BEC Affiliated by Dhaka University
Department of EEE
PHY 1101: Electricity and Magnetism, Modern Physics and Mechanics.
Name:
 Modern Physics
Modern physics: Modern physics is a branch of physics that developed in
the early 20th century and onward or branches greatly influenced by early
20th century physics. The branches of modern physics include Atomic
Physics, Nuclear Physics, Quantum mechanics, Special relativity, and
General relativity. Albert Einstein is the Father of Modern Physics.
Book Reference:
1. Fundamentals of Physic, 10th edition by Resnick & Halliday.
2. Concept of Modern Physics, 16th edition by Arthur Beiser, Shobhit
Mahajan & S Rai Choudhury.
3. Physics Volume 1 & Physics Volume 11, Latest edition by David
Halliday.
4. University Physics, Latest sedition, by Jeff Sanny & Samuel Ling.
5. Physics for Engineers, Part 1 & Part 2, 1st edition by Gias Uddin
Ahmad.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Galilean relativity: Galilean relativity states that the laws of motion are the
same in all inertial frames.
Let us consider two inertial frame of references S and S, where the frame S is at
rest and the frame S is moving along x-axis with velocity v with respect to S
frame. The space and time coordinates of S and S are (x, y, z, t) and (x, y, z,
t) respectively. Thus
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Einstein’s special theory of relativity: Special relativity is an
explanation of how speed affects mass, time, and space. Einstein's special
theory of relativity states that the same laws of physics hold true in all inertial
reference frames and that the speed of light is the same for all observers.
Lorentz
transformation:
The
transformation
which
relates
the
observations of position and time made by the two observers in two different
inertial frames is known as Lorentz transformation.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Lorentz transformation equations:
“Science can amuse and fascinate us all, but it is engineering that
changes the world.”
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Length contraction: The length of any object in a moving frame will appear
foreshortened in the direction of motion or contracted. The amount of
contraction can be calculated from the Lorentz transformation. The length is
maximum in the frame in which the object is at rest.
If the length l0 = x2 − x1 is measured in the moving reference frame and the
length l= x2 – x1 is measured in the rest reference.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Time dilation of special Lorentz transformation:
A clock in moving frame will be seen to be running slow or “dilated” according
to Lorentz transformation. The time will always be shortest as measured in the
rest frame. The time measured in which the clock is at rest called the “proper
time”.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
“As engineers, we were going to be in a position to change the world – not
just study it.” By Henry Petroski
“The engineer has been, and is, a maker of history.” By Isaac Asimov
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
“The best among you is the one who doesn’t harm others with his tongue and
hands.”
“The greatest of richness is the richness of the soul.”
“The strongest among you is the one who controls his anger.”
“The best among you are those who have the best manners and character.”
By Prophet Muhammad Sallallahu Alaihi Wasallam
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Relativity of mass and mass energy relation or Einstein mass energy
relation:
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Photoelectric effect:
“The best of houses is the house where an orphan gets love and kindness.”
“A father gives nothing better than good education.”
“Speak good or remain silent.”
By Prophet Muhammad Sallallahu Alaihi Wasallam
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Characteristic of Photo-electric effect (PEE):
The photo electric effect will be started with threshold frequency, The emission of electron will be
stopped as the light ceases to fall on the metal.
The KE of the photoelectrons depend on the frequency of the incident light does not depend on
the intensity of light.
Photoelectric emission does not depend on temperature.
The maximum velocity of ejected photo electron is proportional to the frequency of incident
radiation.
The rate of emitting electron is proportional to the intensity of the radiations.
The stopping potential V is independent of the intensity of incident radiations.
s
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
H.W: Explain the relevance of the special theory of relativity. Why the special theory of
relativity is considered revolutionary as far as the space and time is considered?
H.W: Beriefly explain why the electromagnetic waves rather than other known forms of
waves are central to the special theory of relativity.
H.W: Why the Compton effect an evidence for the particle nature of electromagnetic
radiation?
