COVENANT UNIVERSITY

ALPHA SEMESTER TUTORIAL KIT (VOL. 2)

P R O G R A M M E : P H Y S I C S

300 LEVEL

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DISCLAIMER

The contents of this document are intended for practice and learning purposes at the undergraduate

level. The materials are from different sources including the internet and the contributors do not

in any way claim authorship or ownership of them. The materials are also not to be used for any

commercial purpose.

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LIST OF COURSES

PHY311: MATLAB for Physicist

PHY312: Seismic Methods

PHY313: Electrodynamics

PHY314: Quantum Physics

PHY315: Electric Circuit Theory

PHY316: Statistics and Thermal Physics

*PHY317: Electronics II

PHY318: Petroleum Geology

PHY331: Digital Electronic and Telecommunications

PHY332: Mathematical Method in Physics II

PHY334: Thin Film Technology and Solar Energy Laboratory

PHY335: Energy Conversion and Storage

PHY336: Electrical and Radiometric Methods

*Not included

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COVENANT UNIVERSITY

CANAANLAND, KM 10, IDIROKO ROAD

P.M.B 1023 OTA, OGUN STATE NIGERIA

TITLE OF EXAMINATION: B.Sc. Degree Alpha Semester Examination

COLLEGE: Science and Technology DEPARTMENT: Physics

SEMESTER: Alpha

COURSE CODE: PHY 311 (1 Unit) COURSE TITLE: MATLAB for Physicist

TIME: 1 hour 30 minutes INSTRUCTIONS: Answer All questions

1. (a) Copy and complete the six scalar arithmetic operations in Matlab form in Table 1 below.

Table 1: Six Scalar Arithmetic Operations in Matlab

Symbol Operation MATLAB form

^ exponentiation:

* multiplication:

/ right division:

\ left division:

+ addition:

- subtraction:

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(b) What Constitutes Hardware? Give any examples you know

(c) What Constitutes Software? Give examples you know

2. (a) What Is MATLAB®? Compare Matlab with other programming language.

(b) Copy and complete Table 2 below which describes the eleven commands used in Matlab

for managing the work session.

Table 2: Eleven Commands in Matlab for Managing the Work Session

COMMAND INTERPRETATION

1) clc

2) clear

3) Clear var1, var2

4) exist(‘name’)

5) quit

6) who

7) whos

8) :

9) ,

10) ;

11) …

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(c) Write a Matlab program to plot the Quadratic equation below

y = x2 +2x +1 from x= -5 to +5 at interval of 0.1 with black colour.

3. (a) Given the following linear algebraic equations, write a Matlab Program

to solve for x, y and z.

6x + 12y + 4z = 70

7x – 2y + 3z = 5

2x + 8y – 9z = 64

(b) Write an interactive Matlab program for any user to solve for the roots of the quadratic

equation of the form, ax2 + bx + c = 0.

(c) Write a MATLAB program that plots the following three graphs on a

single page of A4 paper. y = six (x), z =cos (x) and w = tan (x) from 0 to 50 at

interval of 2π using equal line space.

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COVENANT UNIVERSITY

Canaan Land, Km 10, Idiroko Road, P.M.B 1023, Ota Ogun State. Nigeria

TITLE OF EXAMINATION: B.Sc. Degree Semester Examination

COLLEGE: Science and Technology DEPARTMENT: Physics

SEMESTER: Alpha

COURSE CODE: PHY 312 (2 Units) COURSE TITLE: Seismic Methods

INSTRUCTIONS: Answer Question One and Any Other Two Questions TIME: 2 hrs

1. (a) Derived the expression for the calculation of depth ( h ) to a flat horizon on a seismic

refraction survey.

(b) The seismic refraction data below were obtained along a geophysical traverse with a flat

topography (shot, S and geophones, G).

S G1 G2 G3 G4 G5 G6 G7 G8 G9 G10

T (ms) 0 5 10 15 17.5 20 22.5 25 27.5 30 32.5

mGS 101 , mGG 10 . Plot the XT graph and determine the:

i. Velocities of the different layers, and

ii. Overburden thickness(es).

2. (a) Discuss different types of elastic waves and their elastic properties.

(b) Explain multiple reflections on seismic reflection events.

3. Explain the following terms:

i. Stacking,

ii. Primary reflection,

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iii. Reflection coefficients,

iv. CDP, and

v. Normal move out (NMO).

4. (a) Explain Fermat’s principle as it relates to seismic wave propagation.

(b) Discuss different types of shooting methods in seismic reflection survey.

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COURSE CODE: PHY 313 (TUTORIAL)

COURSE TITLE: Electrodynamics

Question 1: Show that ∇. 𝐸(𝑟, 𝑡) = 4𝜋𝜌(𝑟, 𝑡) is a Maxwell's equation

Solution: Let us assume that the change is with respect to distance and time as shown in the

question. Therefore equation [1] can be written as

∇. 𝐸(𝑟, 𝑡) =𝜌(𝑟, 𝑡)

휀0

Recall in coulombs law, there is a constant of proportionality 'k' which is dependent on the

electrical system. Generally, it is denoted as 𝑘 =1

4𝜋 0.When a charge exerts a force of one dyne

on an equal point charge located one centimeter away, 𝑘 = 1. This simply means that under this

condition, 휀0 =1

4𝜋 which changes the outlook of equation [1] to

∇. 𝐸(𝑟, 𝑡) = 4𝜋𝜌(𝑟, 𝑡)

Question 2: Using the Maxwell’s equation, derive the poything theorem

Question3: Show that 𝑐𝑢𝑟𝑙 𝑩(𝑟, 𝑡) − 𝜕𝑬(𝑟,𝑡)

𝑐2𝜕𝑡=

4𝜋

𝑐2 𝑱(𝑟, 𝑡) is a Maxwell's equation

Solution: Make curl B the subject of formular

𝛻×𝐵(𝑟, 𝑡) =4𝜋

𝑐2𝐽(𝑟, 𝑡) +

𝜕𝐸(𝑟, 𝑡)

𝑐2𝜕𝑡

∇×𝐵(𝑟, 𝑡) =1

𝑐2(4𝜋𝐽(𝑟, 𝑡) +

𝜕𝐸(𝑟, 𝑡)

𝜕𝑡)

Applying the idea of example [1]

∇×𝐵(𝑟, 𝑡) =1

𝑐2(

1

휀0𝐽(𝑟, 𝑡) +

𝜕𝐸(𝑟, 𝑡)

𝜕𝑡)

Applying equation [5b] i.e. Maxwell‘s equation

∇×𝐵 = 𝜇0𝐽 + 𝜇0휀0

𝜕𝐸

𝜕𝑡

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It is a Maxwell equation.

