Concepts in PhysicsMike Hobson

Lent Term 2007

Aims and Objectives

• Non-examinable, but . . .

• Consolidation of Core Material

• How Physics Really Works

• What the text-books don’t tell you

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The Case Studies

• Galileo and the Nature of the Physical Sciences

• The Origin of Maxwell’s Equations

• Thermodynamics and Statistical Physics

• Dimensional Analysis

• Chaos and Self-organised Criticality

• The Origins of the Concept of Quanta (1). Up to 1905.

• The Origins of the Concept of Quanta (2). After 1905.

• A Brief History of 20th Century Cosmology

• Relativity – Special and General

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The Books of the Course

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Case Study 1

The Galileo Caseand the Nature of the Physical Sciences

127 BC: Hipparchus, the first of the great astronomers – Catalogue of 850 stars in theNorthern Sky.

2nd century AD: Claudius Ptolemeaus, or Ptolemy, wrote the 13 volumes of The GreatComposition, which became known as the Almagest.

The Ptolemaic system of the world dominated astronomical thinking until the 16thcentury. The system had to account for the observed motions of the planets on the sky.The concept of epicyclic motion was needed to explain the motions of the planets.

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Epicyclic MotionThe motion of Saturn in AD 133

as observed by Ptolemy

Circular epicyclic motion

The system was based upon two keyconcepts from Greek mathematics. Theonly allowable motions were• uniform motion in a straight line• uniform circular motion

Although complicated, the systemworked well in order to predict thepositions of the Sun, Moon and Planets,which were needed for the preparationof almanacs and to determine thecorrect dates for religious festivals.

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Evidence for Circular Motion

It was obvious from the motionsof the stars that their motionsdefined perfect circles on the sky.

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Evidence for Circular Motion

It was obvious from the motionsof the stars that their motionsdefined perfect circles on the sky.

There was an alternative view:

3rd century BC: Letter from Archimedes toKing Gelon.

‘Aristarchos of Samos has publishedcertain writings on the (astronomical)hypotheses. The presuppositionsfound in these writings imply that theuniverse is much greater than wementioned above. Actually, he beginswith the hypothesis that the fixed starsand the Sun remain without motion.As for the Earth, it moves around theSun on the circumference of a circlewith centre in the Sun’.

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The Rejection of the Heliocentric Hypothesis

This picture was rejected by the followers of Aristotle for two physical reasons:

• If the earth rotated, why do you not observe an object thrown vertically upwards toland at a different spot?

• If objects are not supported, they fall under gravity. Therefore, if the Sun were thecentre of the Universe rather than the Earth, everything ought to fall towards thatcentre. But, if objects are dropped, they fall towards the centre of the Earth and nottowards the Sun.

Thus, religious belief was supported by scientific rationale. Hence the need to developthe picture of the epicyclic motion of the heavenly bodies about the stationary earth.

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The Ptolemaic System of the World

The basic Ptolemaic system with circularorbits.

With the passage of time, morecomplex epicyclic motions wereneeded to account for the details ofthe planetary orbits.

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The Copernican System of the World

Up till the time of Copernicus, the motions of the Moon and the planets were taken fromthe Alphonsine Tables prepared by the Rabbi Isaac Ben Sid of Toledo under thepatronage of Alfonso the Wise in 1277.

In the early 16th century, Aristarchus’s model was revived by Copernicus as providing asimpler description of the motions of the Sun, Moon and planets. The orbits were takento be circular about the Sun.

Copernicus’s work was presented to Pope Clement VII in 1533, who made a formalrequest for their publication. The first edition of De Revolutionibus Orbium Coelestium(On the Revolutions of the Celestial Spheres) was brought to Copernicus on hisdeathbed in 1543.

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The Copernican Model of the Solar System

The Copernican Universe fromCopernicus’s De Revolutionibus OrbiumCelestium

In his forward to De Revolutionibus,Osiander remarked that the Copernicanmodel was simply a calculating device,but, in the text, it is clear thatCopernicus really believed that theplanets orbit the Sun. This had twoimmediate implications:

• The size of the universe isincreased since no parallax of thefixed stars had been observed.

• Shouldn’t all objects fall to thecentre of the Universe?

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Thomas Digges’ Model of the Universe

In England, the Copernicanpicture was enthusiasticallyadopted by Thomas Digges. InThomas Digges’ version of theCopernican picture, the SolarSystem is embedded in aninfinite distribution of stars,which were assumed to beobjects like the Sun.

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Tycho Brahe (1543 – 1601)

Tycho Brahe was inspired to begin his great series of observations of the stars andplanets by the inaccuracies in the Alphonsine Tables.

