A summary for the Final exam:

Topics: Fluids, Oscillations, Waves, Electricity andMagnetism, Quantum physics, Electrical properties of matter– Xerox machine, Magnetism and MRI, EM radiation – sunlight,spectrum, scattering, rainbow, Radio and TV and Nuclear reactors.

I. Fluids:

A. PRESSURE

Due to Random motion of constituents:kinetic theory of gases.

Pressure = Force per unit area = Newtons per square meter. = Joules per cubic meter.

Pressure is a scalar, it is the same in all directions in a fluid.

P=F/A= Nm2

=Nxmm3

= joulesm3

Hence pressure is energy density Atmospheric pressure = weight of 1m2column of air above you

Atmospheric Pressure = 105Newtonsm2

or 105Pascals

where 1 Pascal =1Nm2

If you go to a depth of H meters below the surface of waterat sea level the pressure increases as :

at depth H P=P0gH whereP0 is atmospheric pressure

Atmospheric pressure is equivalent to .76 m of mercury or10 meters of water. So at a depth of 30 meters the pressureis 4 atmpospheres.

B. Density:Concept of density is very useful in describing nature of matter.

Mass of a substance per unit volume is density. Density =ML3

=kgm3

Typical densities:

Air at sea level air=1.3kgm3

Water water=1000kg

m3

For other substances look up the lecture on fluids

C.Temperature:

Temperature is a measure of the average kinetic energy of molecules ofthe substance (either gas or liquid or solid). It is measured in degreesCelsius, Farhenheit or absolute or Kelvin scales. The relation betweenthe average kinetic energy and temperature is

Definition: k T=<12

Mv2 >=12

M<v2>

k is a conversion factor which connects T in 0K to energy in Joules

the value is :k=1.38110−23 Joulesparticles0K

D.Ideal Gas Equation:

Pressure is energy density. Multiplying average kinetic energyper molecule by the particle density one obtains also an energydensity. Hence the relation between Pressure and temperature is

If there are N molecules in volume V

the particle density is :particle=NV

The energy density is : NVk T

HenceP=NV

k T=particlek T

or PV=Nk T

k is called the Boltzmann constant.

Some observations about the ideal gas equations:

1. It describes equilibrium of an ideal gas – ideal gas is one inwhich the molecules only interact perfectly elastically when theycollide and otherwise move freely between collisions.

2. For an ideal gas confined to a fixed (constant) volume thepressure of the gas increases with temperature.: P∝T

3. If temperature is fixed the pressure of the gas varies inversly

as the volume: P∝ 1V

E. Buoyancy:

An object, wholly or partially immersed in a fluid is acted upon by anupward buoyant force equal to the weight of the fluid it displaces.

If the buoyant force is less than the force of weight of the objectitself it will sink, otherwise it will float, neutrally buoyant at any depthor will float to the surface.

Balloons: Consider a balloon which weighs WB and has avolume V, then the buoyance conditions are:

 Net force on the balloon is (downward is positive)  F net=W BW gas−W air

If the volume of the balloon is V then W gas=V gas g ;W air=V air g So if  the magnitude of  gas−airV g  is greater than W B

 the balloon will rise.

Note also that the density of gas inside the balloon depends on

1. what type of gas it is – mass of its molecules compared withthat of air molecuels

2. what is the temperature of the gas inside the balloon, like in case of hot-air balloons.

The height to which ballon will rise to is determined by when thebuoyant force equals the weight of the balloon and the weight of thegas inside it.

F. Fluid Flow: Equation of continuity and Bernoulli's energyequation.

If the density of the flowing fluid is constant we haveincompressible fluid, for example that of a liquid.

If the flow is non-turbulent then we have streamlined flow.

To get a flow there has to be a differential of pressure energy or adifferential of potential energy (water flows downhill). There are twoconservation laws which determine streamline flow:

1. Conservation of mass – equation of continuity2. Conservation of energy -Bernoulli's equation.

Equation of continuity: Consider flow through a pipe of varying diameters or cross sections. Amount of fluid flowing through must bethe same at any place along the pipe. Take two points along the flow, 1and 2. If the crossectional areas at these two points are A1and A2

and the speed of flow is v1andv2 respectively then equation ofcontinuity is

A1 v1=A2 v2 which gives the relation between speeds at differentpoints .