Question 4: At the upper surface of the Earth’s atmosphere, the time-averaged magnitude of the

Poynting vector is given ⟨𝑆⟩ = 2.47×105 𝑊/𝑚2

i. What are the magnitudes of the electric and magnetic fields if the Sun’s electromagnetic

radiation is a plane sinusoidal wave?

ii. What is the total time-averaged power radiated by the Sun?

iii. What is the intensity at a distance r from the source?(Take mean Sun-Earth distance is

𝑅 = 1.53×1011𝑚, 휀0 = 8.85×10−12 𝐶2/𝑁. 𝑚2)

Question 5: Show that ∮ 𝐻. 𝑑𝑐

𝑙 = ∫ 𝐽 +s

dD

dtdS is a valid Maxwell's equation

Solution: First, the interpretation is that the magnetic field in an enclosure is directly proportional

to the electric current and changing flux at the surface of the material. Therefore applying equation

[6] & [7] to the equation below,

∮ 𝐻. 𝑑

𝑐

𝑙 = ∫ 𝐽 +

s

dD

dtdS

∮𝐵

𝜇0. 𝑑

𝑐

𝑙 = ∫ 𝐽 +

s

d휀0E

dtdS

Since 𝜇0 and 휀0 are constants,

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𝜇0∮ 𝐵. 𝑑

𝑐

𝑙 = ∫ 𝐽dS + 휀0 ∫dE

dtdS

ss

Since current do not act the surface alone,

∮ 𝐵. 𝑑

𝑐

𝑙 = 𝜇0𝐽 + 𝜇0휀0 ∫dE

dtdS

s

∮ 𝐵. 𝑑

𝑐

𝑙 = 𝜇0𝐽 + 𝜇0휀0

d

dt∫ E. dS

s

It is a Maxwell equation

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Question 6:Using the Maxwell equation show that ∇2𝐴 −1

𝑐2

𝜕2𝐴

𝜕𝑡2− ∇ (∇. 𝐴 +

1

𝑐2

𝜕Φ

𝜕𝑡) = −𝜇0𝐽

Question 7:Using the Maxwell equation show that ∇2Φ +𝜕

𝜕𝑡(∇. 𝐴) = −

𝜌

0

Solution: Substitute equations [30] into [1]

∇. (−∇Φ −𝜕𝐴

𝜕𝑡) =

𝜌

휀0

∇. (−∇Φ) − ∇. (𝜕𝐴

𝜕𝑡) =

𝜌

휀0

−∇2Φ −𝜕

𝜕𝑡(∇. 𝐴) =

𝜌

휀0

Multiply through with (-)

∇2Φ +𝜕

𝜕𝑡(∇. 𝐴) = −

𝜌

휀0

Question 8:Using the Maxwell equation show that ∇2𝐴 −1

𝑐2

𝜕2𝐴

𝜕𝑡2 − ∇ (∇. 𝐴 +1

𝑐2

𝜕Φ

𝜕𝑡) = −𝜇0𝐽

Question 9: Proof that the electric and magnetic fields influences propagation direction of each

other. Assume that the +𝑞𝑥 traveling uniform plane wave is defined by an electric field of 𝐸 =

𝐸𝑧𝑞𝑧 = 𝐸0𝑒−𝜉𝑦𝑞𝑧 and is related by ∇×𝐸 = −𝑗𝜔𝐵

Solution: 𝐵 =1

−𝑗𝜔∇×𝐸

𝐵 =1

−𝑗𝜔[∂Ez

∂y𝑞𝑥 −

∂Ez

∂x𝑞𝑦]

𝐵 =1

−𝑗𝜔[

∂

∂y(𝐸0𝑒−𝜉𝑦𝑞𝑧)𝑞𝑥]

Note that the direction of propagation for this wave is in the same direction as 𝐸×𝐵, therefore

(𝑞𝑦 = 𝑞𝑧×𝑞𝑥; 𝑞𝑧 = 𝑞𝑦×𝑞𝑥; 𝑞𝑥 = 𝑞𝑧×𝑞𝑦)

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𝐵 =1

−𝑗𝜔[

∂

∂y(𝐸0𝑒−𝜉𝑦𝑞𝑦)]

𝐵 =1

𝑗𝜔𝜉𝐸0𝑒−𝜉𝑦𝑞𝑦

𝐵 = 𝐵𝑦𝑞𝑦

A one dimensional wave equation is derived from equations [above] on the condition that it's an

isotropic, homogeneous, dielectric medium without free electric currents can be written as

𝜕2𝐻(𝑥,𝑡)

𝜕𝑥2 = 휀0𝜇0𝜕2𝐻(𝑥,𝑡)

𝜕𝑡2 [i]

𝜕2𝐸(𝑥,𝑡)

𝜕𝑥2 = 휀0𝜇0𝜕2𝐸(𝑥,𝑡)

𝜕𝑡2 [ii]

𝜕2𝐻(𝑥,𝑡)

𝜕𝑥2 =1

𝑣2

𝜕2𝐻(𝑥,𝑡)

𝜕𝑡2 [iii]

𝜕2𝐸(𝑥,𝑡)

𝜕𝑥2 =1

𝑣2

𝜕2𝐸(𝑥,𝑡)

𝜕𝑡2 [iv]

Question 10:Find the integral form of Maxwell equations in a vacuum.

Question 11: Uniform plane wave of frequency of 1MHz travels through an air/steel interface.

Find (i) Position, wavelength and vector velocity in air

(ii)Skin depth, position, wavelength and vector velocity in steel

(𝜇0= 3 X 10-7 Henry-metre and𝜎= 4.2 X 107 Siemens per metre)

Solution: Assume c=v because its air, then

ia) 𝑣 = 𝑐 =𝜔

𝛽→ 𝛽 =

𝜔

𝑐

𝛽 =2×𝜋×106

3×108

𝛽 = 0.021𝑟𝑎𝑑/𝑚

𝑦 =1

𝛽= 47.62𝑚

ib) 𝜆 =𝑐

𝑓

𝜆 =3×108

1×106

𝜆 = 300𝑚

ic) 𝑣 = 𝑐 = 3×108𝑚/𝑠

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iia) 𝛿 =1

√𝜋𝑓𝜇0𝜎

𝛿 =1

√227 ×106×3×10−7×4.2×107

𝛿 = 1.59×10−4𝑚𝑚

iib) 𝛼 = 𝛽 =1

𝛿

𝛽 =1

1.59×10−4

𝛽 = 6,292.85 𝑟𝑎𝑑/𝑚

∴ 𝑦 =1

𝛽= 1.59×10−4𝑚

iic) 𝜆 =2𝜋

𝛽

𝜆 =2𝜋

6,292.85

𝜆 = 9.986×10−4𝑚

iid) 𝑣 = 𝑓𝜆 = 1×106×9.986×10−4

𝑣 = 998.6𝑚/𝑠

Question 12: Find the phasor state of the integral form of Maxwell equations in a vacuum

Question 13:Proof that at total reflection with inversion of E, the ratio of electric field to magnetic

field is 𝐸𝑇

𝐻𝑇= 𝑗𝜂1tan (𝛽1𝑧)

𝑞𝑥

𝑞𝑦

Solution: The total electric field and magnetic field in the medium 1 (reflection) is given as

𝐸1𝑇 = 𝐸(𝑒−𝜉1𝑧 + Γ𝑒𝜉1𝑧)𝑞𝑥

𝐻1𝑇 =𝐸

𝜂1(𝑒−𝜉1𝑧 − Γ𝑒𝜉1𝑧)𝑞𝑦

Applying the conditions 𝜂2 = 0, 𝜏 = 0 and Γ = −1 where 𝜉1 = 𝑗𝛽1

𝐸1𝑇 = 𝐸(𝑒−𝑗𝛽1𝑧 + 𝑒𝑗𝛽1𝑧)𝑞𝑥

𝐻1𝑇 =𝐸

𝜂1(𝑒−𝑗𝛽1𝑧 − 𝑒𝑗𝛽1𝑧)𝑞𝑦 = −2𝑗

𝐸

𝜂1cos (𝑗𝛽1𝑧)𝑞𝑦

Divide 𝐸1𝑇 by 𝐻1𝑇 = −2𝑗𝐸sin (𝑗𝛽1𝑧)𝑞𝑥

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𝐸𝑇

𝐻𝑇= 𝑗𝜂1tan (𝛽1𝑧)

