All About Inverse Square Law

Universe and Inverse are closely related, for the effects of known forces of universe are inverse in nature. Ever since Newton made a bold proposition that there is a force of attraction amongst the planets of planetary system, amongst different planetary systems and amongst the different galaxies and varies as Inverse Square of their distances, people began to correlate every activities of nature and predict its effect accurately. 1. Kepler’s laws

Tycho Brahe’s largest and most accurate collections of planetary observations helped Kepler to present the first ever natural laws First law: Two planets P1 and P2 move on separate ellipses with the Sun at a common focus. Second law: For planet P1 the line SP1 sweeps out equal area in equal intervals of time. Third law: The square of the time taken by P1 and P2 to complete their respective orbits are in proportion to the cube of their respective semi-major axes. First law indicates clearly what path the orbiting planets follow. Second law explains about the speed of a planet i. . when the planet is nearest to the Sun (focus) speed is less e. g. at perihelion. When planet is farthest to the Sun e. g. at the aphelion the speed is more, so that at equal intervals of time the line joining the Sun and planet sweeps out equal areas. Third law is purely qualitative and compares the orbital motion of the different planets i. e. the ratio of square of period of revolution to the cube of semi-major axes is constant. If T1 and T2 are periods of two planets P1 and P2 and a1 and a2 are their respective semi-major axes then T12a13 = T22a23 =constant

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The Kepler’s laws are a-posteriori i. e. derived by experience. Hence none of the laws state why planets follow elliptical orbit why not straight lines. In those days one of the theories proposed was that the planets went around the elliptical orbits because behind them were the invisible angels beating their wings and driving the planets forward. Newton freed nature from an all embracing god-relatedness of Kepler’s era by making postulates which allow the determination of concepts such as mass, inertia, force, cause, space, time and motion.

Newton had become the first systematizer of modern science. Before Newton Galileo discovered a very remarkable fact about motion, that is the principle of inertia- if something is moving with nothing touching it and completely undisturbed, it will go on for-ever, coasting at a uniform speed in a straight line. Newton modified his idea saying that the only way to change the motion of a body is to use force. If the body speed up, a force has been applied sideways, i. e. a force is needed to change the speed or the direction of motion of a body. . Inverse square law of gravitation The fact that the planets does not move in a straight line with uniform speed indicates that there is a force acting on it and it is proportional to the acceleration according to first and second laws of Newton. Newton found the acceleration of a planet by two methods: the first conventional method he used was the method of geometry, the second method was the use of new technique of mathematics, which Newton himself was invented, the calculus. Figure 1. Illustration of Kepler’s laws.

Two planets P1 and P2 are shown to move on separate ellipses with the Sun S as a common focus. For planet P1, for example, the second focus is at S’. The line joining S to S’ and extended in both directions meets the orbit of P1 at A and A’. The point A is closest to the Sun and is called Perihelion. While A’ is the farthest point and is called aphelion. The length AA’ is called the major axis of the ellipse and is denoted by 2a. The ratio of lengths SS’ to AA’ is called the eccentricity of the ellipse. Refer to fig-1 and denote the distance SP1 by r and the angle P1SA’ by ?.

Then according to Kepler’s first law P1 moves along an ellipse with focus at S, its equation is given by 1r = 1l1-ecos? Where l is constant known as semi-latus rectum of ellipse, e is the eccentricity. The radial and transverse accelerations of P1 are given respectively by (fig-2) fr=d2rdt2-rd? dt2 (1) ft=1rddtr2d? dt (2) The rate at which SP1 sweeps out the area is simply 12 r2d? t which is constant according to Kepler’s second law. i. e. 12 r2d? dt=constant Or r2d? dt=constant (say) h (3) Since r2d? dt is constant, from (2) it is clear that ft=0 and the force on P1 must be radial, i. e. along SP1. Using equation (3) the value of fr comes out to be fr=-h2lr2 (4) Negative sign indicates that the force is directed towards S. Area of the ellipse traced by P1 is given by A=-? l21-e232 Hence the time taken to trace this area must be T=2Ah

