Nuclear physicist Ernest Rutherford (1871–1937) conducted experiments with scattered alpha particles that showed that Coulomb’s Law is accurate even for charged particles having nuclear dimensions and even forr val- ues as low as 10−12 centimeters. (Alpha particlesare helium nuclei, and they consist of two protons and two neutrons bound together.) In fact, today, experiments have demonstrated that Coulomb’s Law is valid over a remarkable range of separation distances, from as small as 10−16meters (a tenth of the diameter of an atomic nucleus) to as large as 106 meters.
Coulomb’s Law is accurate only when the charged particles are stationary because movement produces magnetic fields that alter the forces on the charges.
Note that a coulomb is an extremely large electric charge compared to the charge of a single electron or proton. To get a feel for the magnitude, consider two objects, each with a net charge of +1 coulomb. If you were to place these objects a meter apart, the repulsive force would be about nine billion newtons, which corresponds to one million tons! Because the coulomb is such a huge charge, scientists sometimes use smaller measure- ment units like the microcoulomb (10−6 C), picocoulomb (10−12 C), or even simply the charge of the electron, e (1.602×10−19C).
Coulomb’s Law and Newton’s Law of Universal Gravitation are exam- ples of what physicists sometimes refer to as “action at a distance” laws—
in the sense that when the laws were formulated, no known media- tor of the interaction existed. Newton’s law describes the gravitational attraction of masses m1 and m2 separated by a distancer and may be written Fg = Gm1m2/r2, where Fg is the magnitude of the force due to gravity.
Even a casual examination of the mathematical formulations of New- ton’s law and Coulomb’s Law reveals that the two formulas bear strik- ing similarities. Both the electrostatic force and gravitational forces are directly proportional to the product of the interacting entities (mass or charge), and both the forces are inversely proportional to the square of the distance of separation.
For both Newton’s Law of Universal Gravitation and Coulomb’s Law, one might think that the respective forces involved are instantaneously affected by a change in locations of the relevant objects. However, this is not the case. For example, for Coulomb’s Law, if one of the two charges is moved, then the force acting on the second charge does not immediately change. We know from Einstein’s Special Theory of Relativity that signals do not propagate faster than the speed of light; thus, if one charge is moved, then a time delay must exist for the second particle to “become aware” of this movement. Moreover, if the first charge were suddenly plucked from the experiment, the second charge would be sensitive to this removal only some time later.
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The same kind of delay applies to masses with a gravitational attraction, such as the case for Earth circling the Sun. If the Sun were suddenly removed, Earth would keep orbiting about the missing Sun for several minutes because the gravitational influence cannot travel faster than light speed. During the time needed for this influence to propagate, one body continues to experience an electrical or gravitational influence from the other body as if the missing object still existed.
Despite some similarities, a noticeable difference exists between the Newton’s Law of Universal Gravitation and Coulomb’s Law—the Coulomb force can be attractive or repulsive while the gravitational force is only attractive. Also, the magnitude of the Coulomb force depends upon the medium separating the charges, while the gravitational force is independent of the medium. For example, our first term in Coulomb’s Law may be written more generally usingεinstead ofε0:
k= 1 4πε,
where the permittivity ε is an electrical property of the medium that surrounds the two charges. The symbolε0 denotes the permittivity when the medium is a vacuum. The value ofk, sometimes known as Coulomb’s constant, is approximately equal to 9×109N·m2/C2whenε=ε0. Electri- cally conducting media have permittivity values greater thanε0. Because a vacuum has no charge carriers, the permittivity is lower for a vacuum than for any other medium. The permittivity value of dry air is so close to that of a vacuum that scientists usually treat experiments in the air as if performed in a vacuum.
The permittivity of a material is usually given relative to that of free space. If the relative permittivity is denoted by εr, permittivity is then calculated by multiplyingε0byεr. Approximate room-temperature relative permittivity values are given in table 6, and the values may vary according to temperature and the precise composition of the material under study.
For example, a range of permittivity values exists for different kinds of paper.
Coulomb’s Law is accurate only for point charges, that is, charges that are localized to an infinitely small region of space. However, all real- world experiments are performed with charges on objects that have finite dimensions. Coulomb’s Law may be used in experiments with such objects if the dimensions of the charged objects are much smaller than the distance between their centers. Note that in modern times, the law has been gen- eralized to integral and differential forms that may be used for nonpoint charges, and often these generalizations are also referred to as Coulomb’s Law.