𝑞𝑥

𝑞𝑦

Question 14:Proof that ∇2𝐸 = 휀0𝜇0𝜕2𝐸

𝜕𝑡2+ 𝜇0𝜎

𝜕𝐸

𝜕𝑡

Question 15:A uniform plane wave in air is normally incident on an infinite lossless dielectric

material having 휀 = 3휀0and 𝜇 = 𝜇0. If the incident wave is 𝐸 = −15cos (𝜔𝑡 − 𝑧)𝑞𝑦 V/m, find

i. Angular frequency and wavelength of the waves in both regions

ii. Reflection coefficient and transmission coefficient

Solution:

i). In region 1,

𝛽1 =2𝜋

𝜆1=

𝜔

𝑢1=

𝜔

𝑐

𝛽1 = 1 𝑟𝑎𝑑/𝑚

𝜆1 =2𝜋

𝛽1

𝜆1 =2𝜋

1= 6.284𝑚

Recall 𝑐 = 𝑓𝜆 and 𝜔 = 2𝜋𝑓

∴ 𝜔 = 2𝜋𝑐

𝜆

𝜔 = 2𝜋×3×108

6.284

𝜔 = 3×108𝑟𝑎𝑑/𝑠

Region 2

Recall 𝜇0휀0 = 𝑐−2

𝛽1 =𝜔

𝑐= 𝜔√𝜇0휀0

By the condition, the lossless dielectric material 휀 = 3휀0

∴ 𝛽2 =𝜔

𝑐= 𝜔√3𝜇0휀0

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In other words,

𝛽2 = √3 𝛽1

𝛽2 = √3𝜔

𝑐= √3 𝑟𝑎𝑑/𝑚

𝜆2 =2𝜋

𝛽2

𝜆2 =2𝜋

√3= 3.63𝑚

𝜔 = 2𝜋𝑐

𝜆

𝜔 = 2𝜋×3×108

3.63

𝜔 = 5.19×108𝑟𝑎𝑑/𝑠

ii). 𝛤 =𝜂2−𝜂1

𝜂2+𝜂1

where𝜂1 = 𝜂0 and 𝜂2 =𝜂0

√3

Therefore Γ =

𝜂0

√3−𝜂0

𝜂0√3

+𝜂0

Γ = −0.268

𝜏 = Γ + 1

𝜏 = −0.268 + 1

𝜏 = 0.732

Question 16:Proof that the phasor vector wave equation for electric field is ∇2𝐸 − 𝜉2𝐸𝑣 = 0

Question 17:An air line has characteristic impedance of 18Ω and phase constant of 5 rad/m at

25.5KHz. Calculate the inductance per meter and the capacitance per meter of the line.

Solution: Recall 𝑍 =𝑉0

𝐼0= √

𝐿

𝐶 and the wave number 𝛽 =

𝜔

𝑣, 𝑣 is the phase velocity and its

written as 𝑣 = 1 √𝐿𝐶⁄

𝛽 = 𝜔√𝐿𝐶 (i)

𝑍 = 𝑅 = √𝐿

𝐶 (ii)

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Make L the subject of formula in equation (i) and substitute it into equation (ii)

𝐶 =𝛽

𝜔𝑅=

𝛽

2𝜋𝑓𝑅

𝐶 =5

2𝜋×25500×18

𝐶 = 1.7𝜇𝐹/𝑚

Substitute C into any of equation (i) or (ii). I prefer equation (i)

𝐿 =𝛽2𝐶

𝜔2

𝐿 =52×1.7×10−6

(2𝜋×25500)2

𝐿 = 1.65𝑓𝐻/𝑚

Question 18:A lossless transmission line whose length is 16-m-long possesses a characteristics

impedance 30Ω operating at 35KHz. If it is terminatedwith a load impedance of 𝑍𝐿 = 49 +

𝑗15 Ω. If the phase velocity is u = 0.6c on the line, find

(a) The reflection coefficient

(b) The input impedance

Question 19:If an electromagnetic wave travels at a voltage of 𝑉(𝑥, 𝑡) = 27𝑒𝛼𝑥 sin(6𝜋×108𝑡 −

17𝑥) (𝑉)on a transmission line whose distance is given as 𝑥. Find

(a) frequency

(b) wavelength

(c) Phase velocity of the wave.

(d) 𝛼, when the amplitude of the wave 8 V at x=4.

Solution: From the equation i.e. compared to 𝑉𝑧(𝑥, 𝑡) = 𝑉0𝑒𝛼𝑥cos (𝜔𝑡 − 𝛽𝑥), 𝛽 = 17, 𝜔 =

6𝜋×108, 𝑉2 = 27

a). 𝑓 =𝜔

2𝜋

𝑓 =6𝜋×108

2𝜋

𝑓 = 3×108 𝐻𝑧

b). 𝜆 =2𝜋

𝛽

16

𝜆 =6.284

17 = 0.37𝑚

c). 𝑢 =6𝜋×108

17

𝑢 = 1.11×108𝑚/𝑠

d). Amplitude (A) from the equation is given as 𝑉 = 27𝑒𝛼𝑥. where𝑉 = 8𝑉 and 𝑥 = 4

8 = 27𝑒4𝛼

0.3 = 𝑒4𝛼

𝛼 = −1.204 𝑁𝑝/𝑚

Question 20: Using 𝐸𝑥 = 𝐸0 𝑠𝑖𝑛𝑘𝑦𝑦 𝑠𝑖𝑛𝑘𝑧𝑧𝑒𝑗𝜔𝑡, 𝐸𝑦 = 0 and 𝐸𝑧 = 0 to solve the Maxwell's

equation ∇×𝐸 = −𝜕𝐵

𝜕𝑡. Hence find 𝐵𝑥 , 𝐵𝑦 and 𝐵𝑧.

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PHY 314 TUTORIAL & MARKING SCHEMES

1. a) Give brief history of discovery of quantum mechanics starting

with black body radiation and then how did de Broglie derive his

famous equation.

Ans: Max Planck explained the energy distribution in black body

radiation spectrum by assuming that all electromagnetic radiations of

frequency ν consists of packets of energy quanta with energy E= nhν

where n is an integer (1,2,3 etc.) and h is a constant now known as

Planck’s constant. Einstein’s considered energy quanta as photon with

particle like behavior, capable of knocking electron out of metal surface

(photoelectric effect) if hν is greater than certain value, characteristic of

the metal, now known as work function(𝑊) of the metal. He

successfully explained all experimental observations of the photoelectric

effect by the equation:

ℎν =1

2𝑚𝑣2 + 𝑊

Photon was considered to have rest mass zero and it always moves with

velocity c, the speed of light. Einstein then h.ad also invented his famous

relativistic equation relating energy E with momentum, p as given

below:

E2= p2c2+ 𝑚𝑜2𝑐4

de Broglie applied eq.(2) to photon with rest mass 0 and obtained p =

E/c = hv/c = h/λ.

de Broglie started thinking that photon has wave like behavior as

exhibited through interference, diffraction, polarization etc. and at the

same time it behaves like particles in photoelectric effect. Now photon

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has momentum given by p = h/λ and its wavelength λ = h/p. He

postulated that all physical particles with momentum therefore can also

have waves of wave length given by:

λ = h/p.

This equation known as de Broglie’s equation was later verified to hold

and it gave birth to modern quantum mechanics or wave mechanics.

7 Marks for Q.1a

1b. Describe Davisson-Germer experiment to confirm de-

Broglie’s hypothesis & equation.

Ans: C. J. Davisson and L. H. Germer performed the critical experiment

for verification of de Broglie’s relation λ = h/p. They directed electron

beam of fixed energy (which they controlled by changing the

acceleration voltage V) in a given direction on to a Nickel crystal plane.