Since the rate of sweeping of area is h/2, we therefore get T=-2? l2h1-e232 The major axis of the ellipse is 2a=2l1-e2 From Kepler’s second law T2a3=constant So we have 4? 2l4h21-e23l31-e23=4? 2lh2=constantsay 1K (5) Therefore h2=4? 2lK . Substituting this in (4) fr=-4? 2Kr2 =-? r2 (6) Where ? constant This is the inverse square law. Note that the acceleration of P1 varies inversely as the square of the distance from S. if mp is the mass of the planet; the second law of motion then gives the force on the planet towards the Sun as FP=mP? fP=? mpr2 (7) From third law of motion the same amount of force planet exerts on the Sun in the opposite direction i. e. FP=GMmPr2? FS Where M is mass of the Sun, G is constant of gravitation. From ( 5) and (6)? 4? 2a3T2 , and does not change from planet to planet, without loss of generality can write as: ?=GM This law of attraction can be generalized to any two bodies A and B of masses mA and mB separated by a distance r. FAB=GmAmBr2 (8) This is the complete mathematical statement of Newton’s law of gravitation. 3. Inverse square law of electricity Though now we have only one electromagnetic field, at the beginning they were separately studied, experimented as electric and magnetic.

These two properties were first observed from an amber and magnetic iron ore. The former, when rubbed, attracts light bodies, the latter has the power of attracting iron. In 1747 Franklin did experiment by using rubber and glass and observed that the total quantity of electricity in any isolated system invariable. This is usually known as principle of conservation of charge, and he is the man who named the two opposite charges, one as positive and another as negative. He also showed that the like charges attracts and unlike charges repel each other.

In 1767 Joseph Priestley published that “the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances….. ” But in 1769 Dr. john Robison of Edinburgh determined experimentally that the force of repulsion between two similar charges varies inversely as 2. 06th power of their distance and the force of attraction between two unlike charges varies inversely as less than the 2th power of their distance and he conjured that the correct power was 2.

In 1771 Henry Cavendish in his paper only mentioned that the force between electric charges is inversely as some less power of the distance than cube. But later he did rediscover the exact inverse square law but not published. In 1785 it is Augustine Coulomb who finally established the inverse square law, which states that the two stationary electric charges repel or attract one another with a force proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. i. e. force exerted by q1 on q2, F12=kq1 q2r212r21

Similarly force exerted by q2 on q1, F21=kq1 q2r122r12 Where r21 and r12 are the unit vectors along the line joining the q1 and q2. The first discoverer of the law of force between the magnetic poles was John Michelle (1724-93), he states “whenever any magnetism is found, whether in the magnet itself, or any piece of iron, excited by the magnet, there are always found two poles, which are generally called North and South; and North pole of one magnet always attracts the South pole and repels the North pole of the another, and vice versa. Each pole attracts or repels exactly at equal distances in every direction.

The attraction or repulsion of magnets decreases as the squares of the distances from the respective poles increase”. In 1777 it is again Coulomb who confirmed Michelle’s law and went beyond this and endeavored to account for the fact that the magnetic poles, unlike the two electric charges, cannot be broken into two pieces one containing its North and the other its South pole, it is found that each piece is an independent magnet possessing two poles of its own, so that it is impossible to obtain a North or South pole in state of isolation. 4. Validity of inverse square laws

Both the inverse square laws, gravitation and electromagnetism, are similar to each other except) that one concerns with the quantity of the body (mass) and other concerns about character of the body (charge). Therefore the domains of their influences are different. The validity of gravitational force between two objects for a given distance solely depends on the bulkiness of the objects (masses). Since the G (gravitational constant) is of the order of 10-11 to feel the gravitational force between the two objects the magnitude of product of their masses should be of the order of 1011 Kg. n the other hand the electronic charge of an electron is 1. 602? 10-19 C. to feel the electromagnetic force between two charges the distance between them must be of the order of10-38 m. Let us compare these forces for the simplest atom that of hydrogen, which contains only two particles: one proton and one electron. The coulomb force of attraction between them is FC=e-er2 =-(4. 8? 10-10esu)2r2 The Newtonian force of attraction between them is FN=-Gmpmer2 =-6. 668? 10-8? 1. 67? 10-24? 9. 11? 0-28r2 FCFN=4. 8? 10-10esu26. 668? 10-8? 1. 67? 10-24? 9. 11? 10-28 =0. 227? 1040 Where G=6. 668×10-8 cm3gm-1s-2 mP = 1. 67×10-24 gm me = 9. 11×10-28 gm The ratio is about 1040, is independent of distance and is a fundamental constant of nature. So the Coulombian force is 1040 times stronger than the Newtonian force. Thus in the atomic scale the Newtonian force is not significant. Similarly Coulombian electric force ceases to be of any importance among large astronomical objects like stars, moons and galaxies.