H.W: Is the particle property of waves restricted only to electromagnatic waves and photons?
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Example 12.1: The work function of potassium is 2.0 eV. When the
ultraviolet light of wavelength 3500 Angstrom unit falls on a potassium
surface, what is the maximum energy of electron volts of the
photoelectrons?
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
We know,
𝐓𝐌𝐚𝐱 = 𝐡𝐯 − 𝐡𝐯𝛐
work= 𝟐 𝒆𝑽
=
𝐡𝐜
𝛌
𝝀 = 𝟑𝟓𝟎𝟎 𝑨𝝄
− 𝐰𝛐
𝟔.𝟔𝟑×𝟏𝟎−𝟑𝟒 ×𝟑×𝟏𝟎𝟖
=(
𝟑𝟓𝟎𝟎×𝟏𝟎−𝟏𝟎 ×𝟏.𝟔×𝟏𝟎−𝟏𝟗
− 𝟐 ) 𝐞𝐕
= 𝟑𝟓𝟎𝟎 × 𝟏𝟎−𝟏𝟎
= (𝟑. 𝟔 − 𝟐)𝐞𝐕 = 𝟏. 𝟔 𝐞𝐕
𝑻𝑴𝒂𝒙 = ?
Example 12.5 : A photo electric surface has a work function of 4 eV. What is
the max velocity of photo electron?
Solution: Given,
𝐰𝛐 = 𝟒 𝐞𝐕 = 𝟒 × 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 𝐉
𝐯 = 𝟏𝟎𝟏𝟓 𝐇𝐳
𝐡 = 𝟔. 𝟔 × 𝟏𝟎−𝟑𝟒 𝐉𝐬
𝐦 = 𝟗 × 𝟏𝟎−𝟑𝟏 𝐤𝐠
We know,
𝐰𝛐 = 𝐡𝐯𝛐
𝟏
𝟐
𝟐
𝐦𝐯𝐌𝐚𝐱
= 𝐡𝐯 − 𝐰𝛐
= 𝟔. 𝟔 × 𝟏𝟎−𝟑𝟒 × 𝟏𝟎𝟏𝟓 − 𝟒 × 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗
= 𝟐 × 𝟏𝟎−𝟐𝟎 𝐉
∴ 𝐕𝐌𝐚𝐱 = √
𝟐×𝟐×𝟏𝟎−𝟐𝟎
𝟗×𝟏𝟎−𝟑𝟏
= 𝟐. 𝟏𝟎𝟖 × 𝟏𝟎𝟓 𝐦𝐬 −𝟏
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Example 12.6: Calculate the threshold frequency and the wavelengths of
radiations where 𝐰𝛐 = 𝟑. 𝟑𝟏 × 𝟏𝟎−𝟏𝟗 , 𝐡 = 𝟔. 𝟔𝟐 × 𝟏𝟎−𝟑𝟒 𝐉𝐬.
Solution: We know,
𝐰𝛐 = 𝐡𝐯𝛐
𝐰𝛐
=>
𝐯𝛐 =
=>
=
=>
= 𝟓 × 𝟏𝟎𝟏𝟒 𝐇𝐳
Again,
𝛌𝛐 =
𝐡
𝟑.𝟑𝟏×𝟏𝟎−𝟏𝟗
𝟔.𝟔𝟐×𝟏𝟎−𝟑𝟒
𝐜
𝐯𝛐
=
𝐇𝐳
𝟑×𝟏𝟎𝟖
𝟓×𝟏𝟎𝟏𝟒
= 𝟔 × 𝟏𝟎−𝟕 𝐦 = 𝟔𝟎𝟎𝟎 𝐀𝛐
The maximum Velocity of the photo-electron can be determined from
the value of the stopping potential, 𝐕𝐬
𝟏
𝟐
𝟐
𝐦 𝐕𝐌𝐚𝐱
= 𝐞 𝐕𝐬
=> 𝐕𝐌𝐚𝐱 = √
𝟐𝐞𝐕𝐬
𝐦
𝐞 = 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 𝐜
𝐦 = 𝟗. 𝟏 × 𝟏𝟎−𝟑𝟏 𝐤𝐠
If stopping potential, 𝐕𝐬 is given we can easily get the maximum velocity.