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table 6 Relative Permittivity Values
Material Approximate Relative Permittivity
Values,εr, at 300◦K
Vacuum 1 (by definition)
Air 1.0005
Polyethylene 2.2
Lucite (trade name for a clear plastic)
2.8
Cocaine 3.1
Paper 3.3
Mica, muscovite 5.4
Rubber, Neoprene 6.6
Bone, cancellous (spongy) 26
Methyl alcohol 32
Brain, gray matter 56
Water (20◦C) 80
Lead titanate 200
Source: Glenn Elert, “Dielectrics,” in The Physics Hypertextbook; see hypertextbook.com/physics/electricity/dielectrics/.
Although the coulomb repulsive force should be quite strong for pos- itively charged protons within a nucleus, the protons do not fly apart because they are held together by another fundamental force, the strong nuclear force, which is stronger than the coulomb force.
I conclude this section with a short problem that shows a practical calculation involving Coulomb’s Law. Imagine two small balls, each with a mass of 0.20 grams. They are each attached to a separate 50-cm-long thread that is tied to the same point in the ceiling. Because the two balls have the same charge, they dangle from the ceiling and do not touch each other. In turns out that in this particular experiment, each
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thread makes a 37-degree angle with respect to a perpendicular to the ceiling. To help visualize this problem, draw a triangle. The topmost point represents the attachment point of the threads, and the left and right vertices represent the positions of the balls. If we assume that the charges on each ball are the same, we can determine how large the charge is.
In order to solve this problem, we can use simple trigonometry, while realizing that the weight of an object is equal to its mass m times the acceleration due to gravity,g(which is 9.8 m/s2). First, consider the ball on the left. Three forces act on the ball: its weight downward (mg), the tension Ton the string, and the repulsion force Fdue to the charge on the ball at the right. Because the balls are not moving, forces in thexandydirections are in balance. Thus, for the force in thexdirection, we haveFx– 0.6T= 0.
Considering the forces in the ydirection, we have 0.8T – (0.2)(10−3 kg) (9.8 m/s2)= 0, which yieldsT= 2.45×10−3N. Then, we can calculate the force Fx = 1.47×10−3N. This is the force of repulsion between the two balls. We may substitute this into the formula for Coulomb’s Law to solve for the charge on the ball:
1.47×10−3=(9×109) q2 (0.60)2
(The distancer between the two spheres is 0.60 m, which can be deter- mined by trigonometry, given the 50-cm length of string and 37◦ angle.) Solving forq, we find thatqis approximately equal to 2.4×10−7coulombs or 0.24µC, whereµC is the symbol for microcoulombs.
In a system of many point-charges, charges exert forces on one another, and the resultant force exerted on any one charge is the vector sum of the individual forces exerted on that charge by all the other charges in the system.
Charles-Augustin de Coulomb (1736–1806), French physicist famous for the law that describes the force between two electrical charges.
CURIOSITY FILE: Coulomb’s engineering skills played a large role in the fortification of Martinique, a Caribbean island. • yC is the official unit for yoctocoulomb, which is 10−24 coulombs. • Coulomb won a prize offered by the Académie des Sciences for the best method of constructing a compass for a ship.
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Coulomb’s contributions to the science of friction were exceptionally great. Without exaggeration, one can say that he created this science.
—I. V. Kragelsky and V. S. Schedrov,Development of the Science of Friction
Coulomb can be considered one of the great engineers in eighteenth-century Europe.
—C. Stewart Gillmor, “Charles Coulomb,” inDictionary of Scientific Biography
Who could forget “Chuck” Coulomb’s 1773 address to the Academy of Science in Paris when he discussed pio- neering soil mechanics theory?
—Terence Meany and Matthew Tirschwell, The Com- plete Idiot’s Guide to Electrical Repair
Charles-Augustin de Coulomb is one of the preeminent physicists and engineers of all time who contributed to the fields of electricity, magnetism, applied mechanics, friction, and torsion. Coulomb was born to a well-to- do family in Angoulême in southwest France. His family later moved to Paris, where he entered the Collège Mazarin. Here he received a good general education in the humanities as well as in mathematics, astronomy, and chemistry.
At some point, his father lost all his money in financial speculations.
This hardship, along with Coulomb’s disagreement over career plans with his mother, caused a split in the family, and Coulomb and his father moved to Montpellier while his mother stayed in Paris. According to some sources, Coulomb’s mother wanted him to be a medical doctor, but her son insisted on studying a more quantitative subject such as engineering or mathematics. The disagreements became heated, and his mother virtually disowned him.
In 1760, Coulomb entered the École du Génie at Mézières and later graduated as an engineer with the rank of lieutenant in the Corps du Génie (Corps of Engineers). Over the next two decades, he was posted to a variety of locations where he was involved in structural engineering, forti- fication design, and soil mechanics—for example, he spent several years in the West Indies as a military engineer—before returning to France, where he would begin to write his important papers on applied mechanics.
