Appendix A Vector Algebra and Vector Calculus978-3-319-21816-8/1.pdfA vector can also be represented...
Transcript of Appendix A Vector Algebra and Vector Calculus978-3-319-21816-8/1.pdfA vector can also be represented...
Appendix AVector Algebra and Vector Calculus
A.1 Exercises
EX. A.1 (VECTOR AND SCALAR QUANTITIES). Many of the quantities with which weoften deal, such as temperature and age, are called scalar quantities. Other quanti-ties, such as velocity and force, are called vector quantities. What distinguishesthese two classes of quantities is that the latter possess an orientation in space, theformer do not. One speaks of a car moving “ten miles per hour eastward,” but notof a person being “ten years old eastward.” A vector quantity, �v, may be representedgraphically by an arrow, as shown in Fig. A.1. The length of the vector provides ameasure of its magnitude, such as 10 miles per hour. The orientation of the vectorrepresents its direction in space, such as eastward. A vector can also be representedmathematically in the form of an equation:
�v = vx. (A.1)
On the right side of this equation, the scalar quantity, v, describes the magnitudeof the vector, and the unit vector, x, describes the direction in which the vectoris oriented. In this case, vector �v points along the x-axis of a particular cartesiansystem of coordinates.
As an exercise describe each of the following using either a scalar or a vectorquantity; if it is a vector quantity, represent it both graphically and mathematically.(a) Water has a density of 1 g per cubic centimeter. (b) A compressed gas exertsa pressure of 100 p.s.i. on the bottom of a cylindrical flask. (c) A ball is whirledaround in a circle by a string which exerts a constant force of 5 N inward.
EX. A.2 (VECTOR ADDITION). The rules of vector algebra are a bit different thanthose of scalar algebra. For example, when adding vector quantities, one must con-sider not only their magnitudes, but also their relative orientations. Only in the casein which they are in the same orientation can their magnitudes be simply added likescalar quantities. Generally speaking, the addition of two vectors, �a and �b, may berepresented graphically by placing the “tail” of one vector against the “tip” of theother, and forming a new vector, �c = �a + �b, as shown in Fig. A.2. Vector addi-tion can be expressed mathematically by selecting a coordinate system and writing
© Springer International Publishing Switzerland 2016 457K. Kuehn, A Student’s Guide Through the Great Physics Texts,Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-21816-8
458 A Vector Algebra and Vector Calculus
Fig. A.1 A vector
Fig. A.2 Vector addition
Fig. A.3 The dot productformed from two vectorsdepends on the angle betweenthem
each vector in terms of this coordinate system. For example, the sum of two vectors,�m = mx and �n = ny, produces a vector quantity, �g which may be written as
�g = mx + ny. (A.2)
Conversely, any vector may be decomposed into a (non-unique) sum of two or morevectors which, when summed, would form the vector.
As an exercise in vector decomposition, suppose that a 100 g block rests atopa wedge whose acute angle is 30◦. Choose a convenient coordinate system, andexpress the force of gravity which is acting upon the block as a sum of two vectors,one which is directed parallel to the surface of the ramp, and one which is directedperpendicular to the surface of the ramp. What are the magnitudes and directions ofthese two vectors?
EX. A.3 (VECTOR MULTIPLICATION, PART 1: THE DOT PRODUCT). There are twomethods of multiplying vector quantities: the dot product and the cross product. Inthis exercise we will consider the first of these. The dot product (or scalar product)of two vector quantities, �a · �b, produces a scalar quantity, c, whose magnitude isgiven by
c = ab cos θab (A.3)
Here, a and b are the magnitudes of �a and �b and θab is the angle between �a and �b,as shown in Fig. A.3.
As an example of the dot product, the work done on an object by a force �F inmoving the object a distance �d is given by the dot product W = �F · �d. Consider
A.1 Exercises 459
Fig. A.4 The cross productformed from two vectors isperpendicular to both of them
once again a 100 g block resting at the top of a 30◦ wedge. Suppose that the surfaceof the ramp is 1 m long. Calculate the work done on the block as it slides from thetop to the bottom of the ramp (a) by the force which the earth exerts on the block,and (b) by the force which the ramp exerts on the block.
