$$ \frac{1}{c}(\rho u_{x} + \frac{\partial X}{\partial t}) =
\frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} $$
$$ \frac{1}{c}(\rho u_{y} + \frac{\partial Y}{\partial t}) =
\frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} $$
$$ \frac{1}{c}(\rho u_{z} + \frac{\partial Z}{\partial t}) =
\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} $$
$$ \frac{1}{c} \frac{\partial L}{\partial t} = \frac{\partial
Y}{\partial z} - \frac{\partial Z}{\partial y} $$
$$ \frac{1}{c} \frac{\partial M}{\partial t} = \frac{\partial
Z}{\partial x} - \frac{\partial X}{\partial z} $$
$$ \frac{1}{c} \frac{\partial N}{\partial t} = \frac{\partial
X}{\partial y} - \frac{\partial Y}{\partial x} $$
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.
But Einstein found a support for the new-born concept in another
direction. It was known that if green or ultraviolet light was allowed
to fall on a plate of some alkali metal, the plate lost electrons. The
electrons were emitted with all velocities, but there is generally a
maximum limit. From the investigations of Lenard and Ladenburg, the
curious discovery was made that this maximum velocity of emission did
not at all depend upon the intensity of light, but upon its wavelength.
The more violet was the light, the greater was the velocity of emission.
To account for this fact, Einstein made the bold assumption that the
light is propagated in space as a unit pulse (he calls it a Light-cell),
and falling upon an individual atom, liberates electrons according to
the energy equation
1
_h_ν = --- _mv²_ + A,
2
where (_m_, _v_) are the mass and velocity of the electron. A is a
constant characteristic of the metal plate.
There was little material for the confirmation of this law when it was
first proposed (1905), and eleven years elapsed before Prof. Millikan
established, by a set of experiments scarcely rivalled for the
ingenuity, skill, and care displayed, the absolute truth of the law. As
results of this confirmation, and other brilliant triumphs, the quantum
law is now regarded as a fundamental law of Energetics. In recent years,
X-rays have been added to the domain of light, and in this direction
also, Einstein’s photo-electric formula has proved to be one of the most
fruitful conceptions in Physics.
The quantum law was next extended by Einstein to the problems of
decrease of specific heat at low temperature, and here also his theory
was confirmed in a brilliant manner.
We pass over his other contributions to the equation of state, to the
problems of null-point energy, and photo-chemical reactions. The recent
experimental works of Nernst and Warburg seem to indicate that through
Einstein’s genius, we are probably for the first time having a
satisfactory theory of photo-chemical action.
In 1915, Einstein made an excursion into Experimental Physics, and here
also, in his characteristic way, he tackled one of the most fundamental
concepts of Physics. It is well-known that according to Ampere, the
magnetisation of iron and iron-like bodies, when placed within a coil
carrying an electric current is due to the excitation in the metal of
small electrical circuits. But the conception though a very fruitful
one, long remained without a trace of experimental proof, though after
the discovery of the electron, it was generally believed that these
molecular currents may be due to the rotational motion of free electrons
within the metal. It is easily seen that if in the process of
magnetisation, a number of electrons be set into rotatory motion, then
these will impart to the metal itself a turning couple. The experiment
is a rather difficult one, and many physicists tried in vain to observe
the effect. But in collaboration with de Haas, Einstein planned and
successfully carried out this experiment, and proved the essential
correctness of Ampere’s views.
Einstein’s studies on Relativity were commenced in the year 1905, and
has been continued up to the present time. The first paper in the
present collection forms Einstein’s first great contribution to the
Principle of Special Relativity. We have recounted in the introduction
how out of the chaos and disorder into which the electrodynamics and
optics of moving bodies had fallen previous to 1895, Lorentz, Einstein
and Minkowski have succeeded in building up a consistent, and fruitful
new theory of Time and Space.
But Einstein was not satisfied with the study of the special problem of
Relativity for uniform motion, but tried, in a series of papers
beginning from 1911, to extend it to the case of non-uniform motion. The
last paper in the present collection is a translation of a comprehensive
article which he contributed to the Annalen der Physik in 1916 on this
subject, and gives, in his own words, the Principles of Generalized
Relativity. The triumphs of this theory are now matters of public
knowledge.
Einstein is now only 45, and it is to be hoped that science will
continue to be enriched, for a long time to come, with further
achievements of his genius.