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?

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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