They received the scattered electrons on to a photographic screen and

observed diffraction rings similar to the diffraction pattern of X-rays, as

if electrons also behaved like waves. They were able to calculate the

wave length of the electron beam by measurement of the angles of the

rings and the lattice spacing of the Nickel crystal plane. They found that

the wave length correspond to λ = h/p when p = √2𝑒𝑉𝑚0

4 Marks for Q.1b

1c. Calculate the De Broglie wavelength of an electron accelerated by voltage 1000 V.

Ans: λ = h/p when p = √2𝑒𝑉𝑚0.

A λ= h/p=h/√2𝑒𝑉𝑚0 =6.627𝑥10−34𝐽.𝑠

√2𝑥1.6𝑥10−19(𝐶)𝑥1000(𝑉)𝑥9.105𝑥10−31(𝑘𝑔.) =

3.88x10-11 m.

6 Marks for Q.1c

19

1d. Derive Schrodinger’s time dependent and time independent equations, writing

clearly the operators for momentum and energy.

Ans. Schrodinger attached wave ψ (r,t) function to the de Broglie waves

associated with particles. From energy conservation,

E = T + V = p2/2m + V(r) (1)

He assumed ψ (r,t) to be given by classical wave equation:

ψ (r,t) = A𝑒𝑖(𝑘.𝑟− 𝜔𝑡) (2)

He considered both energy, and momentum to be as operators operating

on ψ (r,t). Thus

Eψ (r,t) = [p2/2m + V(r)]ψ (r,t) (3)

Now, ∇ψ (r,t) = 𝑖𝑘 A𝑒𝑖(𝑘.𝑟− 𝜔𝑡) = 𝑖𝑘ψ (r, t) = 𝑖2𝜋

𝜆 ψ (r, t) (4)

Thus the operator ∇ acting on ψ (r,t) is producing ψ (r,t) back with a

multiplicative constant 𝑖2𝜋

𝜆 . He then used de Broglie’s relation 𝜆 =

ℎ

𝑝.

Thus ∇ψ (r,t) = 𝑖2𝜋𝑝

ℎ ψ (r, t) =

𝑖2𝜋𝑝

ħ ψ (r, t) or p ψ (r, t) = −𝑖ħ∇ψ (r,t)

As if the operator p can be given by p = −𝑖ħ∇. Now

p2 = p.p =−𝑖ħ∇. −𝑖ħ∇= −ħ2∇2 (5)

Again 𝜕

𝜕𝑡 ψ (r, t) = −𝑖𝜔ψ (r, t).

Using Planck’s energy quanta relation 𝐸 = ħ𝜔 we then have from the

above relation

20

𝐸 ψ (r, t) = 𝑖ħ𝜕

𝜕𝑡 ψ (r, t). This relation shows that the energy operator E

is given by,

E = 𝑖ħ𝜕

𝜕𝑡 (6)

Substituting the equations (5) and (6) for the operators p2 and energy E

in equation (1) above we have:

𝑖ħ𝜕

𝜕𝑡 ψ (r, t) = [−

ħ2

2𝑚∇2 + V(r)] ψ (r, t) (7)

Equation (7) gives the time dependent Schrodinger equation.

The time independent equation for a particle with constant energy E is

obtained from (7) by replacing ψ (r, t) 𝑏𝑦 ψ (r) and

𝑖ħ𝜕

𝜕𝑡 𝑏𝑦 𝐸 𝑖𝑛 𝐸𝑞. 7.

Thus the time independent Schrodinger equation is:

𝑖ħ𝜕

𝜕𝑡 ψ (r) = [−

ħ2

2𝑚∇2 + V(r)] ψ (r) (8)

6 Marks for Q.1d

______________END OF MARKING SCHEME FOR Q.1_____

2a. What is the physical meaning of normalization and orthogonality condition in quantum mechanics?

Ans: The two conditions can be written mathematically by:

∫ ψ𝑛∗ ψ𝑚 𝑑𝑉 = 𝛿𝑛𝑚

where 𝛿𝑛𝑚 = 1 𝑓𝑜𝑟 𝑚 = 𝑛 𝑎𝑛𝑑 𝛿𝑛𝑚 = 0 𝑓𝑜𝑟 𝑚 ≠ 𝑛

21

This physically means that the probability of finding a particle

simultaneously at two states is zero(orthogonality) and the probability of

finding the particle at a given state over the entire universe is

unity(normalization). It is also given by <nlm> =𝛿𝑛𝑚 .

3Marks for Q.2a

2b. If a wave function is given by the following matrix: 𝜓 = 𝐴 [1𝑖

−1] find its normalization constant.

Ans: Here we use the relation: < 𝜓𝜓 > = 1; Now 𝜓 > = 𝐴 [1𝑖

−1]. < 𝜓 = A[1 –I -1]

Thus < 𝜓𝜓 > = 𝐴2[1 –i -1] [1𝑖

−1] = 𝐴2[1 + 1 + 1] = 3 𝐴2. Now for normalization condition < 𝜓𝜓 ≥ 1

𝐴2 = 1/3 or A =1/√3.

3 Marks for Q.2b

2. c) What is an Hermitian operator? Which of the following two operators is Hermitian. State your reason clearly.

A = [1 3 10 2 00 1 4

] and B = [3 1 01 2 𝑖0 −𝑖 1

] .

Ans: An Hermitian operator H is hermitian if H = H† where H† represents the transpose and complex conjugate of the matrix that represents the operator H.

Obviously B is the hermitian operator of the two. It is because B† = [3 1 01 2 𝑖0 −𝑖 1

] = B.

4 Marks for Q.2a

2.d) Why in Quantum Mechanics do we emphasize so much on Hermitian operators? If two operators commute what do they have in common? Ans: In quantum mechanics we always emphasize on Hermitian operators because Hermitian

operators have real eigen values and eigen vectors.

When two operators communicate they have common eigen values and eigen vectors(a multiplicative

constant may be there ).

22

3Marks for Q.2d

2.e) Give the expressions for the components of angular momentum in Cartesian coordinates. Prove the commutation relations between the x & y components of angular momentum.

Ans: Angular momentum L = rxp. Now p = −𝑖ħ∇ and r = ix + jy + kz

and p = ipx + jpy + kpz

𝑝𝑥 = -−𝑖ħ𝜕

𝜕𝑥 etc. Thus Lx = −𝑖ħ[y

𝜕

𝜕𝑧− 𝑧

𝜕

𝜕𝑦] etc. Ly and Lz are written

with cyclic permutation of x,y and z.

Commutation relations between Lx and Ly .

[Lx , Ly] = LxLy – LyLx = (−𝑖ħ)2 [y𝜕

𝜕𝑧− 𝑧

𝜕

𝜕𝑦] [z

𝜕

𝜕𝑥− 𝑥

𝜕

𝜕𝑧] −

[z𝜕

𝜕𝑥− 𝑥

𝜕

𝜕𝑧] [y

𝜕

𝜕𝑧− 𝑧

𝜕

𝜕𝑦] = −ħ2 [𝑥

𝜕

𝜕𝑦− 𝑦

𝜕

𝜕𝑥] = 𝑖ħ𝐿𝑧.

The students must show the working how the final result 𝑖ħ𝐿𝑧 is arrived

at.

10 Marks for Q.2e

Type equation here.

__________________END of Marking Scheme for Q.2______

3. a) (i)What is Compton effect? (ii) Write with explanations the main three equations for the Compton scattering and then express the change in wavelength in terms of scattering angle of the scattered photon. (iii) What is Compton wavelength?