EX. A.4 (VECTOR MULTIPLICATION, PART 2: THE CROSS PRODUCT). The cross prod-uct (or vector product) of two vector quantities, �a × �b, produces a vector quantity,�c, whose magnitude is given by
c = ab sin θab (A.4)
and whose direction is perpendicular to the plane defined by vectors �a and �b, asshown in Fig. A.4. The relative orientations of �a, �b and �c may be expressed usingthe (second) right hand rule:1 if you point the index finger of your right hand in thedirection of �a and extend the remaining fingers of your right hand in the directionof �b, then your thumb, when extended, points in the direction of �c.
As an example of a cross product, the Lorentz force acting upon a charge, q,which is traveling at velocity �v through a magnetic field �B is given by the crossproduct
�F = q �v × �B. (A.5)
Suppose that a singly ionized cosmic particle travels horizontally and northwardsabove Ellesmere Island in Canada (the location of Earth’s magnetic pole) at a speedof approximately 400 km/s. The magnetic field strength here is approximately 5 n-T.(a) Calculate the magnitude of the Lorentz force acting on this ion. (b) Describe thetrajectory of the ion; does it continue in a straight line, or does it curve? If it curves,in which direction? (c) How much work is done on the ion by the earth’s magneticfield? Does the magnitude of its velocity change? Does its kinetic energy change?
EX. A.5 (VECTOR FIELDS). A vector field is a function which assigns a vectorquantity to each location of space. For example, the fluid velocity at each spatialpoint (x, y, z) within a stream of flowing water can be described by a vector field
1 Recall that we used the first right hand rule to determine the orientation of the magnetic field inthe vicinity of a current-carrying wire; see Ex. 7.4 of the present volume.
460 A Vector Algebra and Vector Calculus
Fig. A.5 A vector fielddepicting water velocity atvarious points in a flowingriver
�v(x, y, z). Vector fields are often represented graphically by selecting a discrete setof points in the region under consideration and drawing a vector at each of thesepoints so as to give a sense of the overall flow within the region (see Fig. A.5). The
electric field, �E, is another example of a vector field. It may be defined in terms ofthe force that would be exerted on a test charge, q, if it were placed in the vicinityof one or more source charges.
�E ≡ �F/q (A.6)
Generally speaking, the force field of a particular configuration of source charges isquite complicated, but the electric field surrounding a single point charge, Q, maybe obtained readily by using Coulomb’s law,
�F (r) = kqQ
r2r . (A.7)
Here k is Coulomb’s constant, r is the magnitude of the distance from the sourcecharge, Q, to the test charge, q, and r is a radially directed unit vector in a sphericalpolar coordinate system centered on Q. Combining Eqs. 30.1 and A.7, we find thatthe electric field in the vicinity of a charge Q is given by
�E(r) = kQ
r2r (A.8)
Notice that the electric field does not depend on the test charge q. In this sense,the electric field caused by the source charge is said to have an existence which isindependent of any other charges which might respond to the electric field.
As an exercise, write down an expression for the gravitational field, �g(r), sur-rounding the earth. The gravitational field of the earth is analogous to the electricfield surrounding a point charge. Can you make a sketch of this gravitational vectorfield? In which direction are the vectors pointing? Is the magnitude of this vectorfield uniform throughout space?
EX. A.6 (VECTOR CALCULUS, PART 1: LINE INTEGRATION). One may integrate avector field along a particular path between two points in space. This is referredto as a line integral of the vector field. Consider a vector field, �F , defined over aparticular region of space. Two points, a and b, lie in this region and are connectedby a curve C (see Fig. A.6). Curve C may be divided into a large number, N , ofconsecutive vectors of length �s which connect points a and b. Now at each pointalong the curve, one may calculate the dot product of the vector field with the vector
A.1 Exercises 461
Fig. A.6 A curve may bedescribed by a set of infinites-imal tangent vectors at eachpoint along the curve
−→�s. In the limit that N → ∞ and �s → 0, the sum of these dot products becomesthe line integral of the vector field along C:
limN→∞�s→0
N∑
i=0
�Fi · −→�si =
∫ b
a
�F · d�s (A.9)
Generally speaking, this integration is difficult to perform. But as a simple example,
suppose that a spatially uniform magnetic field, �B, is directed along the x-axis of aparticular cartesian system of coordinates. We wish to calculate the line integral ofthe vector field along a straight line, C, from x = 0 to x = xo. The line integral maybe written as:
∫
C
�B · d�s =∫ xo
0(Box) · (dx x)
=∫ xo
0Bo dx
= Boxo
(A.10)
In the second line of Eq. A.10, we have written the magnetic field as �B = Box
and the displacement vector along the x-axis as d�s = dx x. We then performedthe dot product and pulled Bo out of the integral, since it is spatially uniform. Asan exercise, consider the line integral of the same magnetic field (Box) around acomplete square loop of side length l lying in the x − y plane. What is the lineintegral along each side of the square? What is the value of the line integral aroundthe entire loop?
