Unit 19: Dawn of Modern Physics — Long Questions

RELATIVE MOTION

The change in position of a body with respect to another body is called relative motion. If a ball is thrown up, the up direction is only for that particular place. It will be down position for a person on opposite of 1st observer. This concept of direction is relative. Similarly, rest or motion of object is not same for different observers. For example Cabin walls of moving train are at rest with the respect to passenger while, they are in motion with the respect to observer are stationary on earth. All motions are relative to a person or instrument observing it.

Examples

(i) If we place a body on Earth, it remains at rest unless an unbalanced force is applied upon it. This observation shows that Earth may be considered as an inertial frame of reference.

(ii) If a body is placed in a car moving with a uniform velocity with respect to Earth also remains at rest. So that a car is also an inertial frame of reference.

FRAME OF REFERENCE

The space bounded by three mutually perpendicular co-ordinate axes with respect to which observations are made is called frame of reference. For example If an observation is made with respect to the wall of the room, the room is the frame of reference. Similarly, the cabin of a moving train is also a frame of reference. If observation are made with respect to it, The frame of reference may be at rest or in motion. They have two types

(1) Inertial Frame of Reference

Those frame of references which are at rest or moving with uniform velocity i.e., it has no acceleration called inertial frame of reference.

Newton's laws of motion are valid in inertial frame of reference. Inertial frame of reference are also called non-accelerated frame of reference.

(2) Non-inertial Frame of Reference

That frame of reference which moves with some acceleration is called non-inertial frame of reference or accelerated frame of reference.

Example

If the moving car is suddenly stopped or accelerated up is a non-inertial frame of reference.

Newton's first law of motion does not valid in the non-inertial frame of reference.

SPECIAL THEORY OF RELATIVITY

In classical mechanics, the velocities of moving objects are negligible as compared to velocity of light. However, relativistics mechanics deals with velocities of moving objects comparable to speed of light.

The part of relativistic mechanics which deals with uniform relativistic velocities is called special theory of relativity. On the other hand, the part of relativistic mechanics which deals with accelerated relativistic velocities is called the general theory of relativity.

Postulates of Special Theory of Relativity

Special theory of relativity is based on the following postulates

(1) The laws of physics are the same in all inertial frame of reference.

(2) The speed of light in free space has the same value for all observers.

Explanation

If the laws of physics is different for different observers in relative motion. The observer could determine from this difference that which of them were stationary in a space and which were moving. But such distinction does not exist so, the 1st postulate show that there is no way to detect absolute uniform motion.

The 2nd postulate states an experimental fact that speed of light in free space is universal constant i.e., C = 3.0 x 10⁸ meter per sec. These simple postulates have far-reaching consequences. These include such phenomena as the slowing down of clocks and contraction of lengths in moving reference frames as measured by a stationary observer. Some interesting results of the special theory of relativity can be summarized as follows without going into their mathematical derivations.

Results of Special Theory of Relativity

According to special theory of relativity, following are the results:

(1) Time Dilation (time increase in motion)

According to special theory of relativity, time is not absolute quantity. It depends upon the motion of frame of reference. Let t₀ be the time interval between two events as measured by an observer in stationary frame of reference. This is known as proper time. If the frame of events is moving with uniform relativistic velocity v with respect to observer. The time measured by the observer would not be t₀, but it would be t given by

t = t₀ / √(1 - v²/c²)

As,

√(1 - v²/c²) < 1

then,

t > t₀

i.e., time has dilated due to relative motion of observer and frame of events. Time dilation is applicable to all timing processes physical, chemical and biological. Even aging process of human body is slowed down by motion at very high speed.

(2) Length Contraction (Length Decreases in Motion)

The distance from earth to star measured by an observer in a moving spaceship would seems smaller than the distance measured by an observer on earth. This effect is known as length contraction. The length of an object or distance between two points measured by an observer who is relatively at rest is called proper length l₀. If the object and an observer are in relative motion with relativistic speed v, then the contracted length is l is given by

l = l₀√(1 - v²/c²)

The length contraction is only along the direction of motion and not at right angle to motion.