Ans: (i)Compton effect: When an electromagnetic wave is scattered by a

free electron, the wavelength of the scattered radiation is somewhat

greater than that of the incident radiation. In other words the scattered

23

photon has less energy than the incident photon and the electron is also

scattered. This is known as Compton effect. 2 marks

(ii)The three main equations are as follows:

From energy conservation and knowing that ℎ𝑐

𝜆 ,

ℎ𝑐

𝜆 ′+ (𝑚 − 𝑚𝑜)𝑐2 are

the energies of incident photon, scattered photon, and the scattered

electron respectively we have:

ℎ𝑐

𝜆 =

ℎ𝑐

𝜆′ + (𝑚 − 𝑚𝑜)𝑐2 (1)

From conservation of momentum along the direction of the incident

photon and perpendicular to the direction:

ℎ

𝜆 =

ℎ

𝜆′ 𝑐𝑜𝑠𝜙 +

𝑚0

√1−𝑣2

𝑐2

𝑣𝑐𝑜𝑠𝜃 (2)

ℎ

𝜆′ 𝑠𝑖𝑛𝜙 =

𝑚0

√1−𝑣2

𝑐2

𝑣𝑠𝑖𝑛𝜃 (3)

Equations (1),(2) and (3) are the three main equations of Compton

scattering.

The change in wave length is derived from the above three equations

and is given by:

𝜆′ − 𝜆 =ℎ

𝑚0𝑐(1 − 𝑐𝑜𝑠𝜙) (4)

6 Marks for for Q.3a(ii)

Compton wavelength: ℎ

𝑚0𝑐 in eq.(4) is known as Compton wavelength

which is equal to 0.024 (angstrom unit= 10-10 m). 1 Mark

3.b) Which laws of Physics are proven by Compton effect?

24

Ans: The following laws of Physics are proven by Compton effect. (i) Planck’s energy quantum relation. (ii) de Broglie’s relation for momentum of a photon. (iii) Einstein’s relativistic equations for kinetic energy and mass of a particle 2 Marks c) Write the laws of Photoelectric emission. What is work function of a metal? What does photoelectric effect prove? Ans: Laws of Photoelectric emission: (i) Photoelectrons are emitted from the surface of a metal by incident radiation of frequency v, only if v > vo where vo is the threshold frequency. (ii) The amount of electrons emitted from the metal surface per unit time, i.e. the photo electron current is independent of the frequency of radiation but depends on the intensity of the radiation. (iii) The photocurrent is approximately independent of the temperature of the metal. 3 Marks Work function: It is the minimum energy a photon must have to remove an electron from a metal surface. 2 marks Ans: It proves the particle nature of light photon and Planck’s energy quanta relation; 1 mark d) Find the kinetic energy of the photo electrons emitted from a metal surface with work

function W = 1.5 eV and when it is irradiated by radiation of wavelength 4500 .

Ans: We use the equation

ℎν =1

2𝑚𝑣2 + 𝑊

Here = ℎν =ℎ𝑐

𝜆=

6.627𝑥10−34𝑗.𝑠.𝑥3𝑥108𝑚/𝑠

4500 𝑥10−10 𝑚 = 4.42x10-19 J.

W = 1.5 eV = 1.5x1.6x10-19 J =2.4x10-19 J

Therefore K.E.= 1

2𝑚𝑣2 = 4.42 -2.4x10-19 J = 2.02x10-19 J. 6 Marks

______________END of marking scheme for Q.3_______

3. a) Derive the expression for the energy and wave function of a particle in an infinitely deep square potential well including the normalization of the wave function.

Ans: For an infinite square potential well of width L and one end situated at x=0, V=0 for

0<x≤L and V=∞ for x>L

25

Figure Square well with infinite potential at walls

The time independent Schrodinger equation for a particle is:

[−ħ2

2𝑚∇2 + V(r)]ψ (r) = Eψ (r) . (1)

Within the well, the equation is:

−ħ2

2𝑚∇2ψ (r) = Eψ (r) (2)

In one dimensional x direction, ∇2=𝜕2

𝜕𝑥2 and ψ (r) = ψ (x) (3)

From (2) & (3) we get −ħ2

2𝑚

𝜕2

𝜕𝑥2 ψ (x) = Eψ (x) (4)

The solution of Eq.(4) is ψ (x) = A coskx + Bsinkx (5)

where

k2 = 2𝑚𝐸

ħ2 . (6)

The Boundary conditions are: ψ (x) = 0 at x=0 and x=L

Using ψ (x) = 0 at x=0 in Eq.(5) we get:

A =0; Then

ψ (x) = A sinkx (7)

Using ψ (x) = 0 at x=L, we get sinkL =0 or kL = nπ, or k = nπ/L ,n=1,2,3….

n=0 is ruled out because it will make ψ (x) = 0 for all values of x, which is not permissible.

Substituting k= nπ/L in Eq.(7) we get

ψ (x) = 𝐴𝑠𝑖𝑛 nπx/L (8)

26

From Eq.(6), E =ħ2k2/2m=ħ2n2π2/2mL2 = ħ2𝑛2𝜋2

2m𝐿2 (9)

En = ħ2𝑛2𝜋2

2m𝐿2 (9)

Eq.(8) and (9) give the wave function and energy of the particle within the well.

The normalization constant A can be obtained from the following normalization equation:

∫ ψ(x)∗ψ(x)𝐿

0𝑑𝑥 = 1 (10)

Or, 𝐴2 ∫ 𝑠𝑖𝑛2nπx/L𝐿

0𝑑𝑥 = 1 (11)

Or, 𝐴2

2∫ (1 − 𝑐𝑜𝑠2 nπx/L)

𝐿

0𝑑𝑥 = 1 (12)

Now ∫ 𝑐𝑜𝑠 (2nπx

L) dx = 0

𝐿

0

Then from Eq.(12) we get 𝐴2𝐿

2= 1 or A =√

2

𝐿 (13)

With Eq.(13) Eq.(8) for wave function becomes:

ψ (x) = √2

𝐿𝑠𝑖𝑛nπx/L (14)

Eqs. (9) and (14) gives the final expressions for the energy and wave function of the particle in

an infinitely deep square potential well. 10 Marks for Q.4a

4. b) Sketch the wave functions and the probability density for three levels.

The sketch of the wave function and its square(the probability density) are shown in Figure

below.

27

Fig

5 Marks for Fig. 4b

c) Find the energies (both in joules and eV) of the first two levels(n=1 and n=2) of an electron in an infinitely deep square well potential of width 0.2 nm. Mass of electron me = 9.105x10-31 kg.

What will be the wavelength of radiλFor n=2; E2 = 4E1 =6.04x10-18 J. hv = E2 – E1 = (6.04 –

1.51)x10-18 = 4.53x10-18 J = 28.2 eV.

Using v= c/λ, hc/λ = E2 – E1 = (6.04 – 1.51)x10-18 = 4.53x10-18 J. λ = 4.4x10-8 m.

8 Marks for Q.4c

5. a) Derive the expression for the energy of a level in Bohr’s theory of atomic spectra explaining all assumptions made by Bohr.

Models depicting electron energy levels in hydrogen, helium, lithium, and neon

Calculation of the orbits requires two assumptions.

28

Classical mechanics- Assumption 1.

The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.

where me is the electron's mass, e is the charge of the electron, ke is Coulomb's constant and Z is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This equation determines the electron's speed at any radius:

Conservation of Energy- Assumption 2.

It also determines the electron's total energy at any radius:

The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton.

A quantum rule- Assumption 3.

The angular momentum L = mevr is an integer multiple of ħ:

Substituting the expression for the velocity gives an equation for r in terms of n:

so that the allowed orbit radius at any n is:

The smallest possible value of r in the hydrogen atom (Z=1) is called the Bohr radius and is equal to:

The energy of the n-th level for any atom is determined by the radius and quantum number:

An electron in the lowest energy level of hydrogen (n = 1) therefore has about 13.6 eV less energy than a

motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n =

3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited

atom with one electron in a large circular orbit around the rest of the atom. The above is the Bohr theory

of hydrogen atom assuming that the nucleus is not moving or rotating.