EX. A.7 (ELECTRIC FIELDS AND ELECTRIC POTENTIAL). Recall from the electricfield mapping laboratory (Ex. 28.2 in the present volume) that electric field linesare always perpendicular to equipotential lines. More specifically, the electric fieldat any location is equal to the gradient (or more simply put, the slope) of the elec-tric potential. Thus, the electric field points in the direction in which the potentialdecreases most rapidly. Conversely, the electrical potential difference, �V , betweentwo locations a and b may be calculated from the line integral of the electric fieldalong a path between these two points:
�V = −∫ b
a
�E · d�s. (A.11)
462 A Vector Algebra and Vector Calculus
Fig. A.7 A surface maybe described by a set ofinfinitesimal perpendicularvectors at each point on thesurface
As an exercise, suppose that the electric field in the region between two narrowlyseparated and oppositely charged parallel plates is given by �E = σ
2εx. Here, σ
is the surface charge density on each plate (in Coulombs per square meter), ε isthe electrical permittivity of the medium between the plates, and x is a unit vectorwhich is normal to the plates and which points from the positively to the nega-tively charged plate. (a) What is the potential difference between the plates in termsof their separation, d , and the surface charge density of each plate, σ? (b) For afixed electric potential difference, which can store more electric charge, a capacitorhaving nothing, or having mylar, filling the space between the plates?
EX. A.8 (VECTOR CALCULUS, PART 2: SURFACE INTEGRATION). One may also inte-grate a vector field over a particular area. This is referred to as a surface integral ofthe vector field. Consider a vector field, �F , defined over a particular region of space.A surface, A, lies in this region (see Fig. A.7). This surface may be divided into anumber, N , of smaller areas which are described by vectors which are perpendicularto the surface and which have length �A. At each location on the surface one maycalculate the dot product of the vector field with the area vector
−→�A. In the limit that
N → ∞ and �A → 0, the sum of these dot products becomes the surface integralof the vector field over A:
limN→∞�A→0
N∑
i=0
�Fi · −→�Ai =
b�a
�F · d �A (A.12)
Generally speaking, this involves a complicated two-dimensional integration. Butas a simple example, consider a spatially uniform magnetic field directed along thex-axis and a rectangular area which is described by the vector d �A = Aox. Thesurface integral is then
�A
�B · d �A =�
(Box) · (dA x)
=�
Bo dA
= BoAo
(A.13)
A.1 Exercises 463
In Eq. A.13, we have again performed the dot product and pulled Bo out of theintegral. As an exercise, calculate the surface integral of the same magnetic field(B0x) through a square area of side length l whose normal is in the direction �n =ax + by. For what values of a and b is this surface integral maximum? minimum?
Bibliography
Biographical Memoir of Albert Abraham Michelson 1852–1931, vol. XIX, National Academy ofSciences, 1938.
Ampère, A. M., New Names and Actions Between Currents, in A Source Book in Physics, editedby W. F. Magie, Source books in the history of science, pp. 447–460, Harvard University Press,Cambridge, Massachusetts, 1963.
Bence, J., The life and letters of Faraday, Longmans, Green and Co., London, 1870.Brewster, D., On the laws which regulate the polarisation of light by reflexion from transparent
bodies., Philosophical Transactions of the Royal Society of London (1776–1886), 105, 125–159, 1815.
Campbell, L., and W. Garnett, The Life of James Clerk Maxwell, Macmillan and Co., London,1882.