With the sponsorship of Frederick II of Denmark, he constructed his great observatoryon the island of Hven. The instruments were the most advanced available, resulting inan improvement in accuracy of a factor of 10 over previous measurements.

Technical innovations:

• Systematic and random errors.

• Effects of the bending of the instruments.

• Effects of atmospheric refraction.

• Importance of the independence of the astronomical data.

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Tycho’s Observatory at Hven

Tycho built two observatories,Uraniborg and Stjerneborg. Thispicture shows the UraniborgObservatory with some of theinstruments he had constructed,including the Great Mural Transit Circleand the 7-foot globe on which thelocations of the stars were plotted.Note the use to numerous clocks toensure the precise measurement ofthe time of transit of the stars andplanets.

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Tycho Brahe’s Observatory Uraniborg on Hven

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Tycho Brahe’s Observatory Stjerneborg on Hven

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Tycho’s Achievement

Over a period of 20 years, Tycho measuredpositions accurate to about 1-2 arcmin for theSun, Moon, planets and 777 stars.He developed his own Tychonic Cosmology, acompromise between the Copernican andPtolemaic pictures. The planets orbit the Sun,but the Sun and planets orbit the Earth which isstationary at the centre of the Universe.According to Tycho, the Copernican

‘system, although not in accord withphysical principles, was mathematicallyadmirable.’

In 1600, the year before his death, he employedKepler to analyse his observations of the motionof Mars.

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Johannes Kepler (1571 – 1630)

Kepler was a mathematician of genius,who had the technical skill to analyseTycho’s magnificant set of data on theplanetary orbits. Unlike Tycho, he was aconvinced Copernican from thebeginning.

In 1597, he published his model for theSolar System, involving the embeddingof the 5 regular platonic solids withinone another. Remarkably, the modelcould account for the orbits to within anaccuracy of about 5%.

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From Kepler’s Mysterium Cosmographicum

“The Earth’s orbit is the measure of allthings: circumscribe around it adodecahedron and the circle containingit will be Mars; circumscribe aroundMars a tetrahedron, and the circlecontaining this will be Jupiter;circumscribe about Jupiter a cube andthe circle containing this will be Saturn.Now inscribe within the Earth anicosahedron and the circle contained init will be Venus; inscribe within Venus anoctahedron, and the circle contained init will be Mercury. You now have thereason for the number of planets.”

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Kepler’s Chronology

The publication of the Mysterium Cosmographicum brought Kepler European fame.Copies were sent to Tycho and Galileo. Central to the story is Kepler’s discovery of hisThree Laws of Planetary Motion.

• 1601 Succeeds Tycho as Imperial Mathematician at Prague.

• ∼ 1603 Discovery of the Second (Areal) law from the Earth’s motion about the Sun.

• 1605 Discovery of First law from study of the orbit of Mars

• 1609 Second law published in The New Astronomy

• 1610 Galileo’s Sidereus Nuncius published.

• 1619 Discovery of the Third Law while writing Harmonices Mundi

Notice that Kepler’s results came from a very deep understanding of geometry. All thecomputations were done by hand using geometric techniques.

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Kepler’s Second Law

‘Divine Providence granted to us such a diligent observer in Tycho Brahe that hisobservations convicted this Ptolemaic calculation of an error of 8 minutes of arc; it isonly right that we should accept God’s gift with a grateful mind ... Because these 8minutes of arc could not be ignored, they alone have led to a total reformation ofastronomy.’

Kepler’s Second Law

Equal areas are swept out by the line from the Sun to a planet in equal times.

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Kepler’s First Law

Kepler attempted to fit ovoids to the orbits of the planets, but still could not obtain agood enough fit. He needed a figure between the circle and the ovoid - the ellipse.

The New Astronomy, Based on Causes, or Celestial Physics was published in 1609,four years after he had made the discovery of the second law.

Kepler’s First Law

The planetary orbits are ellipses with the Sun in one focus.

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Galileo’s Observations of the Satellites of Jupiter

Two pages from Galileo’s SidereusNuncius

Galileo observed the motions of thesatellites of Jupiter each clear night from7 January to 2 March 1610. Kepler waselated by the discovery of this mini-solarsystem:

‘The conclusion is quite clear.Our moon exists for us onEarth, not for the other globes.Those four little moons exist forJupiter, not for us. Each planetin turn, together with itsoccupants, is served by its ownsatellites. From this line ofreason we deduce with thehighest degree of probabilitythat Jupiter is inhabited.’