29

12 Marks Q.5a 5b. Find the wavelength of radiation emitted by an electron in an helium atom when it jumps from n= 2 to n=1 level.

Ans: Here Z=2; 𝐸𝑛 = −13.6𝑍2

𝑛2 𝑒𝑉. For n =1; E1 = -54.4 eV; E2 = -13.6 eV; hv = -13.6 eV –( -

54.4 eV) = 40.8 eV; = 40.8 eVx1.6x10-19 J/eV. = 65.28x10-19 J.

Frequency of radiation, v = 65.28x10−19 J

6.627𝑥10−34𝐽.𝑠= 9.8506𝑥1015 Hz.

Wave length of radiation λ= 𝑐

𝑣 =

3𝑥108

9.8506𝑥1015 Hz.= 3.045𝑥10−8 𝑚.

6 Marks for Q.5b 5c. What is the correspondence principle?

Ans. The Correspondence principle says that electron transitions from orbits with very high

quantum number results in radiations having continuous frequencies or wavelength as was

assumed in classical physics. 2 Marks

5d. Write the Schrodinger equation for hydrogen atom in Cartesian coordinates only.

Ans. The Schrodinger equation for hydrogen atom is:

[−ħ2

2𝑚∇2 + V(r)]ψ (r) = Eψ (r) where V(r) = -kZe2/r . k = 1/4πεo = 9x109 Nm2/C2.

∇2 =[1

𝑟2

𝜕

𝜕𝑟(𝑟2 𝜕

𝜕𝑟) +

𝑙(𝑙+1)

𝑟2 ]

Also ∇2=𝜕2

𝜕𝑥2 + 𝜕2

𝜕𝑦2 +𝜕2

𝜕𝑧2 and r =√𝑥2 + 𝑦2+𝑧2. 3 Marks for Q.5d

30

COURSE CODE: PHY 315 (TUTORIAL)

COURSE TITLE: Electric Circuit Theory

1. Find the resistance between the terminals X and Y for the circuit shown in the figure

below

2. Using source transformations, determine the voltage cross 5ohm resistor for the circuit

shown below

3. Use Kirchhoff’s laws to find the current that flows in 2 ohm resistor for the circuit

shown below

4. Apply Thevenin’s theorem to calculate the current in 6 ohm resistor for the circuit shown

below

31

5. Apply superposition theorem to calculate the current in 10 ohm resistor for the circuit

shown below

6. Calculate the current delivered by each of the voltage source for the circuit below

7.

8. Determine the current supplied by each battery in the circuit shown in the diagram below

by using Kirchhoff’s laws

32

9. Calculate the effective resistance between points A and B in the given circuit

10. Find the amount of electrical energy expended in raising the temperature of 45 litres of

water by 75 0C. To what height could a load 5 tonnes be raised with the expenditure of

the same amount of energy? Assume efficiency of heating and lifting equipment to be

90% and 70% respectively. Assume the specific heat capacity of water to be 4186J/kg

and 1 litre of water to have a mass of 1 kg and 1 tonne is equal to 1000kg.

11. Find the magnitudes of total current, current through 𝑅1 and 𝑅2 if, 𝑅1= 10 Ω,

𝑅2= 20 Ω and V = 50 V.

12. Convert the given star in the diagram below into an equivalent delta

33

13. Determine the resistance between the terminal X and Y for the circuit shown below

Note: All the resistors in ohm.

14. Find the current in 4 Ω resistors by Norton’s theorem

15. In the circuit shown below, find the source current by the method of simplification of

network

16. State and explain the following laws and theorems

34

I. Kirchhoff’s current law

II. Kirchhoff’s voltage law

III. Thevenin’s principle

IV. Norton’s theorem

V. Superposition theorem

17. Define the following

(i) Network

(ii) Network element

(iii) Branch

(iv) Junction point

(v) Node

(vi) Mesh

(vii) Loop

(viii) Linear network

(ix) Non linear network

(x) Bilateral network

(xi) Active network

(xii) Passive network

(xiii) Lumped network

(xiv) Distributed network

(xv) Energy source

(xvi) Voltage source time

(xvii) Invariant source

(xviii) Time variant source

(xix) Current source

(xx) Dependent source

(xxi) Voltage dependent voltage source

(xxii) Current dependent current source

(xxiii) Current dependent voltage source

(xxiv) Voltage dependent current source

18. Calculate the average and effective values of the saw tooth waveform shown below. The

voltage completes the cycle by falling back to zero instantaneously after regular interval

of time

35

19. A 60 Hz sinusoidal voltage v=141sin𝜔𝑡 is applied to a series R-L circuit. The values of

the resistance and the inductance are 3 Ω and 0.0106 H respectively.

(i) Write the expression for the instantaneous current in the circuit

(ii) Compute the r.m.s. value of the current in the circuit and its phase angle with

respect to the voltage.

(iii) Compute the r.m.s. value and phase of the voltages appearing across the

resistance and the inductance.

(iv) Find the average power dissipated by the circuit.

(v) Calculate the power factor of the circuit

20. An alternating current varying sinusoid-ally with a frequency of 50 Hz has a r.m.s. value

of current of 20A. At what time measured from negative maximum value, instantaneous

current will be 10√2A

36

COURSE CODE: PHY 316 (TUTORIAL)

COURSE TITLE: Statistical and Thermal Physics

1. What is statistical ensemble and when is it considered to be stationary?

2. Explain the term density function (q,p;t) in statistical physics and relate it with volume

element around a point (q,p). At time t = 0s, the density function of a stationary system is

0.78kgm-3, determine its density 30s after.

3. Using the Liouville’s theorem, derive that the equation of continuity for a swarm of

representative points.

4. Illustrate the validity of Liouville theorem on a one-dimensional movement of material

points in a gravity field with acceleration g (constant). List the relevant equations of

conservation of energy and equations of motion for the particle trajectory. Assume that at

t=0 the particle represented by our point has an initial position x0 and momentum p0

5. Describe in details and differentiate between classical statistical mechanics and quantum

statistical mechanics?

6. What are meant by the following: zero law of thermodynamics, equilibrium, local

equilibrium and relaxation time?

7. What are the properties of entropy?

8. Consider a normal deck of 52 distinct playing cards. A new deck is always prepared in

the same order (A♠2♠ · · · K♣).

(a) What is the information entropy of the distribution of new decks?

(b) What is the information entropy of the distribution of completely randomized

decks?

9. Starting with the Hamiltonian formulation of classical mechanics, derive the expression

for the law of conservation of energy

10. Using the principle of a system in thermal equilibrium, derive the equation for a

microcanonical ensemble

11. Explain briefly the ergodic and the quasi ergodic hypothesis

12. (a) Describe briefly what are meant by systems in thermal contact, mechanical contact

and material-transfer contact.

(b) Derive the equations for the three various systems above, when at equilibrium.

13. A system of N free phase particles, each with mass m in a volume V yields a

microcanonical distribution with energy E. Define for the system a phase space volume

, entropy S, and temperature T.

14. ….

15. A system of N non-interactive spin particles in a magnetic field B has N+ spins parallel to

the field B and N- spins anti-parallel to the field. The system is in thermodynamic

equilibrium at temperature T. Define the entropy S and the relation N- / N+ for the

system.

16. For a Fermi-Dirac Distribution with eigen states 1 and 2 show that the partition

function Z of the system is Z e e 1 2

37

17. (a) Starting with the Helmoltz free energy F(V,), deduce the equation for pressure p and statistical entropy of a system. (b) Given the change in Helmoltz free energy of a system was 13.325Joules and a corresponding change in volume -0.009m-3 and temperature 283K. Calculate the pressure of the system. (c) With a simple diagram illustrate the variation of Helmoltz free energy with volume.