(3) Mass Variation

According to special theory of relativity an object whose mass when measured at rest is m₀, will have an increased mass m when moving at relativistic velocity v is given by

m = m₀ / √(1 - v²/c²)

The increase in mass indicates the increase in inertia, so large force is required to accelerate the body.

When v = c then 1 - v²/c² = 0 and m = ∞

So an infinite force is needed to accelerate the mass since infinite force cannot be applied then a body cannot be accelerated to a speed of light in free space.

Discussion on above Results

In our everyday life, we deal with velocities which are extremely small as compare to velocity of light. Even the earth orbital speed is only 30 km/sec. Whereas velocity of light in free space is 300,000 km/sec. So, under this situation

v/c ≈ 0 and hence

t = t₀ , l = l₀ , m = m₀

No relativistic effects are observe in everyday life and newtanion physics is valid in everyday situation. However, dealing with atomic particles moving with velocities approaching with speed of light, the relativistic effects cannot be neglected. The experimental results are explained by taking Einstein's equations into account.

(4) Mass Energy Relation

According to Einstein special theory of relativity, mass and energy are interconvertible and are related by the relation

E = mc²

where m depends upon the speed of object. Energy E = m c² correspond to rest mass m is called rest-mass energy.

As mc² is greater than m₀c², the difference of energy is due to motion and is called K.E of the mass.

Thus K.E is equal to

K.E = mc² - m₀c²

K.E = (m - m₀)c²

The change in mass is due to change in K.E.

ΔE is given by

Δm = ΔE/c²

As c² is very large quantity, so sizeable change in mass requires very large change in energy. In our everyday world, energy changes are too small to provide measurable mass changes. However, energy and mass changes in nuclear reactions are found to be exactly in accordance with the above mentioned equations.

NAVSTAR Navigation System

The results of special theory of relativity are put to practical use even in everyday life by a modern system of navigation satellites called NAVSTAR. The location and speed anywhere on Earth can now be determined to an accuracy of about 2 cm s^-1. However, if relativity effects are not taken into account, speed could not be determined any closer than about 20 cm s^-1. Using these results the location of an aircraft after an hour's flight can be predicted to about 50 m as compared to about 760 m determined by without using relativistic effects.

BLACK BODY RADIATION

Radiation

The amount of energy emitted from hot body in the form of electromagnetic waves is called radiations or thermal radiations.

When a body is heated, it emits radiations. The radiation emitted depend upon the temperature and nature of material of body. However, radiations from black body depend only on the temperature of a black body.

Black Body

It is a cavity in a solid with small opening. The walls of the cavity are made black with suitable black material to make it good absorber. The small opening appears black because the radiations that enters is reflected from the inside walls many times untill completely absorbed i.e., cannot escapes out. Thus black body is a good absorber of radiation. If a black body is heated to higher temperature, the opening of cavity emits radiation like perfect emitter. The radiations from such body are called black body radiations or cavity radiations.

Intensity Distribution in Black Body Radiation

Lummer and Pringsheim measured the intensity of radiation emitted from the black body in different wavelengths when the black body was heated through different temperatures. Their experimental arrangement is shown in figure.

The amount of radiation emitted with different wavelength when black body is heated at different temperature have been shown graphically.

The following effects can be studied from the graphs.

(i) At a given temperature, the energy is not uniformly distributed in black body radiation spectrum.

(ii) Wien Displacement Law This law states that; for a particular temperature, the wavelength of maximum intensity (λ_max) is inversely proportional to the absolute temperature of the black body that is

λ_max ∝ 1/T

λ_max = Constant / T

λ_max × T = Constant

The value of constant is about (2.9 x 10^-3 mK is called Wein's constant.

This law tells us that as temperature increases, wavelength λ shifts to shorter values.

(iii) For all wavelengths, an increase in temperature causes an increase in energy emission. The radiation increases with increase in wavelength and at particular wavelength λ_max, it has maximum value. With further increase in wavelength, the intensity of radiation decreases.

(iv) Stephen's Boltzman Law This law states that area under each curve represents the total energy emitted over all wavelength at a particular temperature and is found to be directly proportional to the fourth power of absolute temperature of black body.

Mathematically

E ∝ T⁴

E = σT⁴

Where σ is constant called Stephen's constant and its value is 5.67 x 10^-8 watt m^-2 K^-4.