Coulomb, C., Law of Electric Force and Fundamental Law of Electricity, in Source Book inPhysics, edited by W. F. Magie, Source books in the history of science, pp. 408–413, HarvardUniversity Press, Cambridge, Massachusetts, 1963.
Descartes, R., The World or Treatise on Light, Abaris Books, 1979.Drosdoff, D., and A. Widom, Snell’s law from an elementary particle viewpoint, American Journal
of Physics, 73(10), 973–975, 2005.Faraday, M., Experimental Researches in Electricity, vol. 1, Taylor and Francis, London, 1839.Faraday, M., Experimental Researches in Electricity, vol. 3, Taylor and Francis, London, 1855.Faraday, M., A Course of Six Lectures on the Forces of Matter and Their Relations to Each Other,
Richard Griffin and Company, London and Glasgow, 1860.Faraday, M., The Chemical History of a Candle: to which is added a Lecture on Platinum, Harper
& Brothers, New York, 1861.Feynman, R.P., Leighton, R.B., and Sands, M.L., The Feynman Lectures on Physics, Commemo-
rative ed., Addison-Wesley Publishing Co., 1989.Franklin, B., Experiments and Observations on Electricity, fourth ed., Henry, David, London,
1769.Franklin, B., Fart Proudly, Enthea Press, Columbus, Ohio, 1990.Fresnel, A., Diffraction of Light, in Source Book in Physics, edited by W. F. Magie, Source books
in the history of science, pp. 318–324, Harvard University Press, Cambridge, Massachusetts,1963.
Gilbert, W., On the Loadstone and Magnetic Bodies and on the Great Magnet the Earth: a NewPhysiology, Demonstrated with Many Arguments and Experiments, Bernard Quaritch, London,1893.
Heaviside, O., Electromagnetic Theory, vol. 1, “The Electrician” Printing and Publishing Com-pany, London, 1893.
Heaviside, O., Electromagnetic Theory, vol. 3, “The Electrician” Printing and Publishing Com-pany, London, 1912.
© Springer International Publishing Switzerland 2016 465K. Kuehn, A Student’s Guide Through the Great Physics Texts,Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-21816-8
466 Bibliography
Helmholtz, H., On the Conservation of Force, in Popular Lectures on Scientific Subjects, edited byE. Atkinson, D. Appleton and Company, New York, 1885.
Helmholtz, H., On the Sensation of Tone as a Physiological Basis for the Theory of Music, fourthed., Longmans, Green, and Co., New York, Bombay and Calcutta, 1912.
Helmholtz, H., Science and Culture, The University of Chicago Press, 1995.Hertz, H., Electric Waves, 2 ed., Macmillan and Co., London, 1900.Hofmann, J. R., André-Marie Ampère: Enlightenment and Electrodynamics, Cambridge Science
Biographies, Cambridge University Press, 1996.Huygens, C., Treatise on Light, Macmillan, London, 1912.Jackson, J. D., Classical Electrodynamics, second ed., John Wiley & Sons, New York, 1975.Lemay, J. A. L., The Life of Benjamin Franklin, vol. 3, University of Pennsylvania Press, 2009.Livingston, D. M., The Master of Light: A biography of Albert A. Michelson, The University of
Chicago Press, 1973.Magie, W. F. (Ed.), A Source Book in Physics, Harvard University Press, Cambridge, Mas-
sachusetts, 1963.Maxwell, J. C., A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of
the Royal Society of London, 155, 459–512, 1865.Maxwell, J. C., On action at a distance, Proceedings of the Royal Institution of Great Britain, VII,
48–49, 1873–1875.Maxwell, J. C., A Dynamical Theory of the Electromagnetic Field, Wipf and Stock Publishers,
1996.McVittie, G., Laplace’s alleged “black hole”, The Observatory, 98, 272–274, 1978.Meleshko, V. V., Coaxial axisymmetric vortex rings: 150 years after Helmholtz, Theor. Com-