18. Differentiate between the wave functions of Fermi-particles and Bose-particles, give

examples of each particle in real life.

19. (a) Consider an ideal quantum gas of Fermi particles at temperature T, write the

probability p(n) that there are n particles in a given single particle state as a function

of mean occupation number n.

(b) Assuming the universe is filled with black body radiation (photons) at a

temperature T. Deduce an approximate photon number density n analytically in

terms of the temperature T and universal constants.

20 Given a subsystem s which can exchange particles and energy with the heat reservoir r,

show that the grand canonical ensemble is kTENeN /)()( , where N is the number

of particles, is the chemical potential,

SOLUTION 1

1. The number of macroscopically identical systems distributed along admissible

microstates with same density is defined as statistical ensemble. For a statistical

ensemble, the statistical average value has the same meaning as the ensemble average

value.

An ensemble is said to be stationary if the density does not depend explicitly on time, i.e. does

not vary with time that is 0

t

SOLUTION 2

2. Density function (q,p;t) represent the manner in which the members of the ensemble are

distributed over various possible microstates at various instants of time.

38

Density function (q,p;t) is defined in such a way that at any time t, the number of

representative points in the volume element’ (d3Nq d3Np) around the point (q,p) of the phase

space is given by the product (q,p;t) d3Nq d3Np.

For a stationary system,

which implies density is constant. So at time t = 30s, density remains 0.78kgm-3.

SOLUTION 3

Given an arbitrary "volume" in the relevant region of the phase space and let the "surface”

enclosing this volume increases with time is given by

(1)

where d(d3Nq d3Np). On the other hand, the net rate at which the representative points ‘’flow’’

out of the volume (across the bounding surface ) is given by

Net rate of flow = (2)

where v is the vector of the representative points in the region of the surface element d, while

is the (outward) unit vector normal to this element. By the divergence theorem,

= (3)

where the operation of divergence means the following:

(4)

Given that the total number of representative points must be conserved

)dσ(ρσ

nν

)dσ(ρσ

nν

ddiv )( v

N

i

i

i

i

i

pp

qq

div3

1

)()()(

v

d

tddiv

)( v

0)(

ddiv

tv

0t

39

As the volume integral vanish for arbitrary volumes , we have

(5)

This is the equation of continuity for the swarm of the representative points. This equation means

that ensemble of the phase points moving with time as a flow of liquid without sources or

sinks.

SOLUTION 5

Quantum mechanics is the theory that every object can be described in terms of a wave function

which contains all the information about the object concerned.

All systems in nature obey quantum statistical mechanics i.e.

their energy levels are discrete,

particles are indistinguishable

Classical statistical mechanics is valid only as a special, limiting case when the average

occupation of any single–particle quantum state is << 1.

For Classical statistical mechanics –

particles are distinguishable,

There is no restriction on the number of particles that can occupy the same state

Classical mechanics deals with translational motion of atoms and molecules by approximation,

non-translational motions of molecules, such as their rotation and vibration, are very poorly

described by classical mechanics.

While quantum mechanics describes it exactly

SOLUTION 7

• Entropy S has the following important properties:

• dS is an exact differential and is equal to DQ/T for a reversible process, where DQ is the

heat quantity added to the system.

• Entropy is additive: S=S1+S2. The entropy of the combined system is the sum of the

entropy of two separate parts.

0)(

v

div

t

40

• S 0. If the state of a closed system is given macroscopically at any instant, the most

probable state at any other instant is one of equal or greater entropy.

SOLUTION 9

The Hamiltonian of a system is the sum of the kinetic K and potential energies U expressed as a

function of positions and their conjugate momenta. Momentum of a particle is defined in

terms of its velocity

In terms of Cartesian momenta,

the kinetic energy is given by

Then, the Hamiltonian, which is defined to be the sum, K+U, expressed as a function of positions

and momenta, will be given by

Where

In terms of the Hamiltonian, the equations of motion of a system are given by Hamilton's

equations:

Because a system described by conservative forces conserves the total energy, it follows that

Hamilton's equations of motion conserve the total Hamiltonian.

Given that:

41

SOLUTION 11

According to the ergodic hypothesis, the trajectory of a representative point passes, in the course

of time, through each and every point of the relevant region of the phase space.

A little reflection in real life shows that the statement as such cannot be strictly true; so it was

replaced with the quasi-ergodic hypothesis, which states that the trajectory of a representative

point traverses, in the course of time, any neighborhood of any point of the relevant region.

SOLUTION 13

From the definition of a micro canonical ensemble there exists a phase volume in which the

distribution of the particles is uniform. The volume is given by the integral:

iidpdq where i ranges from 1 to N (1)

the limits of the integral are defined by the relationship

otherwise 0=

)2

1+(E),()

2

1-(E if ),( pqconstpq H

(2)

42

In the thermodynamic limit, constVNVN ,, , the volume defined above can be

equated with the volume of phase space bounded by the surface defined in (2).

The Hamiltonian for a set of free particles, each having the same mass, is simply:

N

i

i

m

pE

1

2

2 (3)

This clearly defines a N-dimensional sphere in the momentum space whose radius is given by:

mER 2 (4)

The volume can now be calculated:

p

NV (5)

where VN is a result of integration of the coordinate space for N particles. p is the volume in

momentum space. From the definition of a N-dimensional sphere its volume will be:

2

32

32

323

2

)2(!

2!

2

N

N

NNN

N

N

p EAEmN

RN

(6)

Therefore the total volume of phase space is:

23N

N

N EAV (7)

where AN is a constant

43

the fundamental volume of phase space is defined. Namely:

Nh

3

0 (8)

The entropy of the system follows directly from the definitions:

N

N

N

N

h

EAVkkkS

3

23

0

ln)ln()ln(

(9)

The thermodynamic relation links the Temperature to the entropy for a constant volume and

pressure:

kN

E

E

ST

PV 3

21

,

(10)

SOLUTION 15

44

SOLUTION

This leads to

N+ is parallel, N

- is anti-parallel

Entropy is a measure of the number of energy states

available to the ensemble given its energy at thermo

dynamical equilibrium.

Given N particles and a choice between up or down

what is the number of available combinations?

The solution is !!

!

NN

Nw

from here the entropy is given directly as

!!

!lnln

NN

NkwkS

45

To solve this we can use stirling’s approximation nnnn

n

ln)!ln(lim

The temperature will be given by

46

SOLUTION 17

17a

Helmoltz free energy F(V, ) is defined as:

Thus we have the derivative of F as:

We have from above

And

The temperature will be given by

Therefore

V

EEEF

dF

dVV

FdpdVdddEdF

V

V

Fp

V

F

47

17b.

Using the equation

17c.

The Helmoltz free energy versus volume

The Helmholtz free energy F is a convex function of volume V.