Bernoulli's equation: If the pressure energy density at a point is p, thepotential energy density due to gravity is gh where h is the heightabove some reference level and the kinetic energy density of flow is

12v2 then energy conservation is

pgh12v2=constant Bernoulli's equation.

Which states that sum of pressure energy density, potential energydensity and kinetic energy density is constant along a streamline.

Some consequences of these equations:

1. Narrower cross sections have a higher speed.2. Where speed is higher, pressure is lower for the same height

or faster the flow lower is the pressure.

Pressure differentials cause by different flow speeds can give rise to airplane lift, roof removal in a hurricane and curved balls in baseball.

II.Oscillations

Most important oscillations are Simple Harmonic oscillations. They arecause by restoring forces which are proportional to the displacementfrom equilibrium position. Such forces are called elastic forces andthese forces obey Hooke's Law:

Hooke's Law: F=−k x where x is measured from the equilibriumposition.

The motion followed is sinusodial oscillation.

Typical examples are mass hung by a spring from a support.Equilibrium position is when restoring force balances the weight of themass.

Important properties of the motion:

1. Motion is periodic – repeats itself.

2. Angular Frequency of oscillations is = km

3. Frequency is f=

2

4. Period of oscillation is T=1f=2mk

5. Units of force constant of the spring is Newtons/meter.6. The equation of motion is: x t=Asin t

where A is the amplitude and the phase at t=0.7. In this motion position, velocity and acceleration are all functions

of time. 8. When x = 0, F =0, v = max, henc KE = max, PE=09. When x = max, magnitude of F = max , PE = max and KE =010. The sum of KE + PE = constant in the motion.

SHM motion is the projection of circular motion (which is periodic) ineither the x or y direction.

T=2Inertiarestoring force constant

For the case of a simple pendulum. Inertia is proportional to mass andrestoring force due to gravity is also proporional to mass henc we get

T=2Lg

A mass attached to a spring will have the same period on earth as onthe moon, while a simple pendulum will move slower on the moon thanthe earth.

III.Wave motion:

1.Wave is a moving disturbance2. Waves can pass through each other3. Continuous waves are oscillatory; which means that

some quantity (displacement, pressure etc) is executingperiodic motion with a period T or frequence f = 1/T

4. Wave pulses do not have a single frequency5. After a wave has passed through a medium it returns to its

original undisturbed state.

To define a wave you must know its wavelength, frequencyamplitude and phase. Velocity. Frequency and wavelength are relatedin a wave: Fundamental relation of wave motion:

v=f ;

f= v

= vf

If the shape of the wave at time t=0 is described by the function f(x)then at a later time t, the wave should be described by f(x-vt), becausethe whole wave moves to the right preserving its shape. Moving to theright means that any location on the wave satistfies the simple relation x = vt+const or x – vt = const

So to completley define a wave we need to know:

1. Wave speed and direction or wave velocity2. Wave frequency or wavelength3. Wave amplitude 4. Wave phase5. Wave polarization for transverse waves.

In addition we must know what is the source generating the waves. The wave is mathematically described by:

y x ,t=Asinkx− t

here notice that the argument of the sign function must be an angle in

radians. So k is 2

, omega is 2f and phi is in radians. The

quantity k is called the wave number.

The quantity y represents what is oscillating in the wave. For waves ona string it is the displacement perpendicular to the string direction. Forsound waves it is pressure or density of air . For EM waves it is eitherthe electric or magnetic field.

These are the characteristics of propagating waves.

IV.Superposition and Characteristic Oscillations:

When waves pass through each other they get superposed. Theresulting amplitude at the positions and times when they overlappis determined by the algebraic sum of the two wave amplitudes. Thiscan lead to constructive (enhancement) and destructive amplitudes.Where a crest of one wave coincides with a trough of the other wavethere can be total cancellations and the medium remains undisturbed.

If two waves of slightly different frequencies are superposed thanphenomenon of beats occurs where the resultant wave has averagefrequency and a modulated amplitude at the difference frequency.

Confined Waves and Characteristic oscillations of a string tied attwo ends (and ofcourse under tension.