Planck's Assumptions

Electromagnetic theory of radiation could not explain the above energy distribution curves successfully. Maxwell Planck proposed quantum theory of radiation to explain energy distribution curves. His theory is based on following assumptions

(i) The energy is radiated or absorbed by a black body is in the form of packets of energy rather than in the shape of continuous wave. The packets of energy are called quanta. Each quantum has an energy given by

E = hf

where h is constant and its value is 6.63 x 10^-34 J sec. and f is the frequency of radiation.

(ii) The atoms of the cavity behave as harmonic oscillator each having a particular frequency. The energy is emitted or absorbed by the harmonic oscillator shifts from one the quantised state to another of quantised state. The energy emitted is according to relation

E = nhf where n = 1, 2, 3, ......

Max Planck proposed this theory in 1900 and got noble prize for it in 1918.

PHOTOELECTRIC EFFECT

The emission of electron from a metal surface when expose to light of suitable frequency is called photoelectric effect.

Demonstration of Photoelectric Effect

The photoelectric effect is demonstrated by the apparatus shown in the figure. An evacuated glass tube X contain two electrode. The electrode A is connected to positive terminal of the battery and is called anode. The electrode C is connected to -ve terminal of battery and is called cathode.

When monochromatic light is allowed to fall on cathode, it begin to emit electrons, these photoelectrons are attracted by the anode and the current is shown by ammeter. The current stops when light is cut off which shows emission of electrons with light.

Calculation of Maximum Energy of Photoelectrons

The max energy of photoelectrons can be calculated by reversing the connection of battery in above circuit. The anode A is at -ve potential and cathode C is at +ve potential. In this condition, the photoelectrons emitted are repelled by anode A and photoelectric current decreases. If the potential is made more and more negative at a certain value of potential which is called stopping potential. The current just stops even the electron of max energy are not able to reach the collector plate C. The maximum energy of photoelectrons is given by

1/2 mv²max = Voe

where m is the mass and e is charge of electron. Vmax is max velocity of electron and Vo is the stopping potential. The above experiment was repeated by changing frequency and intensity of incident light, and the following results were obtained.

(1) There is minimum frequency (called threshold frequency) below which no electron are emitted whatever may be the intensity of light. The threshold frequency depends upon the nature of the metal.

(2) The electrons are emitted with different energies. The maximum energy of photoelectron depends upon particular metal surface and frequency of incident light. If the frequency is made different keeping the intensity as constant, the current is same but stopping potential is different as shown for different frequency which shows that maximum K.E of the photoelectrons is proportional to the frequency of incident light.

(3) The number of electron emitted is directly proportional to the intensity of light e.g., when light beam of higher intensity is made incident on metal surface the amount of current increase but current stops for same value of stopping potential Vo.

Quantum Theory of Light

Einstein using the idea of Maxwell Planck, proposed that light is emitted or absorbed in the form of quanta called photons. The energy of each photon is given by

E = hf

where h is the Planck's constant and f is the frequency of light.

Failure of Electromagnetic Theory to Explain Photoelectric Effect

According to electromagnetic theory of light, the increase of intensity of incident light should increase the K.E of emitted electron which experimentally is not correct. Also classical theory of light, fails to explain the threshold frequency.

Explanation of Photoelectric Effect on the Basis of Quantum Theory

When photon falls on metal surface, it disappears after giving its energy to the electron of the metal. If this energy is more than the energy required to pull out an electron from the metal surface, the electron is instantaneously emitted. But if the energy is less than that value, no emission of electron takes place from the metal surface whatever may be the intensity of light. The minimum energy required to pull out an electron from metal surface is called work function of metal surface and is expressed as φ (work function) = hf.

Einstein Photoelectric Equation

Let a photon of energy "hf" falls on the metal surface. A part of photon energy is used to eject the electron from the metal surface and remaining energy is taken up by the electron in the form of K.E.

Incident photon energy - Work function = K.E of electron

hf - hfo = K.E of electron

hf - φ = 1/2 mv²max

This is known as Einstein photoelectric equation.

If (K.E)max = 0 then f = fo (threshold frequency).