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The Harmony of the World

In 1618, Kepler began writing asummation of all his previous work inthe influential treatise entitled ‘Harmonyof the World’. This may be consideredthe ‘Grand Unified Theory’ of its day, thesubjects including:• Geometry• Architecture• Harmony• Metaphysics• Psychology• Astrology• Astronomy• Metaphysics• Macrocosmos• Microcosmos

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Discovery of Kepler’s ThirdLaw of Planetary Motion

Kepler had reached the writing of Book V, Chapter III, 8th Division of the Harmony ofthe World, when he discovered the third law of planetary motion.

Kepler’s Third Law The period of a planetary orbit is proportional to the three-halvespower of the mean distance of the planet from the Sun.

This was the crucial discovery which eventually led to Newton’s law of gravity.

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Galileo Galilei (1564 – 1642)

Galileo was a physicist in the sense wewould recognise today. He was stronglyopposed to Aristotelian physics.

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Galileo Galilei (1564 – 1642)

Galileo was a physicist in the sense wewould recognise today. He was stronglyopposed to Aristotelian physics.

For example, according to Aristotle,‘If a certain weight move acertain distance in a certaintime, a greater weight will movethe same distance in a shortertime, and the proportion whichthe weights bear to each other,the times too will bear to oneanother; for example, if the halfweight cover the distance in x,the whole weight will cover it inx/2.’

This is just wrong. Galileo may wellhave tested the idea by droppingweights from the leaning tower of Pisa.

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Galileo Galilei as Physicist

Galileo began to take the Copernican theory seriously in order to explain the origin oftides in the Adriatic. A huge force is needed to raise the tide by 5 feet.

He began a magnificent set of experiments to eludicate the nature of motion. The threegreat achievements were:

• The law of acceleration, x =1

2at2.

• Galileo’s law – the time for free fall down the diameter of a circle equals the time toroll down a chord.

• The period of the swing of a long pendulum is independent of its amplitude.

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Galileo’s Physics Experiments

The law of acceleration, x =1

2at2

Galileo’s law

The period of the swing of a longpendulum is independent of itsamplitude.

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Galileo’s Telescopic Discoveries 1609 – 1612

The invention of the telescope by Hans Lipperhey was announced in 1608. Galileoconstructed his own telescopes which had magnifying power up to a factor 30 by early1610.

The Moon is mountainous rather than aperfectly smooth sphere.

The Milky Way was resolved into stars,rather than a uniform distribution of light.

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Galileo’s Telescopic Discoveries 1609 – 1612

The rings of Saturn, which he took tobe close satellites of the planet.

Jupiter has four satellites orbiting theplanet, like a miniature CopernicanSolar System (see before).

The phases of the planet Venus wereconsistent with the Copernican picture.

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The Galileo Affair

1615 Galileo’s letter to the Grand Duchess Christina.

1616 Galileo accused of suspicion of heresy.

1616 Galileo acquited, but the Congregation of the Index ruled that Copernicanism wasphilosophically and scientifically untenable and theologically heretical.

1624 Urban VIII concluded that Copernicanism could be discussed, provided it was onlyconsidered hypothetically.

1632 Galileo’s Dialogue on the Two Chief World Systems, Ptolemaic and Copernican

1632 Galileo condemned by the Inquisition.

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The Galileo Affair

Problems with the GeokineticHypothesis

1. The Deception of the Senses2. Astronomical Problems3. Physical Arguments4. The Authority of the Bible5. The Hypothetical Nature of the

Copernican theory

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Physical Problems

1. If the Earth moves, falling bodies should not fall vertically.

2. The speed of projectiles fired in the direction of motion of the Earth and in theopposite direction should be different if the Earth were rotating.

3. The Extruding Power of Whirling. Objects placed on a rotating potter’s wheel areflung off if they are not held down. Why are we not flung off the Earth if it isrotating?

4. Copernican picture was inconsistent with Aristotelian physics.

5. If Aristotelian physics was to be rejected, what was going to replace it?

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Hypothetical Nature of the Copernican Picture

In 1616, Cardinal Roberto Bellarmine wrote

‘. . . it appears to me that Your Reverence (Foscarini) and Signor Galileo didprudently to content yourself with speaking hypothetically and not positively, asI have always believed Copernicus did. For to say that, assuming the Earthmoves and the Sun stands still serves all appearances better than eccentricsand epicycles, is to speak well. This has no danger in it, and it suffices formathematicians.

But to wish to affirm that the Sun is really fixed in the centre of the heavens. . . is a dangerous thing, not only by irritating all the theologians and scholasticphilosophers, but also by injuring our holy faith . . . .’

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Deduction and InductionDeduction

• If it is raining, the streets are wet.

• It is raining.

• Therefore, the streets are wet.

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Deduction and InductionDeduction

• If it is raining, the streets are wet.

• It is raining.

• Therefore, the streets are wet.

Induction

• If it is raining, the streets are wet.

• The streets are wet.