On the other hand Newtonian gravitational force builds up to enormous strength because it is additive in terms of the accumulation of the matter therefore the gravitational force is a long range force, and its effects, although diluted by the inverse square law of distance, extends over hundred thousands of light years. The experiments carried out in 1772 by Henry Cavendish to test inverse square has an accuracy of 2%. Modern repetitions of Cavendish’s test have, in effect, checked the inverse square law over the distances of the order of nches or feet, to an accuracy of a few parts in 109. However the real question is not whether the exponent of distance is 2 or some other number like 1. 9998 is correct, but rather at what range of distances inverse square law breaks down. The experiments at higher precision and involving different regimes of size have been performed to study the distance dependence of the electrostatic law of force. The test of inverse square law is now quoted in one of the two ways: 1. Assume that the force varies as 1/r2+? and quote a value or limit for ?. 2.

Assume that the electrostatic potential has the Yukawa form r-1e-µr and quote a value or limit for µ or µ-1. Since µ=m? C/?, where m? is the assumed mass of the photon. The inverse square law is sometimes phrased in terms of an upper limit on m?. Laboratory experiments usually give ? and µ or m? , geomagnetic experiments give µ or m?. The original experiments with concentric spheres by Cavendish in 1772 gave an upper limit on ? of | ? |= 0. 02. About 100 years later Maxwell performed a very similar experiment and set an upper limit of | ? | ? 5×10-5.

The laboratory experiment performed by Plimpton and Lawton based on Gauss’s law gave | ? |< 2×10-9. And recent one by William, Filler and Hill gave a limit of ? = (2. 7±3. 1) x10-16. Measurement of Earth’s magnetic field, both on the surface and out from the surface by satellite observation permit the best limits to be set on ? or equivalently the photon mass m?. The surface measurements of Earth’s magnetic field gave the value m? < 4×10-48 gm or µ-1 > 1010 cm The laboratory and geophysical test show that on length scales of the order of 1 to 109 cm, the inverse square law holds with extreme precision.

At smaller distances we turn to less direct evidence often involving additional assumption. Rutherford’s analysis of scattering of alpha particles by thin foils substantiates the Coulomb’s law of force down to distances of the order of 10-11 cm, provided alpha particles and nucleus can be treated as classical point charges interacting statically, and the charge cloud of the electron can be ignored. All of these assumptions can be and have been tested within the framework of the validity of quantum mechanics, linear superpositions and other assumptions.

At still smaller distances relativistic quantum mechanics is necessary and strong interaction effects enter to obscure the questions as well as answers. Nevertheless elastic scattering experiments with positive and negative electrons at centre of mass energies 5 GeV have shown that quantum electrodynamics holds distances of the order of 10-15 cm. hence we can conclude that the photon mass cannot be taken to be zero over the whole classical range of distances and deep into the quantum domain as well. The inverse square is known to hold over at least 24 orders of magnitude in length scale. 5.

Inverse square law and relativity It was electromagnetic theory discovered by James Clerk Maxwell (1860) which necessitated to modify the Newton’s laws of dynamics or the electromagnetic theory itself. Einstein took the bold step and modified the former. He says “Enough of this, Newton forgive me; you found the only way which in your age was just possible for a man of highest thought and creative power. The concepts, which you created, are even today still guiding our thinking in physics, although we now know that they will have to be replaced by others farther removed from the sphere of immediate experience.