Waves on a string, characterized by a tension (Newtons) and a lineardensity (or mass in kg/meter) move with a speed which is

v= T which is meters per second

Note that the ratio T=kgms−2

kg /m=m2

s2

The end points of the string must be unmoving hence the oscillations ofthe string must be such as to have the wave form go to zero at theends. This leads to the following allowed wavelengths:

Longest wavelength1=2L; frequency f1=v 12L

Next wavelength2=L; frequency f2=2 v2L

=2f1

Next wavelength3=23

L;frequency f3=3f1

Next wavelength 4=24

L; frequencyf4=4f1

These are called the characteristic frequencies or harmonics ofoscillations of string tied at both ends.

Confined waves in vibrations of air columns:

There are two cases to be considered: (a) air column open at both ends and (b) air column closed at one end and open at the other.

(a) Here the characteristic modes of oscillations are the same as thatfor string tied at two ends.

(b) In this case the lowest wavelength is 1=4Landf1=v

4L. The next

allowed oscillations corresponds to a the following:

second harmonic 2=43

L;f2=3f1

third harmonic3=45

L;f3=5f1

only odd harmonics are allowed for this calse. These tubes (organ pipes or flute etc) are excited into vibrations by

blowing into them.

Standing waves pipe open at both ends:

Standing waves tube open at one end open at the other.

V.Electricity:

New force of nature, new property of matter: electric charge, Q

Comes in multiples of a basic quantum of charge, the charge on anelectron. Called negative by convention: - e

Coulomb's law: F=kQ1Q2

r2 attractive for unlike charges and

repulsive for like charges. Similar to Law of Universal Gravitation,

which is always attractive (indicated by the – sign) Fg=−Gm1m2

r2 and

here the fundamental property is gravitaional mass. The mass whichappears in Newton's II law is called the inertial mass. Equivalenceprinciple states that gravitational mass is the same as inertial mass.The constants k and G have to be determined by experiment – they areuniversal constants.

Fields

Surrounding a mass there is an gravitational field whose presence canbe detected by measuring the force on another mass placed in the field.

Surrounding an electrical charge is an electric field whose presencecan be detected by measuring the force on a positive charge placed inthe field.

Unit of mass is a kg.Unit of charge is called a Coulomb.

If we call electric field E then Force exerted on another charge Q bythis field is QE = Force = Newtons. Hence units of electric field areNewtons per Coulomb.

If a charge which is free to move is placed in a field it will move as thefield does work on the charge. After moving a distance d , the workdone in the field is QEd Joules. Work done on 1 Coulomb of charge inmoving through a distance d is called the Voltage (analogue ofgravitational potential energy) and Voltage is Joules per Coulomb.

Sources of Voltage:

A source of steady voltage is a chemical battery. It generates the

voltage which can drive a current through a circuit by electro-chemicalprocesses in the batteries molecules and atoms.

Voltage source can be non-steady of alternating, when it is generatedby electrical generators using electro-magnetic effects.

Electric Current:

Flow of electric charge per second: Coulombs/seconds ; unit is anampere = 1 Coulomb /sec. Circuit elements which allow current to floware called conductors.

Electrical Circuits:

Electrical circuits are made up of the following basic elements:

1. Source of electrical power : Power Supply (e.g. Battery) – defines aVoltage V (units Volts)

2. Resistance: R (units Ohms) offers resistance to current: I = V/R, orproduces a voltage drop given by V = IR. Unit: Ohm = 1 Volt/(1 Amp)

3. Capacitor: C (units Farads) stores electrical charge: If a capcitor ischarged with Q Coulombs it has a voltage V(capacitor) = Q/C, it playsthe role of a spring. Unit: 1 Farad = 1 Coulomb/ 1 Volt.

4. Inductor: L (units Henrys) offers electrical inertia to variations of

current through it . It develops a voltage: V=Ldidt

5. Switch: A device to open or close a circuit.

If a current I is being delivered by a source of voltage V, then as V isJoules per Coulomb, Power dissipated in the circuit (because of theresistance) is V I = [Joules/Coulomb] [Coulomb/sec] = Joules/sec. Andas

V=IR , V×I=I2R or V×I=V2

Rwatts or joules/sec

Fuses or circuit breakers : are protective devices to prevent the circuitfrom drawing too much current.

Example: V = 100 volts, I = 2 amperes , Power = 200 watts

Different charge distributions generate different electric fieldconfigurations:

Electric Dipole: + and – charges separated by a small distanceproduce a dipole field.

Parallel conducting plates charged + and – respectively produce auniform field between them.