• Therefore, it is raining.

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The Nature of the Physical Sciences

Galileo could not prove that the Copernican model was correct on the basis of theobservations he had available. For example, his observations of Venus would havebeen consistent with Tycho’s picture of the world. Even a sufficiently complicatedPtolemaic model could be devised to explain the observations.

The key point is that physics is a hypothetico-deductive process. We make hypothesesand see how economically we can explain observed physical phenomena. The besttheories are those which can explain large amounts of independent data quantitativelyand make predictions to new circumstances.

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The Nature of Physics

A scientifically satisfactory model has the capability of making predictions aboutapparently unrelated phenomena.

I use the word model in describing this process rather than asserting that it is in anysense truth. Galileo’s enormous achievement was to realise that the models to describenature could be put on a rigorous mathematical basis. In perhaps his most famousremark, he stated in his treatise Il Saggiatore (The Assayer) of 1624:

Book of Nature . . . is written in mathematical characters

This was the great achievement of the Galilean revolution. Notice that even theapparently elementary facts established by Galileo required a remarkable degree ofimaginative abstraction.

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Two New SciencesFollowing his condemnation in 1632, Galileo was placed under house arrest for the restof his life. He wrote Two New Sciences, summarising all the work he had accomplishedon the laws of motion.

• The relativity of motion – GalileanRelativity.

x′ = x − vt

y′ = y

z′ = z

t′ = t

• The Law of Inertia – this was tobecome Newton’s First Law ofMotion.

• The application of the laws ofacceleration and inertia to themotion of projectiles.

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Isaac Newton (1641 – 1727)

At the age of 22, while at home atWoolsthorpe because of the GreatPlague, Newton made the followingdiscoveries:

• The Binomial Series• The Differential Calculus• The Integral Calculus• The Theory of Colour in Optics• The Unification of Celestial

Mechanics and the Law of Gravity

Note: Newton was also a brilliantexperimental physicist.

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Newton and the Law of Gravity

Kepler’s Harmony of the World was in Trinity Library and the laws of planetary motionwere probably brought to Newton’s attention by Isaac Barrow. While at Woolsthorpe, hewrote that

The notion of gravitation [came to me] as I sat in contemplative mood [and]was occasioned by the fall of an apple.

First, he rederived the expression for the centripetal acceleration of an object moving ina circle of radius r at speed v

a =dv

dt=

v2

r.

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The Law of Gravity

There must therefore be a centripetal force, f = ma, holding the planets in their orbits.

Kepler’s third law states that the period T of the planet’s orbit is proportional to r3/2.The speed of a planet in its orbit is v = 2πr/T and so

v ∝1

r1/2

Therefore, the force which keeps the planets in their orbits must be

f = ma =mv2

r∝

1

r2

This is the primitive form of Newton’s inverse square law of gravity.

Now, if gravity is universal, the same force which binds the planets in their orbits, shouldalso cause apples to fall to the ground.

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Universal Gravity

The Moon is 60 times further away from the centre of the Earth than the apple is.Hence the centripetal acceleration of the Moon is only 1/602 = 1/3600 times that due togravity at the surface of the Earth.

The acceleration due to gravity at the surface of the Earth is 9.80665 m s−2.

The Moon’s centripetal acceleration is v2/r. The orbital period of the Moon about theEarth is 27.32 days. Its mean distance from the Earth is r = 384,408,000 m.Therefore, the mean speed of the Moon is v = 1,023 m s−1 andv2/r = 2.72 × 10−3 m s−2.

The ratio of the Moon’s acceleration to the local acceleration due to gravity is therefore

9.80665

2.72 × 10−3= 3,600

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Universal Gravitation

This is the origin of Newton’s law of gravity

f = −GM1M2

r2ir

There were, however, problems

• The orbits of the planets are ellipses, not circles

• What is the influence of the other planets?

• Is it correct to place all the mass of the Earth at its centre?

• Newton could not account for the details of the Moon’s motion.

By 1684, he had solved these problems.

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Newton’s Principia Mathematica

The results of Newton’s researches were eventually published in 1687 as hisPhilosophiae Naturalis Principia Mathematica.

Newton’s Laws of Motion

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After Newton’s Principia Mathematica

This was only one side of Newton’s character.

• Newton as Master of the Mint

• Newton the Alchemist

• Newton and the Interpretation of the Scriptures

His writings on Alchemy and the Scriptures were each as extensive as his researchesinto mathematics and natural philosophy.

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Alchemy and the Scriptures

Newton’s Alchemy

Newton’s classification of chemicalsubstances.

Newton and the Interpretation of theScriptures

Between 1733 and 1922, there weretwelve editions of Newton’s biblicalstudies.

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