Therefore

hfo - φ = 0

hfo = φ

Hence Einstein equation can be written as

(K.E)max = hf - hfo

It is to be noted that all the emitted electrons do not possess the maximum kinetic energy, some electrons come straight out of the metal surface and some lose energy in atomic collisions before coming out. The above equation holds good only for those electrons which come out with full surplus energy.

Albert Einstein was awarded Nobel Prize in physics in 1921 for his explanation of photoelectric effect.

Note The phenomenon of photoelectric effect cannot be explained if we assume that light consists of waves and energy is uniformly distributed over its wavefront. It can only be explained by assuming light consists of corpuscles of energy known as photon. Thus it shows the corpuscular nature of light.

PHOTOCELL

A photocell is based on photoelectric effect. A simple photocell is shown in figure. It consists of an evacuated glass bulb with a thin anode rod and a cathode of an appropriate metal surface. The material of the cathode is selected to suit to the frequency range of incident radiation over which the cell is operated. For example sodium or potassium cathode emits electrons for visible light, cesium coated oxidized silver emits electrons for infrared light and some other metals respond to ultraviolet radiation. When photo-emissive surface is exposed to appropriate light, electrons are emitted and a current flows in the external circuit which increases with the increase in light intensity. The current stops when the light beam is interrupted. The cell has wide range of applications. Some of these are to operate

(1) Security systems

(2) Counting systems

(3) Automatic door systems

(4) Automatic street lighting

(5) Exposure meter for photography

(6) Sound track of movies

COMPTON EFFECT

Arthur Holly Compton at Washington University in 1923 studied the scattering of X-rays by loosely bound electrons from a graphite target. He measured the wavelength of X-rays scattering at an angle θ with the original direction. He found that wavelength λs of the scattered X-rays is larger than the wavelength λi of the incident X-rays. This is known as Compton effect. The increase in wavelength of scattered X-rays could not be explained on the basis of classical wave theory. Compton suggested that X-rays consist of photons and in the process of scattering the photons suffer collision with electrons like billiard balls. In this collision, a part of incident photon energy and momentum is transferred to an electron. Applying energy and momentum conservation laws to the process, he derived an expression for the change in wavelength Δλ, known as Compton shift for scattering angle θ as

Δλ = h/(mec) (1 - cos θ) ... (i)

where mec is the rest mass of the electron. The factor h/(mec) has dimensions of length and is called Compton wavelength and has the numerical value.

h/(mec) = (6.63 × 10^-34 Js)/(9.1 × 10^-31 kg × 3 × 10⁸ ms^-1)

= 2.43 × 10^-12 m

If the scattered X-ray photons are observed at θ = 90°, the Compton shift Δλ, equals the Compton wavelength. The eq. (i) was found to be in complete agreement with Compton's experimental result, which again is a striking confirmation of particle like interaction of electromagnetic waves with matter.

Arthur Holly Compton was awarded Nobel Prize in physics in 1927 for his discovery of the effect named after him.

PAIR PRODUCTION

If a very high energy photon such as γ-rays photon attempted to be slowed down by introducing heavy nucleus in its path, but without any physical interaction, the photon energy is changed into an electron positron pair.

The phenomenon of creation of electron and positron in the process of slowing down of an energetic photon is called pair production or materialization of energy.

Positron is a positive electron. The creation of two particles with equal and opposite charge is essential for charge conservation. Positron is also called anti-electron. The process of pair production takes place in the electric field close to the nucleus. This process also takes place according to Einstein mass energy equation that is E = mc². For an e+ or e+, the rest mass energy is equal to 0.511 MeV. Thus to create two positron, 2 × 0.511 MeV or 1.02 MeV energy is required. So, pair production takes place when photon energy hf > 2mec² or hf > 1.02 MeV is used pair and the remaining energy is taken up by the pair in the form of K.E. So,

Energy of incident photon = 2mec² + (K.E)e- + (K.E)e+

hf = 2mec² + (K.E)e- + (K.E)e+

ANNIHILATION OF MATTER

The phenomenon in which electron-positron pair combine together to form two called annihilation of matter.

(OR)

The reverse process of pair production is called annihilation of matter.