Coulomb created a torsion balance around 1777 in order to measure electrostatic forces. The torsion balance contains two metal balls attached to an insulating rod. The rod is suspended at its middle by a nonconducting filament or fiber. To measure the electrostatic force, one of the two balls is charged. A third ball with similar charge is placed near the charged ball of
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the balance, causing the ball on the balance to be repelled. This repulsion causes the fiber to twist by a certain amount. If we measure how much force is required to twist the wire by the same angle of rotation, we can estimate the degree of force caused by the charged sphere. In other words, the fiber acts as a very sensitive spring that supplies a force proportional to the angle of twist. Coulomb showed that the force varied as 1/r2for repulsion between like charges, and attraction between unlike charges, separated by an initial distancer. It appears that he never actually demonstrated that the force between charges is proportional to the product of the charge values—
he simply asserted this to be true. C. Stewart Gillmor, writing inDictionary of Scientific Biography, indicates the degree to which Coulomb’s balance affected science for many generations:
Coulomb’s simple, elegant solution to the problem of torsion in cylinders [with graduated scales] and his use of the torsion balance in physical applications were important to numerous physicists in succeeding years. . . . Coulomb developed a theory of torsion in thin silk and hair threads. Here he was the first to show how the torsion suspension could provide physicists with a method of accurately measuring extremely small forces.
In particular, Coulomb showed with his experiment that the exponent of r(the charge separation distance) was 2 within a few percent uncertainty.
Today, we know that the exponent is 2 within about 2 parts in 109. In 1779, Coulomb began his research into friction, which eventually led to his important publication Théorie des machines simples, en ayant égard au frottement de leurs parties et à la raideur des cordages(“Theory of Simple Machines, with Regard to the Friction Between their Parts and the Rigidity of the Linkages”). This work was followed twenty years later by a memoir on viscosity. Coulomb’s Law of Friction states that for two surfaces in relative motion, the kinetic friction is almost independent of the relative speed of the surfaces.
Coulomb’s research in friction was stimulated by a prize offered by the Academy of Science in Paris for “the solution of friction of sliding and rolling surfaces, the resistance to the bending in cords, and the application of these solutions to simple machines used in the navy.” According to Peter J. Blau, writing inFriction Science and Technology,
Coulomb’s researches and conclusions about the nature of friction dominated thinking in the field for over a century and a half, and many of his concepts remain in use. In fact, the term “Coulombic friction” is still found in publications that interpret the results of recent experiments. . . .
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Between 1785 and 1791, Coulomb wrote seven important papers on electricity and magnetism, which he submitted to the Académie des Sci- ences. Topics included his continued use of the torsion balance to under- stand attraction, repulsion, distribution of electricity on the surface of charged objects, and his demonstration of the inverse square law for magnetic poles. His 1785 work, which described the use of his different kinds of torsion balances, was published in hisRecherches théoriques et expérimentales sur la force de torsion et sur l’élasticité des fils de metal (“Theoretical and Experimental Studies on the Twisting Strain and Elas- ticity of Metal Wires”). Here, he showed that the torsion balance could be used to accurately measure extremely small forces.
In 1802, Coulomb married Louise Françoise LeProust Desormeaux, the mother of his two sons who were born before marriage. Louise was in her twenties. Toward the end of his life, Coulomb particularly enjoyed being in the country and teaching science to his youngest son, Charles.
During his last days, Coulomb contracted a fever that finally killed him.
His funeral services were held at Abbaye de St.-Germain-des-Prés.
Of Coulomb’s scientific prowess, Ioan James writes in Remarkable Physicists: From Galileo to Yukawa:
He has been described as the complete physicist, rivaled in the eighteenth century only by Henry Cavendish, combining experi- mental skill, accuracy of measurement, and great originality with mathematical powers adequate to all his demands.
A lunar crater with a diameter of 89 kilometers was named after Coulomb and approved in 1970 by the International Astronomical Union General Assembly.
Throughout his career, Coulomb conducted a variety of research and made contributions to our understanding of
r
rupture of masonry piers and beamsr
the sheer of brittle materialsr
the physics of vaulted archesr
friction of machinery and fluid resistancer
design of windmillsr
elasticity of metal and silk fibers and of soil mechanicsr
magnetic compass designr
efficiency of human and animals workers (ergonomics)Several people discovered aspects of Coulomb’s Law before Coulomb. As far back as 1750, the British Reverend John Michell (1724–1793) published studies that showed that attraction and repulsion between the poles of magnets varied inversely as the square of the distance between them.