Compton effect: Compton effect is defined as the effect that is observed
when x-rays or gamma rays are scattered on a material with an increase in
wavelength. Compton found that wavelength is not dependent on the intensity
of incident radiation. It is dependent on the angle of scattering and on the
wavelength of the incident beam.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
What is de Broglie wave? Deduce the expression of wavelength of it.
Every moving particle has dual nature like radiant energy, particle and wave.
This wave is called de Broglie wave.
The wavelength of de Broglie matter is called de Broglie wavelength. Now we
have to find out its expression.
From Planck’s Quantum theory of light. The energy of photon
𝑬 = 𝒉𝝂
(i)
Where h is the Planck’s constant and 𝝂 is the frequency of photon.
If m be the mass of photon then from Einstein’s mass energy equation we
have
𝑬 = 𝒎𝒄𝟐
(ii)
Where c is the velocity of light.
From equation (i) and (ii) we have
𝑬 = 𝒎𝒄𝟐 = 𝒉𝝂
𝒐𝒓, 𝒎 = 𝒉𝝂/𝒄𝟐
(iii)
Let, the momentum of photon = P
𝑷 = 𝒎𝒂𝒔𝒔 𝒐𝒇 𝒑𝒉𝒐𝒕𝒐𝒏 × 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒐𝒇 𝒑𝒉𝒐𝒕𝒐𝒏
𝒉𝝂
𝒉𝝂
𝒄
𝒄
= 𝒎𝒄 = 𝟐 × 𝒄 =
(iv)
Again, 𝒄 = 𝝀𝝂
(v)
Equation (iv) and (v) implies 𝑷 =
So, 𝝀 =
𝒉𝝂
𝒄
=
𝒉𝝂
𝝀𝝂
=
𝒉
𝝀
𝒉
𝐏
Which is the required de Broglie wave equation or expression of wavelength.
De Broglie wavelength of electron
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Let, an electron be accelerated with a potential difference V volt, the kinetic
𝟏
energy of the electron is 𝒎𝒗𝟐 = 𝒆𝒗
𝟐
=> 𝒎𝟐 𝒗𝟐 = 𝟐𝒎𝒆𝒗
=> 𝒎𝒗 = √𝟐𝒎𝒆𝒗
Hence, the wavelength associated with the wave is
𝝀=
𝒉
𝒎𝒗
==
𝒉
√𝟐𝒎𝒆𝒗
Ex 15.1. Compute the de Broglie wavelength of the following
(i)
(ii)
(iii)
(iv)
(v)
A 1000kg automobile traveling at 100m/s
A 10 gm bullet traveling at 500m/s
A smoke particle of mass 𝟏𝟎−𝟗 gm moving at 1cm/s
An electron with a K.E. of 1 ev
An electron with a K.E. of 100 Mev
Solution
(i)
We know that 𝝀 =
𝒉
𝑷
=
𝒉
𝒎𝒗
=
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
𝟏𝟎𝟎𝟎𝒌𝒈×𝟏𝟎𝟎𝒎/𝒔
= 𝟔. 𝟗 × 𝟏𝟎−𝟑𝟗 𝒎
= 𝟔. 𝟗 × 𝟏𝟎−𝟐𝟗 𝑨̇
(ii)
𝝀=
𝒉
𝑷
=
𝒉
𝒎𝒗
=
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
𝟏𝟎×𝟏𝟎−𝟑 𝒌𝒈×𝟓𝟎𝟎𝒎/𝒔
= 𝟏. 𝟑 × 𝟏𝟎−𝟑𝟒 𝒎
= 𝟏. 𝟑 × 𝟏𝟎−𝟐𝟒 𝑨̇
(iii)
𝝀=
𝒉
𝑷
=
𝒉
𝒎𝒗
=
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
𝟏𝟎−𝟗 ×𝟏𝟎−𝟑 𝒌𝒈×𝟏×𝟏𝟎−𝟐 𝒎/𝒔
= 𝟔. 𝟔 × 𝟏𝟎−𝟐𝟎 𝒎
= 𝟔. 𝟔 × 𝟏𝟎−𝟏𝟎 𝑨̇
(iv)
𝑷 = 𝒎𝒗 = √𝟐𝒎𝒆𝒗 = √𝟐𝒎𝒌
= √𝟐 × 𝟗. 𝟏 × 𝟏𝟎−𝟑𝟏 𝒌𝒈 × 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 𝑱
= 𝟓. 𝟒 × 𝟏𝟎−𝟐𝟓 𝒌𝒈. 𝒎/𝒔
∴𝝀=
𝒉
𝒎𝒗
=
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