The two γ-ray photons produced move in the opposite direction, so, that momentum is conserved. Each photon has energy 0.511 MeV equivalent to rest mass energy of a particle.

e- + e+ -----> γ + γ

Each particle has antiparticle. For example, proton has antiparticle antiproton. Similarly, antineutron and so on, when particle and antiparticle combine together. They annihilate each other to form another form of energy. The existence of positron was predicted by Dirac in 1928 and it was discovered in the cosmic radiation in 1932 by Carl Anderson. It gradually became clear that every particle has a corresponding antiparticle with the same mass and charge (if it is a charged particle) but of opposite sign. Particles and antiparticles also differ in the signs of other quantum numbers that we have not yet discussed. A particle and its antiparticle cannot exist together at one place. Whenever they meet, they annihilate each other. That is, the particle disappear, their combined rest energies appear in other forms. Proton and antiproton annihilation has also been observed at Lawrence Berkeley Laboratory.

WAVE NATURE OF PARTICLES

It has been observed that light shows a dual nature. It acts as waves in interference and diffraction and particle nature in photoelectric and Compton effect.

Loius de-Broglie proposed in 1924 that particles like electrons or protons should also possessed wave like particle. As momentum of photon is given by

P = h/λ

de-Broglie suggested that the momentum of particle of mass m moving with velocity v may also be given by similar expression

P = h/λ = mv

mv = h/λ

λ = h/mv

This is known as de-Broglie equation. It gives wavelength λ associated with a particle of mass m moving with velocity v.

Example

An object of large mass and ordinary speed has such a small wavelength that its wave effects such as interference and diffraction are negligible. When a rifle bullet of mass 20 g and flying with speed 330 ms^-1 will have a wavelength λ given by

λ = h/mv

= 6.63 × 10^-34/(2 × 10^-2 × 330)

= 1 × 10^-34 m

This wavelength is so small that it is not measurable or detectable by any of its effects.

On the other hand for an electron moving with a speed of 1 × 10⁶ ms^-1.

λ = 6.63 × 10^-34/(9.1 × 10^-31 × 1 × 10⁶)

= 7 × 10^-10 m

This wavelength is in the X-rays range. Thus, diffraction effects for electrons are measurable whereas diffraction or interference effects for bullets are not.

Davisson and Germer Experiment

The de-Broglie equation was confirmed by Davison and Germer. They showed that electrons are diffracted from metal crystals in the same way as x-rays or any other wave. The apparatus used is shown in figure. The electrons are obtained by heating a filament and are accelerated in electron gun by potential difference applied between filament and anode of electron-gun.

Thus, 1/2 mv² = Ve

mv² = 2Ve

m²v² = 2Vem

mv = √(2Vem)

Putting the value in de-Broglie equation

λ = h/√(2Vem)

λ = 6.63 × 10^-34/√(2 × 54 × 1.6 × 10^-19 × 9.1 × 10^-31)

λ = 1.66 × 10^-10 m

The electron beam of energy Ve is made incident on a nickel crystal (Ni). The beam is diffracted and is recorded by moveable detector. The first order (n = 1) was obtained for an angle 65°. According to Bragg's equation

2d sin θ = nλ

where, n = 1

θ = 65°

d = 0.91 × 10^-10 m

2 × 0.91 × 10^-10 × sin 65° = 1 × λ

λ = 1.65 × 10^-10 m

Thus experimental result of wavelength agrees with the theoretical result. The wave like property of diffraction pattern have also been observed for protons, neutrons, hydrogen atoms and helium atoms. For his work on the dual nature of particles, de-Broglie awarded a Nobel Prize in 1929 in physics Davison and Germer also shared the Nobel Prize in 1937 for this experimental conformation of the wave nature of particle.