put. Fluid Dyn., 24, 403–431, 2010.Michelson, A. A., Light Waves and Their Uses, University of Chicago Press, Chicago, IL, 1903.Newton, I., Opticks: or A Treatise of the Reflections, Refractions, Inflections & Colours of Light,
4th ed., William Innes at the West-End of St. Pauls, London, 1730.Oersted, H. C., Selected Scientific Works of Hans Christian Oersted, Princeton University Press,
1998.Ørsted, H. C., Experiments on the effect of a current of electricity on the magnetic needle, Annals
of Philosophy, 16(4), 273–276, 1820.Schneider, N. H., Induction Coils: How to Make, Use and Repair Them, Spon & Chamberlain,
E. & F.N. Spon, New York and London, 1901.Smith, A. M., Descartes’s Theory of Light and Refraction: A Discourse on Method, Transactions
of the American Philosophical Society, 77(3), 1–92, 1987.Tait, P. G., James Clerk Maxwell, Proceedings of the Royal Society of Edinburgh, 10, 331–339,
1878–1880.Tricker, R. A. R., Early Electrodynamics, Pergamon, New York, 1965.Tyndall, J., Address Delivered Before The British Association Assembled At Belfast with Additions,
Longmans, Green, and Co., 1874.Tyndall, J., Six lectures on light delivered in America in 1872–1878, D. Appleton and Company,
New York, 1886.Wallace, W. A., The Problem of Causality in Galileo’s Science, The Review of Metaphysics, 36(3),
607–632, 1983.Young, T., Course of lectures on natural philosophy and the mechanical arts, vol. 1, Printed for
Joseph Johnson, London, 1807.
Index
Aaction-at-a-distance, 93, 347, 348, 381, 394,
395Ampère, André Marie, 83, 99, 299, 321, 356,
357, 364, 385Andersen, Hans Christian, 75animal
design, 228electricity, 100, 312, 366eye, 100, 146, 161motion, 217, 388
apparatusantenna, 426capacitor, 45, 51, 55, 58charge producers, 35compass, 24, 77crabwinch, 109current balance, 88, 92electric kite, 62electric pinwheel, 37electric spider, 39electrometer, 56, 87Faraday ice pail, 57frictional machine, 88galvanic battery, 77galvanometer, 75, 87, 302, 324generator, 132, 321interferometer, 449lever, 108Leyden jar, 39, 44, 55, 451lightning rod, 36, 65machine, 103opthalmoscope, 100overshot wheel, 107pendulum clock, 105, 116, 157pulley, 107steam engine, 117, 124telegraph, 90, 133
torsion balance, 69, 73voltaic pile, 85, 87, 131
Arago, Dominique François, 299, 301, 321,337
Aristotle, 1astronomy, 228
Jupiter, 163Orion Nebula, 157Saturn, 157stellar aberration, 238, 444
BBartholinus, Erasmus, 210, 260Bencora, Thebit, 4Bernoulli, Daniel, 102birefringence, 191, 259, 263
and crystal structure, 259, 275, 289Iceland crystal, 210, 211, 238particle theory, 214wave theory, 214
Boscovich, Roger, 234, 385Boyle, Robert, 165, 216, 384Bradley, James, 444Brewster, David, 262
CCarnot, Sadi, 125Cassini, Giovanni, 158Cavendish, Henry, 385, 389Cavendish, Thomas, 4circuit
Kirchoff’s rules, 437Ohm’s law, 314RC, 436resonance, 436RLC, 438
Clausius, Rudolph, 125, 129Collinson, Peter, 34Copernicus, Nicholaus, 162
© Springer International Publishing Switzerland 2016 467K. Kuehn, A Student’s Guide Through the Great Physics Texts,Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-21816-8
468 Index
Cortesius, Martinus, 3cosmology
celestial spheres, 384geocentrism, 1, 3, 290heliocentrism, 1vortex theory of gravity, 227
Costa, Philip, 29Cotes, Roger, 384Coulomb, Charles, 68, 385, 389
DDavy, Humphry, 102, 126, 299, 356Descartes, René, 161, 170, 198, 384diffraction, 253