Charged sphere: generates a radial electric field.

Currents and magnetic fields:

Although a conductor (wire) carrying a current is net electricallyneutral it generates a magnetic field around it. Hence moving chargesgenerate magnetic fields. Changing electric fields can also generatemagnetic fields.

Other moving charges (or other currents) can experience this magneticfield – magnetic fields exert forces on moving charges or currents.

A bar magnet is made up of aligned atomic current loops. One side ofthese curent loops correspond to North magnetic pole and the otherside to South magnetic pole.

Changing magnetic fields through a wire loop can create currents inthe wire, currents which have to be driven by an induced electric fieldin the wire. This process is called magnetic induction.

Magnetic north poles repel magnetic north poles and attractmagnetic south poles ! Same holds true for south poles whereone exchanges north and south in this statement.

Compass needles can trace out magnetic fields.

Magnetic fields are measured in units of Tesla. Earth's magnetic fieldin Irvine has a strength of approximately 0.0002 Tesla. Highestmagnetic field produced in the laboratory is in Mega Tesla.

Magnetic fields deflect electron beams in television sets. Earth'smagnetic field protects us from solar magnetic storms and also frombombardment of the earth by charged cosmic particles.

VI.EM radiation:

Oscillators, angtennas and EM waves: Electrical oscillators: Circuits containing an Inductor and a capacitor,called Tank Circuits in the book. Inductor provides “inertia” andcapacitor is the “spring”. The natural frequency of the circuit is:=LC

A tank circuit when connected to an antenna (see notes) will radiateEM energy in form of EM waves of the same frequency as the tankcircuit.

An EM wave consists of propagating, oscillating Electric and Magneticfields. As changing electric field generates a changing magnetic fieldand a changing magnetic field generates a changing electric field, theEM wave can propagate by self generating itself as it move.

EM waves can be detected by a piece of wire connected to a tankcircuit whose natural frequency matches that of the wave. The varyingelectric field in the wave induces a varying electrical current or voltagein the antenna which can be detected.

EM waves are transverse waves, the electric and magnetic oscillationsare in a plane perpendicular to the direction of propagation of thewave.

Different types of EM waves:

EM waves in space travel with a unique speed : c= 3x108 ms

EM waves can have any wavelength.

Frequency of an EM wave is f =c

EM spectrum extends from very long wavelengths (km) to very smallwavelengths 10−18m ; from Radio waves to gamma rays.

Visible light is EM radiation, with wavelengths ranging from 700 nm to400 nm .

Interaction of EM waves with matter ( atoms):

EM waves can interact with charged atomic matter. Processes ofinteraction involved are:

Absorption. Emission, Scattering, polarization, reflection, refraction

to mention a few. To correctly describe these one needs quantumphysics.

Scattering by nitrogen molecules (Rayleigh scattering) is responsiblefor the sky away from the sun looking blue. This is because blue light isscattered more than red light.

Rainbow is developed from the property that different wavelength lightwaves when travelling through glass or water have different speeds, aphenomenon called dispersion.

In the sun, atoms are excited as the sun is at a high temperature. Theatoms emit EM radiation producing the solar spectrum.

Earth recieves about 1.3 kilowatt of energy per 1 square meter of theearth's surface from the sun's radiation.

When EM radiation interacts with atoms, it does so in quanta ofenergy , called photons. Energy of each photon for EM radiation offrequence f is E = hf, which is the Einstein relation. Thisquantization property of nature is discussed next.

VII. Quantum Physics and Electrical properties of Materials:

Quantum mechanics was developed to explain the structure of matterat the atomic level. It was able to not only provide for stability, identityand regeneration properties of atoms, but it was able to provide thebasis of all of chemistry, understanding of solid state or condensedmatter physics and the basis of nuclear physics – how the sun shines.

The fundamental assumptions of quantum mechanics are :

1. Wave particle duality of matter: When travelling freely all objectsbehave like a wave and when interacting with each other all objectsbehave like particles. Thus in quantum physics the basic descriptionof nature is not in terms of matter particles and waves but in termsof a mathematical quantity called a wave function.

(a) So to moving particles we assign a wavelength:

de Broglie relation: =hp

where p = mv.

(b) And to all waves we assign an energy:

Einstein relation: E = hf, where f is the frequency of the wave.

