But to the serious student of science, he has been known from the beginning of the current century, and many dark problems in physics has been illuminated with the lustre of his genius, before, owing to the latest sensation just mentioned, he flashes out before public imagination as a scientific star of the first magnitude.
Einstein is a Swiss-German of Jewish extraction, and began his scientific career as a privat-dozent in the Swiss University of Zürich about the year 1902. Later on, he migrated to the German University of Prague in Bohemia as ausser-ordentliche (or associate) Professor. In 1914, through the exertions of Prof. M. Planck of the Berlin University, he was appointed a paid member of the Royal (now National) Prussian Academy of Sciences, on a salary of 18,000 marks per year. In this post, he has only to do and guide research work. Another distinguished occupant of the same post was Van’t Hoff, the eminent physical chemist.
It is rather difficult to give a detailed, and consistent chronological account of his scientific activities,—they are so variegated, and cover such a wide field. The first work which gained him distinction was an investigation on Brownian Movement. An admirable account will be found in Perrin’s book ‘The Atoms.’ Starting from Boltzmann’s theorem connecting the entropy, and the probability of a state, he deduced a formula on the mean displacement of small particles (colloidal) suspended in a liquid. This formula gives us one of the best methods for finding out a very fundamental number in physics—namely—the number of molecules in one gm. molecule of gas (Avogadro’s number). The formula was shortly afterwards verified by Perrin, Prof. of Chemical Physics in the Sorbonne, Paris.
To Einstein is also due the resuscitation of Planck’s quantum theory of energy-emission. This theory has not yet caught the popular imagination to the same extent as the new theory of Time, and Space, but it is none the less iconoclastic in its scope as far as classical concepts are concerned. It was known for a long time that the observed emission of light from a heated black body did not correspond to the formula which could be deduced from the older classical theories of continuous emission and propagation. In the year 1900, Prof. Planck of the Berlin University worked out a formula which was based on the bold assumption that energy was emitted and absorbed by the molecules in multiples of the quantity _h_ν, where _h_ is a constant (which is universal like the constant of gravitation), and ν is the frequency of the light.
The conception was so radically different from all accepted theories that in spite of the great success of Planck’s radiation formula in explaining the observed facts of black-body radiation, it did not meet with much favour from the physicists. In fact, some one remarked jocularly that according to Planck, energy flies out of a radiator like a swarm of gnats.