If we aim at a profounder understanding of relationships. ” Similarly Coulomb’s law also had to be modified when experiments in the nineteenth with rapidly moving charges began to produce results inconsistent with its predictions. But the modification needed in the law of gravitation goes deeper than in the case of Coulomb’s law. After having made an historical announcement of special theory of relativity in 1905 Einstein discovered that inverse square law of gravitation cannot co-exist consistently with the special theory of relativity.

The planetary orbits calculated according to general relativity using the Schwarzschild solution differ from the orbits calculated by the Newtonian theory by small amounts. The biggest estimated difference between the two theories is for the planet Mercury and is that the perihelion of Mercury advances at an angular rate of 43 angular seconds per century. The smallness of the effect, however, illustrates the fact that the difference between the predictions of the two approaches to gravity are expected to be small within our planetary system.

Apart from the precession of the perihelion of the Mercury two other tests were proposed and carried out soon after Einstein put forward the general theory of relativity in 1915. It was Sir Arthur Eddington’s effort that prompted astronomers to conduct observations in respect of the above two tests. One test involves the bending of light by massive objects. The second test involves the possible effects of non-Euclidean geometry on the measurement of time. However the difference between the predictions of general theory of relativity and Newtonian gravity is minute.

The general rule of thumb is that the gravitational effects at a distance R from a gravitating mass M are considered small if the parameter ?=GMC2R is small compared to unity- say less than 1 percent and in the cases of such small effects the prediction of the two theories differ by very small amounts. In such cases the simpler Newtonian framework is preferable to relativity from practical point of view. Nevertheless we must not lose sight of the conceptual shortcomings of Newtonian gravity, which led Einstein to think of general relativity.

In particular, whenever we encounter situations of strong gravitational effects (where 0. 1 < ? < 1) we should begin to suspect calculations based on Newtonian gravity and switch over to general relativity. If we look back to the history from Aristotle to Einstein, the interaction between man and nature has been changed right from 400 BC to 20th century. The nature was the laboratory and natural events were the experiments. As subtlety of the experiment increased precision in measurement also increased.

In the process of subtlety and precision one law could take precedence over other but fact that without the earlier one latter would not have discovered at all!. It said that in the beginning of 20th century, the situation was ripe for a major review of Newtonian mechanics, and, if not Einstein somebody else would have soon developed special theory of relativity. But the general theory of relativity is Einstein’s own creation, a gem emerging out of his geniuses. Acknowledgements. Author is grateful to Prof. A. W.

Joshi, retired Professor of Physics, University of Pune; for his guidance in writing this article. Key words: Inverse square, Kepler’s law, gravitation, relativity Suggested readings: [1] Jayant Narlikar, Violent Phenomenon in the Universe, Oxford University Press. 1989 [2] Jayant Narlikar, The primeval Universe, Oxford University Press. 1988 [3] Richard P Feynman, Feynman Lectures on Physics; vol. 1, Narosa Publishing House. 1998 [4] Edward M Purcell, Electricity and Magnetism; vol. 2; 2nd edition, Berkeley Physics course,

Mc Graw Hill International Book Co. Singapore. 1985 [5] Keith Gibbs, Advanced Physics, Cambridge University Press. 1988 [6] Robert Resnick, Introduction to Special Relativity, Wiley Eastern Limited New Delhi. 1990 [7] J. D. Jackson, Classical Electromagnetism, Wiley Eastern, 1978 [8] Werner Heisenberg, Physicist’s Conception of Nature, Hutchinson, London, 1958 [9] A. W. Joshi(Ed), A World View of Physics, South Asian Publications Pvt. Limited. New Delhi. 1999 About author Teaching Physics for undergraduate students since twenty four years.

Presently working towards Ph. D. in the field of Physics education at Indira Gandhi National Open University, New Delhi. Developing innovative experiments for the undergraduate laboratory courses. Umapati Pattar Associate Professor in Physics Vijayanagar College, Hospet – 583201 Karnataka State. e-mail: umapati. [email protected] com ——————————————– [ 2 ]. Transforming the coordinate system from Cartesian to elliptical and finding the rate of change of displacement of r [ 3 ]. in his autobiographical notes (18)


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