The consequences of these assignments leads to new quantumproperties of all materials: Existance of energy levels. This isseen by the following discussion:

1.Electrons moving around in atoms are considered as waves confined to the environs of the atomic nucleus. We know that confinedwaves can only oscillate at certain frequencies or wavelengths.Wavelengths must have the order of magnitude of thedimensions of confinement. 2.Now for each allowed wavelength we can calculate the momentumusing the de Broglie relation, which in turn tells you what energy theparticle has. So one finds that confined electrons can only possesscertain well defined energy levels – Quantum energy levels.

3. EM radiation is emitted by accelerated electrons. Quantum physicsmake one more assumption to prevent electrons in a particularenergy level from radiating energy away. When an electron isresiding in a particular energy level it does not radiate any energy(in the form of radiation) . It only emits or absorbs energy whenit makes a transition from one level to another. Photon isemiitted when the transition is from a level of higher energy down toa lower energy level. Absorption moves an electron from a lowerenergy level to one which has higher energy. So each atom emitsor absorbs its own characteristic spectrum of EM radiation,which can be used to identify the elements in far distantastrophysical objects.

4. Standing waves are also generated for electrons confined in solids.The energy levels corresponding to these standing waves in this caseoccur in bands. The highest energy band which has electrons iscalled the conduction band.

5. Characteristic energy level bands in a solid are separated by bandgaps or forbidden region in energy. The energy width of the bandgap above the conduction band determines whether a material is aninsulator (non-conductor of electricity) or a semi-conductor. If theconduction band(see 4 above) is partially filled then the material is aconductor.

6. Pauli Exclusion Principle: This principle applies to particles like

electrons, those which have an intrinsic spin of 12

integer. Such

particles are called Fermions, in honor of Enrico Fermi. The principleis that only distinguishable electrons can occupy an energy level.All electrons are identical except for the orientation of their spin -orientation can take only two distinct values : up or down. Thusaccording to the Pauli Principle one can place only two electrons in a

given energy level – one with spin up and the other with spin down.This quantum property must be used to find out whether a substancebehaves electrically like a conductor, insulator or semi-conductor.

7. Photo-conductor, which is central to the operation of Xerox copiers,is a material which has a gap which is small enough so that lightshining on the material can lift electrons from the valence band intothe conduction band. So light makes photoconductors conduct.

VII.Quantum Physics and MRI:

MRI uses quantization of the intrinsic spin and magnetic moment ofnuclei of atoms ( e.g. Protons for H atoms, and phosphorous nucleus forPhosphorous atoms ) in a strong magnetic field to generate a signalfrom the “ flipping “ of the nuclear spins by radio-frequency (RF)signals. The regions from which these signals are detected is found bystudying the “ relaxation “ of the spins back to their original state. Athree dimensional picture of the biological medium is generated bycomputer analysis. Please see my notes for details.

VIII.Nuclear chain reaction, radioactivity etc.

I have a discussion of this in the notes titled “ Quantum bullets”. I summarize it briefly here: Nucleus is kept to gether by strong forces(previously called nuclear force). Nucleus contains both protons andneutrons. Protons are all positively charged while neutrons areelectrically neutral. So a nucleus, which is tiny, tries to blow upbecause of electrical repulsion amongst the proton. Attractive nuclearforce keeps the nucleus bound and stable.

Some nuclei, which are rather massive -like Uranium – can be unstableand can spontaneously break up into two parts, a process callednuclear fission. A light isotope of Uranium, U 235 , can be induced tofission by bombarding it with slowly moving neutrons. Each time U235fissions it emits another two neutrons which can be slowed down andcan cause other U235 nuclei to fission. A cascades of fission processesfollow. This generates energy which can heat up water or which canblow up like a bomb. The energy generated per kg U235 is enormousas compared to burning 1 kg of coal.

Radioactivity is the sponatneous emission of energetic particles,electrons, positrons, alpha particles and gamma-rays from nuclei.Emission of charged particles like electrons, positrons and alphaparticles cause transumutation of the emitting nucleus to another one.Emission of gamma rays does not transmute the nucleus but brings itdown to its ground state. This is a statistical process and as more and

more radioactive nuclei decay, their number decreases exponentiallywith time. Half life is the time it takes to reduce the number ofradiaoactive nuclei by a factor of 2. Emission from Radioactivesubstances provide important medical tools to alleviate suffering.