Radius vector

“ Also the motion of the planet in its orbit was such that equal areas were described about the Sun by the radius vector of the planet in equal times.—Kepler, Astronomia Nova αἰτολογητός (Prague) , 1609. ”

“ At first, we want to circumscribe the formulas (34b) , so that only four-vectors related to the light-point occur
is the vector of to , the velocity vector in (see equation (33a) ) and the acceleration vector in . In the previous definition (equation (29b) ) of by the world-line element , since in our cyclic coordinates applies, it evidently becomes , thus according to equations (31)
consequently is follows:
thus has in the light-point the direction of radius and the magnitude . ”

“ From the mathematical standpoint the paper is a comprehensive statement, without a suggestion of theory, describing the known facts of Electro-magnetism in terms of a system of vectors supposed to exist at all points of the field. Precision is here given to Faraday’s idea of lines of [Pg 35] force, whether electric or magnetic, by means of the analogy of the motion of an incompressible fluid. The new vector here introduced into Electro-magnetism is that of momentum, and its rate of change is shewn by a dynamical, argument to be responsible for electro-magnetic induction. ”

“ If x, y, z, it (space co-ordinates, and time it) is subjected to a Lorentz transformation, and at the same time (convection-current, and charge density × i) is transformed as a space time vector of the 1st kind, further (magnetic force, and electric induction × i) is transformed as a space time vector of the 2nd kind, then the system of equations (I) , (II) , and the system of equations (III) , (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced info the system. ”

“ A straight line conceived in thisway as having a definite length, direction and sense, but no definite location in space, is called a vector. It may be denoted by AB (or CD) , or (when no confusion is likely to arise) simply by'AB. Thus a vector may be used to specify a displacement of translation (without rotation) of a rigid body. Again, a force acting on a particle, the velocity or momentum of a particle, the state of electric or magnetic polarization at a particular point of a medium, are examples of physical entities which are naturally represented by vectors. ”

“ There, denotes the boundary of the field, which can be determined by other bodies or by an (only imagined) surface; is the inner (scalar) product of vectors and , and the first member of the left-hand side is the "power of the inner forces". The second member contains the normal component of "Poyntings radiation vector".
and gives the radiation sent through the boundary by the electron. Power of the inner forces and radiation appear at the cost of the scalar quantity , which we will call "electromagnetic energy", and which occupies the field with the density ”

“ For a far distant point, one can view and as approximately constant for all charge elements and execute the three-times integration, where the total charge of the system emerges. Thus one has for the limiting case:
In consequence of Fig. 6 one can easily convince himself, that direction and magnitude of vector are only expressed by the state of motion at light-point . ”

“ The equations which express the relations between on one hand and x, y, z, t on the other, may be replaced by other equations, containing the vectors defined by (25) and the quantities x',y',z',t' defined by (4) and (5) . Now, by the above assumptions a and b, if in a particle A of the moving system, whose coordinates are x, y, z, we find an electric moment at the time t, or at the local time t', the vector given by (26) will be the moment which exists in the other system at the true time t' in a particle whose coordinates are x', y', z' . ”

“ We shall now attempt a similar experiment with the electric ray. A block of paraffin is fashioned into the shape of a semi-cylinder, flat on one side and convex on the other, and this plano-convex paraffin lens has a convex surface having a radius of 6 inches. If I place the [246] radiator A and receiver B about 4 feet apart, then by making a few little adjustments it is possible to so arrange matters that the radiation which proceeds from the radiator is not powerful enough at a distance of 4 feet to sensibly affect the coherer and make the bell ring (see Fig. 78) . ”

“ The vector-potentials of these currents are also radial, and their tensors are and . We have now merely to find their resultant when the linear element is indefinitely shortened, add on to the former qu/r, and multiply by μ0, to obtain the complete circuital vector-potential of qu, viz.:—
where r is the distance from q to the point P when A is reckoned, and the differentiation is to s, the axis of the convection-current. ”

“ Taking the second point first, it is, I think, clear that if by the propagation of vector-potential is to be understood that of electric and magnetic disturbances, it is merely the mode of expression that is in question. I am myself accustomed to mentally picture the electric and magnetic forces or fluxes, and their propagation, which takes place at the speed of light or thereabouts, because they give the most direct representation of the state of the medium, which, I think, must be agreed is the real physical subject of propagation. ”

“ We also arrive at a possible motion, when we take the mirror image of an arbitrary, given motion in relation to a stationary plane, namely in such a way that both the translational velocity, as well as all light-vectors are replaced by the mirror images.
If we are dealing with the pure aether, then we satisfy these conditions if we choose the dielectric displacement as the light vector. ”

“ The resultant of all the vectors thus found is the potential of the whole current. Since the current is a vector quantity, its potential is also a vector. See Art. 422.
When the distribution of electric currents is given, there is one, and only one, distribution of the values of , such that , is every where finite and continuous, and satisfies the equations
and vanishes at an infinite distance from the electric system. This value is that given by equations (5) , which may be written
Quaternion Expressions for the Electromagnetic Equations. ”

“ The transformation equation (11) is only applicable to the pure space components of the six-vector and is not reversible, i.e., it cannot serve to express the components of vector by those of vector . [17] There (as well as in the following) we understand under the total torque, thus the sum of torques of all individual forces. We see, that is not a six-vector like the six-vector of electrodynamics. ”