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Michell’s torsion balance was subsequently used by Henry Cavendish (1731–1810) to help measure the density of Earth. Coulomb had invented his torsion balance around 1784, and although Michell’s work probably preceded Coulomb’s, their discoveries were made independently.
John Robinson (c. 1725–?), a British doctor, measured the electrostatic forces of attraction and repulsion in 1769, and his experiments suggested that electrical repulsion had an 1/r2.06dependency, and electrical attraction had an 1/rcdependence in whichc<2. From these results, he suggested that 1/r2was probably correct. Also, the English chemist Joseph Priestley (1733–1804) had suggested the 1/r2law of electric force. Priestly wrote in The History and Present State of Electricity:
May we not infer from this experiment [with charged hollow con- ductors] 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; since it is easily demonstrated that were the Earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another?
Although Priestly offered no convincing proof for Coulomb’s Law, his speculations were essentially correct. Priestly also independently invented the torsion balance and used it to show that the force between two mag- netic poles varies as the inverse square of the distance between the poles.
Today, we call the 1/r2 law Coulomb’s Law in honor of Coulomb’s independent results gained through the evidence provided by his torsional measuring system. In other words, Coulomb provided convincing quanti- tative results for what was, up to 1785, often a good guess.
The Coulomb force is relevant at atomic-size scales, and in fact, it is instructive to compare the gravitational forces and Coulomb forces for a hydrogen atom. As an approximation, if we think of the electron as a point particle orbiting a point-particle proton, with the electron separated from the proton by a distance of about 5.3×10−11meters on average, the Coulomb force can be calculated by
kq2
r2 = (9×109)(1.6×10−19)2
(5.3×10−11)2 =8.2×10−8N.
The magnitude of the gravitational forceFg between the proton and elec- tron can be approximately determined using the mass of the electronme
and protonmp: Fg = Gmemp
r2 = (6.67×10−11)(9.1×10−31)(1.67×10−27) (5.3×10−11)2
=3.6×10−47N 160 | a r c h i m e d e s t o h a w k i n g
Notice that the Coulomb force is significantly greater than the gravitational force between the two subatomic particles.
In closing, note that the Eiffel Tower in Paris features the names of 72 great French scientists and other thinkers—including Coulomb. Gustave Eiffel’s original engraving of these names was painted over in the early 1900s but restored in 1987. The letters in the names are approximately 60 centimeters tall. In the following list, I have highlighted in boldface those people listed on the tower who also have main entries in this book (only last names appear on the tower):
1. Ampère(André-Marie Ampère, mathematician and physicist) 2. Arago (Dominique François Jean Arago, astronomer and physi-
cist)
3. Barral (Jean-Augustin Barral, agronomist, chemist, physicist) 4. Becquerel (Antoine Henri Becquerel, physicist)
5. Bélanger (Jean-Baptiste-Charles-Joseph Bélanger, mathemati- cian)
6. Belgrand (Eugene Belgrand, engineer) 7. Berthier (Pierre Berthier, mineralogist)
8. Bichat (Marie François Xavier Bichat, anatomist and physiolo- gist)
9. Borda (Jean-Charles de Borda, mathematician)
10. Breguet (Abraham Louis Breguet, mechanic and inventor) 11. Bresse (Jacques Antoine Charles Bresse, civil engineer and
hydraulic engineer)
12. Broca (Paul Pierre Broca, physician and anthropologist) 13. Cail (Jean-François Cail, industrialist)
14. Carnot (Nicolas Léonard Sadi Carnot, mathematician) 15. Cauchy (Augustin Louis Cauchy, mathematician)
16. Chaptal (Jean-Antoine Chaptal, agronomist and chemist) 17. Chasles (Michel Chasles, geometer)
18. Chevreul (Michel Eugène Chevreul, chemist) 19. Clapeyron (Émile Clapeyron, engineer)
20. Combes (Émile Combes, engineer and metallurgist) 21. Coriolis (Gaspard-Gustave Coriolis, engineer and scientist) 22. Coulomb(Charles-Augustin de Coulomb, physicist)
23. Cuvier (Baron Georges Leopold Chretien Frédéric Dagobert Cuvier, naturalist)
24. Daguerre (Louis Daguerre, artist and chemist) 25. De Dion (Albert de Dion, engineer)
26. De Prony (Gaspard de Prony, engineer)
27. Delambre (Jean Baptiste Joseph Delambre, astronomer) 28. Delaunay (Charles-Eugène Delaunay, astronomer)
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