𝟓.𝟒×𝟏𝟎−𝟐𝟓 𝒌𝒈.𝒎/𝒔
= 𝟏. 𝟐 × 𝟏𝟎−𝟗 𝒎 = 𝟏. 𝟐 𝒏𝒎 = 𝟏𝟐𝑨̇
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
(v)
𝑷 = 𝒎𝒗 = √𝟐𝒎𝒆𝒗 = √𝟐𝒎𝒌
= √𝟐 × 𝟗. 𝟏 × 𝟏𝟎−𝟑𝟏 𝒌𝒈 × 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 × 𝟏𝟎𝟎 × 𝟏𝟎𝟔 𝑱
= 𝟓. 𝟒 × 𝟏𝟎−𝟐𝟏 𝒌𝒈. 𝒎/𝒔
∴𝝀=
𝒉
𝒎𝒗
=
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
𝟓.𝟒×𝟏𝟎−𝟐𝟏 𝒌𝒈.𝒎/𝒔
= 𝟏. 𝟐 × 𝟏𝟎−𝟏𝟑 𝒎
Obtain the expression for the radius of n-th permitted orbit of hydrogen atom.
Solution: Consider an electron of charge e- and mass m moves round the nucleus in an orbit of
radius r in a Hydrogen atom.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
The electrostatic force between the nucleus & the electron is
F= 1 / 4πε0× Ze. e /r2
= (1 /4πε0 )× (Ze2/r2)
… … … (1)
The centrifugal force of repulsion by an electron is
F=mv2/r
… … … (2)
The system will be stable if,
mv2/r = 1/4πε0 × (Ze2/r2)
… … … (3)
the angular momentum of the electron moving in a circular orbit of radius r.
I.w = mr2 × v/r =mvr
According to Bohr’s postulate,
I . w = mvr = n×(h/2π)
or, v = nh/2πm r or, v2 = n2h2/4π2m2r2
So, from (3),
m/r . n2h2/4π2m2r2= 1/4πε0 . Ze2/r2
r = (n2h2ε0/πZe2m)
So,
Hence,
… … … (4)
r ∝ n2
Constituent of atomic nucleus: The atomic nucleus consists of nucleons—
protons and neutrons.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Nuclear binding energy: Nuclear binding energy is the energy required to
separate an atomic nucleus completely into its constituent protons and
neutrons.
Or
Binding energy, amount of energy required to separate a particle from a
system of particles or to disperse all the particles of the system. Binding
energy is especially applicable to subatomic particles in atomic nuclei, to
electrons bound to nuclei in atoms, and to atoms and ions bound together in
crystals.
Mass defect is the difference between the predicted mass and the actual mass
of an atom's nucleus. The binding energy of a system can appear as extra
mass, which accounts for this difference.
“A person who never made a mistake never tried anything new.”
“We cannot solve our problems with the same thinking we used when we
created them.”
“In the middle of difficulty lies opportunity.”
Albert Einstein
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Different types of radioactivity: Radioactivity is the process by which a
nucleus of an unstable atom loses energy by emitting particles of ionizing
radiation. The three most common types of radiation are alpha particles, beta
particles, and gamma rays.