“ A radio-frequency band of 46.0 to 56.0 kilomegahertz. See also FREQUENCY, FREQUENCY BAND.
vector, n. Any quantity, such as a force, velocity, or acceleration, which has both magnitude and direction, as opposed to a SCALAR which has magnitude only. Such a quantity may be represented geometrically by an arrow of length proportional to its magnitude, pointing in the given direction.
vector, adj. A type of computerized display which consists of layers of differentiated data, each with discreet features. Individual data files can be independently manipulated. ”

“ We call and the rest vectors belonging to the spacetime vector with respect to velocity , because in that coordinate system in which a point moving with velocity appears to be at rest, the first three components of and are identical with the space vectors and , which compose the transformed vector of second kind . ”

“ The only use we have made of experimental knowledge is to recognise, in the abstract quantities deduced from the theory, the concrete quantities discovered by experiment, and to denote them by names which indicate their physical relations rather than their mathematical generation.
In this way we have pointed out the existence of the electromagnetic momentum as a vector whose direction and magnitude vary from one part of space to another, and from this we have deduced, by a mathematical process, the magnetic induction, , as a derived vector. ”

“ If we suppose the field to terminate at the surface of a conductor, inside which the vectors vanish, we must see that the "curl" relations of the field are not violated for circuits which lie partly inside the empty space enclosed by the conductor. In particular, if there is a vector whose line integral round every circuit in the field vanishes, the lines of this vector must meet the surface at right angles. Otherwise we should have a finite value for the integral round a circuit drawn close to the surface outside and completed inside. ”

“ Suppose the β-rays to be projectiles (electrons) negatively charged with electricity and shot from the radium with great velocity. Let be the mass of a projectile, its charge, its initial velocity, the radius of curvature of its path, the intensity of the magnetic field normal to the initial velocity, the magnetic permeability of the medium. We then have the easily established relation
The β-rays are also deviated in an electric field. ”

“ This physicist subjected a narrow beam of radium rays to the simultaneous action of an electric field and a magnetic field, the two fields being uniform and having a similar direction, normal to the original direction of the beam. The impression produced on a plate normal to the primitive beam and placed beyond the limits of the field with reference to the source, has the form of a curve, each point of which corresponds to one of the original beam. The most penetrating and least deflected rays are at the same time those with the greatest velocity. ”

“ During this variation the angular momentum of each of the electrons round the line connecting the nuclei will remain constant. If the distance apart of the nuclei increases, the radius of the ring will obviously also increase; the radius, however, will increase at a slower rate than the distance between the nuclei. For example, imagine a displacement in which the distance as well as the radius are both increased to a times their original value. In the new configuration the radial force acting on an electron from the nuclei and the other electrons is times that in the original configuration. ”

“ Dick, looking monstrous in heavy leaden surplice, goggles and gloves, grasped the projector controls and swept the tubes below with the ray. Peering over his shoulder Stan watched the vacuum tubes begin to glow as the V-rays bombarded, with billions of electrons, their radium-coated elements. First a point at the base of each tube became incandescent, grew, and finally resolved itself into a ruddy ball of fire and rolled to the top of the plates, where it exploded and spread over the quartz surfaces as an opal luminescence, turning them into swirling blazes of color— lavender, gold, red. ”

“ For, in general, the electromagnetic energy does not always flow in the outward direction, but flows alternately outwards and inwards, and we should obtain in this manner for the quantity of the radiation outwards, values which are alternately positive and negative, and which also depend essentially upon the radius of the supposed sphere in such manner that they increase toward infinity with decreasing radius—which is opposed to the fundamental conception of radiated energy. ”

“ If a thick layer of radio-active matter is employed, only the α particles from the surface have a range a. Those which reach the surface from a depth d have their range diminished by an amount ρd, where ρ is the density of the radio-active matter compared with air. This is merely an expression of the fact that the absorption of the α rays is proportional to the thickness and density of matter traversed. The rays from a thick layer of active matter will thus be complex, and will consist of particles of different velocity whose ranges have all values between 0 and a. ”

“ The β rays are the same as the cathode rays, while the γ rays resemble the Röntgen rays. In a vacuum tube, a large amount of electric energy is expended in producing the rays, but, in the radio-active bodies, the rays are emitted spontaneously, and at a rate uninfluenced by any chemical or physical agency. The α and β rays from the active bodies are projected with much greater velocity than the corresponding rays in a vacuum tube, while the γ rays are of much greater penetrating power than Röntgen rays. ”

“ It is thus probable that in all cases the α rays from the different radio-active substances consist of positively charged bodies projected with great velocity. Since the rays from radium are made up in part of α rays from the emanation stored in the radium, and from the excited activity which it produces, the α rays from each of these products must consist of positively charged bodies; for it has been shown that all the α rays from radium are deviated in a strong magnetic field.
157The kinetic energy of each projected particle is enormous, compared with its mass. ”

“ In general, however, the exerted radiation of a resting body has a resulting momentum and accordingly exercises a motion drive, as is can be most easily seen by the example of a cavity (of perfectly reflecting substance) filled with radiation, having a small opening only at a single point. But if the body is isotropic, homogeneous and limited by a convex surface everywhere, it suffers a normal pressure from the radiation which is everywhere the same, not resulting in a net force. If one such body is at rest in the primed system, then it has a constant speed in relation to the unprimed system. ”