aperture, 243corrugated surface, 243grating, 247radio waves, 428single slit, 178, 242solar corona, 242water waves, 214
doppler shift, 166Drake, Francis, 4
EEarth
diameter, 164electric field
cause, 349displacement, 412effect on matter, 397propagation, 349
electric potential, 55, 461electric tension, 84electromotive force, 84, 85, 314, 397hydrostatic analogy, 56
electricityarc discharge, 36attraction of parallel currents, 89, 93, 356charge conservation, 37, 48charge distribution, 58charging by induction, 52conventional current direction, 86Coulomb’s law, 68, 71current, 47, 48, 84electrolysis, 130electrostatics and electrodynamics, 84, 349frictional, 37ground, 65inertia, 407Joule heating, 342lightning, 64, 389one-fluid theory, 35
piezoelectricity, 84repulsion, 36, 52, 53, 71, 85striking distance, 64thermo-electric current, 133two-fluid theory, 34vitreous and resinous, 34
electromagnetic induction, 319, 323Arago’s wheel, 320discovery, 302, 310, 332eddy currents, 343Faraday’s law, 316generator, 338Lenz’s law, 309, 317motor, 343mutual, 399self-inductance, 408
energychemical potential, 116, 129conservation, 102, 123, 134, 344, 395elastic potential, 116, 122electromagnetic, 396, 397, 420, 421kinetic, 114, 123mechanical advantage, 108mechanical theory of heat, 117, 124perpetual motion, 100, 135potential, 123work, 103, 113work-kinetic energy theorem, 120
Epicurus, 384ether
composition, 170, 190, 442drag, 216, 443Kelvin endorses, 452Maxwell endorses, 399Michelson endorses, 451necessary, 165Tyndall endorses, 272Young endorses, 253
Euler, Leonhard, 83, 236
FFaraday, Michael, 99, 282, 299, 381, 386, 396Fay, Charles François du, 34Fermat, Pierre de, 198Ficinus, Marsilius, 3Fizeau, Hippolyte, 446fluid
Bernoulli’s principle, 382density, 215laminar and turbulent drag, 215non-Newtonian, 444Pascal’s principle, 214surface tension, 225
Index 469
viscosity, 69, 216, 226vortices, 100, 452
Foucault, Léon, 388, 441Franklin, Benjamin, 33, 67Fresnel, Augustin, 280, 389, 446
GGauss, Carl Friedrich, 81Gilbert, William, 1, 75, 99Guericke, Otto von, 33
HHamilton, William, 379heat
caloric theory, 125conduction, 125latent, 125mechanical equivalent, 127mechanical theory, 102, 129, 165, 214produced by friction, 125radiation, 125
Heaviside, Oliver, 411Helmholtz, Hermann von, 99, 423, 441Herschel, John, 321Hertz, Heinrich, 100, 423, 451Hipparchus, 290Hire, Philippe de la, 158Hooke, Robert, 174
Hooke’s law, 121Hopkinson, Thomas, 36Huygens, Christiaan, 99, 157, 210, 260
Huygens’ principle, 173, 177, 181, 184
Iinterference, 176
acoustic grating, 277constructive and destructive, 239microwave, 427Newton’s rings, 250, 383thin fibre, 241thin film, 249, 254, 255, 272Young’s two-slit experiment, 240
JJoule, James, 102, 127, 389
KKepler, Johannes, 1kinetic theory of gases, 129Kirchoff, Gustav, 423Krönig, August, 129
LLaplace, Pierre-Simon, 236Liebniz, Gottfried, 158light
absorption, 398blue eyes and blue sky, 292electromagnetic theory, 79, 395, 411, 412,
451intensity, 421particle theory, 185, 213, 217, 224, 234photoelectric effect, 425Poynting vector, 421radiation, 165, 225, 424scattering, 291, 296speed, 140, 161, 164, 171, 191, 236, 258,
388, 401, 427, 442, 446sunlight, 246wave theory, 161, 234, 250, 270, 290
Lorentz force, 93, 341, 357Lorentz, Hendrik, 449
MMüschenbroek, Pieter van, 34magnetic field, 81
Biot-Savart law, 81, 94created by electric current, 77, 81, 87exists, 93, 347, 364, 374, 375, 386Oersted’s ideas, 79right hand rule, 82units, 81