Radioactive decay law:
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
What is meant by half-life?
Half-life is the interval of time required for one-half of the atomic nuclei of a radioactive
sample to decay.
Mean life: Mean life in radioactivity is average lifetime of all the nuclei of a particular
unstable atomic species
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Nuclear chain reaction: A chain reaction refers to a process in which
neutrons released in fission produce an additional fission in at least one
further nucleus. This nucleus in turn produces neutrons, and the process
repeats.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Atomic Power Plant: A nuclear power plant is a thermal power station in
which the heat source is a nuclear reactor. As is typical of thermal power stations,
heat is used to generate steam that drives a steam turbine connected to a generator
that produces electricity.
In a nuclear power plant, this heat is used to boil water to produce steam that can
be used to drive large turbines. This, in turn, activates generators to produce
electrical power. Atomic energy is more correctly called nuclear energy.
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
State and Explain Heisenberg’s Uncertainty Principle
Statement: It is not possible to determine simultaneously both the position and
momentum of a particle with unlimited precision.
If the uncertainty of position is ∆𝑥 and the momentum is ∆𝑃 at a particular
time then their product will be greater than or equal to Planck ’s constant.
Mathematically ∆𝑥. ∆𝑃 ≥ ℎ̅
Ex. 15.2 Calculate the de Broglie wavelength of a proton with
𝟏
𝟐𝟎
𝒕𝒉 of the
velocity of light.
Solution
We know
𝝀=
𝒉
𝒎𝒗
=
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
𝟏.𝟔𝟕𝟑×𝟏𝟎−𝟐𝟕 𝒌𝒈×
𝟑×
𝟏𝟎𝟖 𝒎
𝒔
𝟐𝟎
= 𝟐. 𝟔𝟒 × 𝟏𝟎−𝟏𝟒 𝒎
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Ex. 15.3 Calculate the wavelength associated of a thermal neutron of energy
0.025ev. Solution
𝟏
The kinetic energy of the neutron 𝟐 𝒎𝒗𝟐 = 𝟎. 𝟎𝟐𝟓𝒆𝒗
𝒎𝟐 𝒗𝟐 = 𝟐 × 𝟏. 𝟔𝟕𝟔 × 𝟏𝟎−𝟐𝟕 𝒌𝒈 × 𝟎. 𝟎𝟐𝟓 × 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 J
∴ 𝒎𝒗 = √𝟐 × 𝟏. 𝟔𝟕𝟔 × 𝟏𝟎−𝟐𝟕 𝒌𝒈 × 𝟎. 𝟎𝟐𝟓 × 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 𝑱
𝒉
Hence, 𝝀 = 𝒎𝒗 =
𝟔.𝟔×𝟏𝟎−𝟑𝟒 𝑱𝒔
√𝟐×𝟏.𝟔𝟕𝟔×𝟏𝟎−𝟐𝟕 𝒌𝒈×𝟎.𝟎𝟐𝟓×𝟏.𝟔×𝟏𝟎−𝟏𝟗 𝑱
= 𝟏. 𝟖𝟎𝟕 × 𝟏𝟎−𝟏𝟎 𝒎 = 𝟏. 𝟖𝟎𝟕𝑨̇
Abdul Kalam, in full Avul Pakir Jainulabdeen Abdul Kalam, (born October 15, 1931,
Rameswaram, India—died July 27, 2015, Shillong), Indian scientist and politician who
played a leading role in the development of India's missile and nuclear weapons programs .
1. Dream is not something that you see while sleeping. It is something
that does not let you sleep.
2. Books become permanent companions. Sometimes, they are born
before us; they guide us during our life journey and continue for
many generations.
3. One of the very important characteristics of a student is to question.
Let the students ask questions.
4. All Birds find shelter during a rain. However, Eagle avoids rain by
flying above the Clouds. Problems are common, but attitude makes
the difference.
5. While children are struggling to be unique, the world around them is
trying all means to make them look like everybody else.