WAVE PARTICLE DUALITY

Interference and diffraction of light confirm its wave nature, while photoelectric effect proves the particle nature of light. Similarly, the experiments of Davisson and Germer and G.P. Thomson reveal wave like nature of electrons and in the experiment of J. J. Thomson to find e/m we had to assume particle like nature of the electron. In the same way we are forced to assume both wavelike and particle like properties for all matter electrons, neutrons, molecules etc. and also light, X-rays, γ-rays etc. have to be included in this. In other words, matter and radiation have a dual 'wave-particle' nature and this new concept is known as wave particle duality. Niels Bohr pointed out in stating his principle of complementarity that both wave and particle aspects are required for the complete description of both radiation and matter. Both aspects are always present and either may be revealed by an experiment. However, both aspects cannot be revealed simultaneously in a single experiment, which aspect is revealed is determined by the nature of the experiment being done. If you put a diffraction grating in the path of a light beam, you reveal it as a wave. If you allow the light beam to hit a metal surface, you need to regard the beam as a stream of particles to explain your observations. There is no simple experiment that you can carry out with the beam that will require you to interpret it has a wave and as a particle at the same time. Light behaves as a stream of photons when it interacts with matter and behaves as a wave in travelling from a source to the place where it is detected. In effect, at micro-particles (electrons, protons, photons, atoms etc.) propagate as if they were waves and exchange energies as if they were particles - that is the wave particle duality.

Use of Wave Nature of Particles

The fact that energetic particles have extremely short de-Broglie wavelengths has been put to practical use in many ultra-modern devices of immense importance such as electron microscope.

ELECTRON MICROSCOPE

Electron microscope makes practical use of the wave nature of electrons which is thousands of time shorter than visible light which enables the electron microscope to distinguish details not visible with optical microscope. In an electron microscope, electric and magnetic fields rather than optical lenses are used to focus electrons by means of electromagnetic forces that are exerted on moving charges. The resulting deflections of the electrons beams are similar to the refraction effects produced by glass lenses used to focus light in optical microscope. The electrons are accelerated to high energies by applying voltage from 30 kV to several megavolts. Such high voltages give extremely short wavelength and also give the electron sufficient energy to penetrate specimen of reasonable thickness. A resolution of 0.5 to 1 nm is possible with a 50 kV microscope as compared to best optical resolution of 0.2 nm. A schematic diagram of the electron microscope is shown in the figure. The magnetic conducting lens concentrates the beam from an electron gun on to the specimen. Electrons are scattered out of the beam from the thicker parts of the specimen. The transmitted beam therefore has spatial differences in density that correspond to the features of the specimen. The objective and intermediate lenses produce a real intermediate image and projection lens forms the final image which can be viewed on a fluorescent screen or photographed on special film known as electron micrograph. A three dimensional image of remarkable quality can be achieved by modern versions called scanning electron microscopes.

UNCERTAINTY PRINCIPLE

This principle states that "it is impossible to measure the momentum and position of particle simultaneously with perfect accuracy". The uncertainty in momentum and position measurements is not due to measuring instrument but is related with wave particle duality of matter and radiation. This uncertainty is completely negligible for measurements of positions and momentum of macroscopic particles but is significant in the atomic particles. For example a stream of light of photons scattering from a flying tennis ball effects its path. But one photon strikes an electron alters its position very much. Since, light has also wave properties, the accuracy of measurement of position is always of the order of one wavelength. Thus in order to measure position of an electron with less uncertainty and to avoid diffraction effect, light of very short wavelength must be used. Hence, for a microscopic particles moving along x-axis, the value of uncertainty in position Δx is of the order of wavelength λ used. So Δx ≃ λ

The light with short wavelength have large momentum. So, uncertainty in momentum of electron is increased. The uncertainty in the momentum of electron can be of the order of momentum of photon. So, the uncertainty in momentum of particle or electron along x-axis can be expressed as

ΔP ≃ h/λ

Multiplying the above relations

ΔP . Δx ≃ h/λ . λ

ΔP . Δx ≃ h

This is the mathematical form of uncertainty principle. It states that the product of the uncertainty Δx in the position of a particle at some instant and the uncertainty ΔP in the x-component of its momentum at the same instant approximately equals Planck's constant h.

There is another form of uncertainty principle which relates the energy of a particle and the time at which it had the energy. If ΔE is the uncertainty in energy and if Δt is the time interval during which the particle had the energy.

then ΔE . Δt ≃ h

Thus more accurately we determined the energy of a particle, the more uncertain we will be of the time during which it has that energy.

According to Heisenberg's more careful calculations, he found that at the very best

Δx . ΔP ≥ h

and ΔE . Δt ≥ h

where h = h/(2π) = 1.05 × 10^-34 Js

Werner Karl Heisenberg received Nobel Prize for physics in 1932 for the development of quantum mechanics.