magnetismAmpère’s theory, 91, 301, 308, 312, 364cartography, 25cause, 27compass, 3demagnetization, 9, 303Earth’s, 5, 26iron, 9, 20, 28, 303law, 12, 21loadstone, 6magnetic flux, 316magnetic torque, 90, 93monopoles, 351, 353, 367polarity, 2saturation, 370
Malus, Étienne, 262mathematics
geometry, 159line integration, 460surface integration, 462vector addition, 73, 457vector cross product, 459vector dot product, 458
470 Index
vector field, 459vectors and scalars, 457
matteratomism, 160, 224, 227, 385chemical affinity, 224, 225, 397coefficient of restitution, 170, 226cohesion, 225combustion, 238compressibility, 191constitutive relations, 413density, 191, 216electric permittivity, 55, 349, 403, 412electrical conductivity, 36, 64, 78, 85, 86,
315four elements, 2frost-ferns, 275inertia, 191, 226inhomogeneous, 204insulator, 38, 397magnetic permeability, 78, 81, 365, 367,
404, 412molecular vortices, 388, 452optical properties, 192, 276, 280, 395paramagnetic and diamagnetic, 352, 367,
375phosphorescence, 238soap bubble, 250structure, 259, 276, 284, 390surface charge, 398vacuum, 45, 165, 190
Maxwell’s equations, 400, 412Ampère’s law, 404Ampère-Maxwell law, 406Faraday’s law, 405Gauss’s law, 403no-name law, 405
Maxwell, James Clerk, 99, 129, 299, 379, 411,449
Mayer, Julius, 102Michelson, Albert, 441Mithradates, 107momentum
conservation, 171Morse, Samuel, 133
NNewcomb, Simon, 441Newton, Isaac, 99, 157, 234, 260, 290
Newton’s cradle, 172rules of reasoning, 53universal law of gravitation, 382
OOersted, Hans Christian, 75, 83, 299, 357, 385Ohm, Georg, 314optics
angle of incidence, 140angle of minimum deviation, 155far-sightedness, 146focal length, 151, 153, 426, 430image, 145, 147, 153lenses, 146magnification, 148, 154near-sightedness, 146object distance, 153optical path length, 254ray diagram, 140, 143, 160, 206thin-lens equation, 153virtual image, 154
PPeregrinus, Petrus, 3Picard, Jean, 164Poisson, Siméon, 385polarization, 389, 443
Brewster’s angle, 262, 266, 269by scattering, 293Faraday rotation of light, 282, 350, 388, 396filter, 269, 422Nicol prism, 270Oersted’s ideas, 79particle theory, 212, 213, 238plane polarization, 261radio waves, 428, 430
Porta, Baptista, 26Ptolemy, Claudius, 25, 290
RRömer, Ole, 158, 163reflection
law of, 141, 154, 159, 182, 184, 237partial, 197, 237, 249radio waves, 429specular and diffuse, 185spherical mirror, 144total internal, 140, 151, 196, 238
refraction, 140, 348angle, 141atmospheric, 204dispersion, 238, 239, 252Fermat’s principle, 198, 201lenses, 143radio waves, 431refractive index, 192, 201, 254, 259, 401Snell’s law, 141, 159, 193, 211, 237, 259
Index 471
wave theory, 193, 206Regnault, Henri, 128
Sscience
causality, 1, 159, 160, 217, 224, 228empiricism, 385hypothesis, 229intelligibility, 162laws, 101, 228method, 160, 228, 229, 253moral, 102, 229music, 100natural theology, 217, 227, 228, 389order and disorder, 226–228pedagogy, 63practical, 101prediction, 384probability, 159reason, 102skepticism, 290speculation, 347, 361, 376teleology, 171truth, 290vitalism, 100, 111
soundphysiology, 161speed, 164, 170, 443
waves, 165Stokes, George, 294
TThompson, Benjamin (Count Rumford), 102Thomson, William (Lord Kelvin), 382, 388,
396, 452Torricelli, Evangelista, 165, 190Tyndall, John, 300, 411
VVolta, Alessandro, 299
Wwave
angular frequency, 420electromagnetic, 435period, 420plane wave, 418standing, 427, 435superposition principle, 172, 176, 245two-slit interference, 247wavelength, 240, 254, 290, 420, 451wavenumber, 420
YYoung, Thomas, 233, 389