6. Do not fear failing in the first attempt because even the successful
math’s starts with zero only?
7. You see, Allah helps only people who work hard. That Principle is
very clear.
8. If you fail, never give up. Because fail means “First Attempt in
Learning”.
9. Education is the most powerful weapon, which you can use to change
the world.
10.The two most important life goals I would like every youth to have:
One, increase the amount of time that you have at your disposal; two,
increase what you can achieve in the time available.
By APJ Abdul Kalam
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Dr. Md. Tarek Hossain, Assistant Professor, BEC
Find the relationship between the total energy, rest energy and the
momentum.
Solution: We know from mass transformation,
m=
𝑚𝑜
2
√1−𝑉2
𝑐
= m0 ( 1- V²/c²)-(1/2)
1 𝑉²
= m0 [1+
2 𝑐²
3 𝑉²
+ ( ) 2+……………….]
8 𝑐²
1
Or, mc2 = m0c2+ m0V2+…………
(1)
2
From the mass energy relationship
E = mc2 =
𝑚𝑜 𝑐 2
(2)
2
√1−𝑉2
𝑐
And the momentum p = mv =
𝑚𝑜 𝑣
2
√1−𝑉2
(3)
𝑐
From equation (2) we have
𝐸 2
(𝑐 ) =
𝑚𝑜 2 𝑐 2
𝑉2
𝑐
1− 2
𝐸 2
𝐸 2 𝑉2
𝑐
𝑐
or, 𝑚𝑜 2 𝑐 2 = ( ) -( )
𝐸 2
= ( ) -(
𝑐
=
𝐸2
𝑐
𝑚𝑐 2
2
)
𝑐
- 𝑚2 𝑣 2
2
𝑐2
𝑉2
𝑐2
=
𝐸2
𝑐2
- 𝑝2
Hence, 𝐸 2 = 𝑚𝑜 2 𝑐 4 + 𝑝2 𝑐 2
momentum relationship.
which is the required energy, rest energy and
Example: 18x10−3 kg of a substance is fully converted to heat energy. Calculate
the amount of heat produced.
Solution:
We know, E = mc²
32
Dr. Md. Tarek Hossain, Assistant Professor, BEC
= 4.18×10-3× (3×108)2 J
... Heat produced =
=
𝐸
4.18
cal
4.18×10−3 ×(3×108 )²
4.18
= 9×1013 calorie. (Ans)
Example: 11.4: What would be the speed of a passenger who wishes to a
spaceship, whose speed with respect to the earth is .9c at a relative speed of 0.5c?
Solution: According to conventional mechanics, the actual speed of the passenger
is 0.9 c + 0.5c = 1.4c more than speed of light. However, according to the realistic
mechanics the required speed of the passenger is
u=
𝑢′ +𝑉
𝑢′𝑉
1+
𝑐²
=
0.9𝑐+0.5𝑐
1+
V = 0.5c
.9𝑐×0.5c
𝑐²
𝑢′ = 0.9c
= 0.9655c
The passenger must move with a speed of 0.9655c in order to have a relative speed
of 0.5c with respect to the spaceship.
Example: (a) Two events occur at the same place but separated in time. S declare
that the two events occur in different places.
S
𝑆′
Rest
Moving
(b) 𝑆 ′ observes that two events occur at the same time but are separated in space.
S will than declare two events occur at different places.
In (a) let the time separation in 𝑆 ′ in 10 minutes. What is the distance separation
observed by S?
In (b) let the distance separation in 𝑆 ′ is 25meter. What is the time separation
observed by S?
Solution:
Take, V = 20 m/s and c = 3×108 m/s
33
Dr. Md. Tarek Hossain, Assistant Professor, BEC
(a)
Here, 𝑡2′ − 𝑡1′ =10 min = 600s
𝑥2 - 𝑥1 = ?
𝑥2 =
𝑥2′ +𝑉𝑡2 ′
√1−𝑉²
𝑥2 - 𝑥1 =
=
√1−𝑉²
𝑐²
𝑥2′ − 𝑥1′ +𝑉(𝑡2′ −𝑡1′ )
𝑐²
[Here , 𝑥2′ = 𝑥1′ ]
√1−𝑉²
𝑐²
𝑉(𝑡2′ −𝑡1′ )
√1−𝑉²
20×600
=
√1−(
𝑐²
(b)
𝑥 ′ +𝑉𝑡1 ′
𝑥1 = 1
&
= 12km
20
)²
3×108
(ans)
𝑥2′ − 𝑥1′ = 25, t2 - t1=?
t2 =
𝑥 ′
𝑡2′ +𝑉 2
𝑐²
√1−𝑉²
t2 - t1 =
,
t1 =
𝑥 ′
𝑡1′ +𝑉 1
𝑐²
√1−𝑉²
𝑐²
′
𝑡2 −𝑡1′ +𝑉(𝑥2′ −𝑥1′ )/𝑐²
√1−𝑉²
𝑐²
×
𝑐²
t2 - t1 =
𝑉(𝑥2′ −𝑥1′ )
𝑐²√1−
𝑉²
𝑐²
=
20×25
20
(3×108 )²√1−(
)²
3×108
= 5.55×10-15s.
Example:11.14: How fast would a rocket ship have to go relative to an observer
for its length to be contracted to 99% of it’s length when at rest?
Solution: Let, l0 = Proper length
l = 0.99 l0
or,
𝑙
𝑙0
= 0.99 =
𝑙
𝑙0
=
or, 1 −
or,
𝑉²
𝑐²
𝑐²
99
100
𝑉²
𝑐²
100
𝑉²
We know, l = l0√1 −
or,
99
= √1 −
=(
= 1-(
99
100
99
100
𝑉²
𝑐²
)²
)²
34
Dr. Md. Tarek Hossain, Assistant Professor, BEC
or, 𝑉 2 = 0.0199 × c²
or, V = 0.1416c = 0.1416×3×108 m/s
= 4.245×107 m/s
The rocket ship have to go with velocity 4.245×107 m/s.
Example: 11.21: A particle is moving with a speed of 0.5c. Calculate the ratio of
its rest mass and the mass which in motion.
Solution:
We know, m =
𝑚0
√1−𝑉²
𝑐²
or,
𝑚0
𝑚
= √1 −
= √1 −
(0.5𝑐)²
𝑐²
𝑉²
𝑐²
= 0.866
(Ans)
H.W: X-ray of wavelength 10.0 pm are scattered from a target. (i) Find the wavelength of the
X-ray scattered through 450. (ii) Find the maximum wavelength present in the scattered x-ray.
(iii) Find the maximum kinetic energy of the recoil electrons.
H.W: Ultraviolet light of wavelength 350 nm and intensity 1.00 W/m2 is directed at a potassium
surface, (a) hind the maximum KE of the photoelectrons. (b) If 0.50 percent of the incident
photons procedure photoelectrons, how many are emitted per second if the potassium surface has
an area of 1.00 cm2?
H.W: Calculate the velocity that one atomic mass until will have if it has a kinetic energy equal
to twice the rest mass energy.
H.W: The photo-electric threshold of copper is 3200 Å. If ultra-volit light of wavelength 2500 Å
falls on it, find (i) the maximum kinetic energy of the photo-electrons ejected, (ii) maximum
velocity of the photo-electron and (iii) the value of the work function.
H.W: What is the threshold wavelength for a tungsten surface whose work function is 4.5 eV?
35
Dr. Md. Tarek Hossain, Assistant Professor, BEC
QM: Find the change in wavelength of an X-ray photon when it is scattered through an angle 900
by a free electron.
Ans: We know,
 

h
1  cos  
m0 c
6.624  10 3 4
1  cos 90 
3 1
8
9.1 10  3  10


 0.0242  10 1 0 m
 0.242 A
Ans : 0.242 A
36
Dr. Md. Tarek Hossain, Assistant Professor, BEC