Pap

p

V

Fp

1472009.0

25.13

009.0

25.13

48

SOLUTION 19 a

49

SOLUTION 19 b

50

COVENANT UNIVERSITY

Canaan Land, Km 10, Idiroko Road, P.M.B 1023, Ota Ogun State. Nigeria

TITLE OF EXAMINATION: B.Sc. Degree Semester Examination

COLLEGE: Science and Technology DEPARTMENT: Physics

SEMESTER: Alpha

COURSE CODE: PHY 318 (2 Units) COURSE TITLE: Basic Petroleum Geology

INSTRUCTIONS: Attempt Question One (25 Marks) and Any Other Three Questions

Time Allowed: 2 hrs

1. (a) Discuss the advantages of seismic reflection method over gravity and magnetic method as an

exploration tool for petroleum. (6 Marks)

(b) What is meant by subsidence? Discuss the factors that control subsidence in basin

development. (5 Marks)

(c) State the characteristics of the ideal animal or plant for biostratigraphic analysis. (3 Marks)

(d) Explain the following: lowstand systems tract (LST), highstand systems tract (HST),

transgressive systems tract (TST) and maximum flooding surface (MFS). (7 Marks)

(e) List the stratigraphic formations, from the oldest to the youngest in the following basins in

Nigeria: Niger Delta, Anambra and Bornu basins. (4 Marks)

2. (a) What are reservoirs, discuss the intrinsic properties of reservoirs. (7 Marks)

(b) What is kerogen and how would you classify kerogen? (8 Marks)

3. (a) Distinguish between seismic stratigraphy and sequence stratigraphy. (7 Marks)

(b) What are traps? Discuss the processes responsible for the formation of structural and

stratigraphic traps. (8 Marks)

51

4. (a) What are seals? Distinguish between membrane and hydraulic seals. (7 Marks)

(b) Discuss the processes involve in acquisition, processing and analysis of cores in petroleum

exploration and production. (8 Marks)

5. (a) Write concisely on the following: play and play fairway, lead and prospect, and risk and

uncertainty. (7 Marks)

(b) What is meant by migration? Discuss the various types of migration and their driving

mechanisms. (8 Marks)

52

COURSE CODE: PHY 331 (TUTORIAL)

COURSE TITLE: Digital Electronic and Telecommunications

Perform each of the following conversions:

(i) 141710 = __________2

(ii) 110100012 = __________10

(iii) 3𝐸1𝐶16 = __________10

(iv) 100101000111𝐵𝐶𝐷 = __________10

(v) 186510 = __________𝐵𝐶𝐷

2. Apply the input waveforms of Fig.1 below to a (i) NAND gate (ii) NOR gate and draw the

output waveform.

3. Simplify the following algebraic expressions

(i) 𝑍 = 𝑋( + 𝑌)( + )

(ii) 𝑍 = ( + ) + 𝑋𝑌

(iii) 𝑍 = (𝑋𝑊 + 𝑌)( + )

4. Draw the circuit diagram that implements the expression 𝑥 = 𝐴𝐵 + 𝐶

5. Give the truth table for the circuit below and determine the output X.

6. x1x0 represents a 2-bit binary number that can have any value (00, 01, 10 or 11); for

example, when x1 = 1 and x0 = 0, the binary number is 10, and so on. Similarly, y1y0

represents another 2-bit bnary number. Design a logic circuit, using x1x0, y1y0 inputs whose

output will be HIGH only when the two binary numbers x1x0 and y1y0 are equal.

U1

AND

U2

AND

U3

AND

1

2

3

U4:A

4071

A

B

C

D

E

X

53

7. A 4-bit binary number is represented as A3 A2 A1 A0 where A3 A2 A1 and A0 represent

individual bits with A0 equal to LSB. Design a logic circuit that will produce a HIGH output

whenever the binary number is greater than 0010 and less than 1000.

8. The Figure below shows waveforms of a SC flip-flop that triggers on positive going

transition (PGT) of a clock signal. Deduce, with a brief explanation the output waveform Q.

9. Reduce the following equations to MSP form using Karnaugh map technique

𝑍 = [𝑄 + 𝑌(𝑄 + 𝑊)] + 𝑋𝑌𝑊

10. Apply the input waveforms of Fig.1 below to a (i) NAND gate (ii) NOR gate and draw the

output waveform:

11. A Boolean function is described by the truth table given below with Z as the output.

X Y W Q Z

0 0 0 0 0

0 0 0 1 0

0 0 1 0 0

0 0 1 1 0

0 1 0 0 1

0 1 0 1 1

0 1 1 0 0

0 1 1 1 0

1 0 0 0 0

1 0 0 1 0

1 0 1 0 1

1 0 1 1 1

1 1 0 0 1

1 1 0 1 1

1 1 1 0 1

1 1 1 1 1

54

(i) Using Karnaugh map techniques, determine the MSP equation that describes this function

(ii) Find the complementary function, 𝑍, by complementing each cell of the Karnaugh map, finding new

enclosures and determining new products.

(iii) Verify that this result is consistent with the result that would have been obtained by complementing

algebraically the MSP equation of (i) and applying De Morgan’s law.

12. Give the truth table for the circuit below and determine the output X.

13. The function depicted by the Karnaugh given below is to be realized with a minimum number of

two-input NAND gates. No other gates are available. Find the minimal network.

𝑄 𝑊𝑄 𝑊

1 0 0 1

𝑌 1 1 1 0

𝑋𝑌 1 1 1 0

𝑋 1 0 0 1

14. (a) Design a logic circuit whose output goes HIGH only when either inputs A or B or

both is LOW while C and D are NOT both HIGH

(b) In a basic counter each FF can provide an output waveform that is exactly half the

frequency of the waveform at its CLK input similar to those in question 5 (b). If the clock

signals entering a counter is 16 kHz

(i) Draw a four stage ripple counter using JK FFs to implement this

(ii) Draw waveforms one over the showing frequency division from 16 kHz right down

to 2 kHz.

15. (a) Write the Boolean equation for the circuit shown. Use De Morgan’s theorem and then

Boolean algebra rules to simplify the equation. Draw the simplified circuit.

U1

AND

U2

AND

U3

AND

1

2

3

U4:A

4071

A

B

C

D

E

X

55

(b) Use De Morgan’s theorem and Boolean algebra to prove that the two circuits (i) and (ii)

shown below are equivalent.

16. Use Karnaugh map to design a circuit that will output a HIGH whenever the 4-bit

hexadecimal input is an even number from 0 to 10.

17. A chemical plant needs a microprocessor-driven alarm system to warn of critical

conditions in one of its chemical tanks. The tank has four HIGH/LOW fluid

switches that monitor the temperature (T), pressure (P), fluid level (L) and weight

(W). Design a system that will notify the microprocessor to activate an alarm

when any of the following conditions arise:

(i) High fluid level with high temperature and high pressure

(ii) Low fluid level with high temperature and high weight

(iii) Low fluid level with low temperature and high pressure

(iv) Low fluid level with low weight and high temperature.

Write the Boolean equation and use Karnaugh mapping and then draw the logic

circuit.

18. The following pulses go into a clocked NAND gate RS flip-flop

1

2

3

1

2

3

1

2

3

1

2

3

A

B

C

D

X

1

2

3

5

6

4

1

2

3

1

2

3

1

2

3

5

6

4

A

B B

A

X1 X2

(i) (ii)

56

Draw the NAND gate equivalent of the clocked RS flip flop and use it to determine

the output waveform Q.

19. In a chemical factory, the contents of a vat must be pumped into a reaction chamber by an

electrically driven pump via sensors. The pumping action must not commence unless:

(i) the pH has reached a specified value and

(ii) the temperature of the contents is correct and

(iii) the vat is full and

(iv) the start button is pressed

The pumping action must stop

(v) immediately the emergency button is pressed or

(vi) if the temperature drops too low a value or

(vii) if the pH becomes too low (or too high) or

(viii) 5 seconds after the level of the liquid in the vat has fallen below a specified level (5 sec.

Delay)

All instructions (i)-(viii) must be sent to the pump driver through a memory bloc.

Design a logic circuit that may describe this operation.

20. Determine the minimum expression for the k-map below and draw the logic circuit

for each final output expression:

𝐶 𝐶D CD C

1 1 0 1

𝐵 0 1 0 0

𝐴𝐵 0 0 0 0

𝐴 1 1 0 1

57

Answers (PHY 331)

1. (i) 141710 = 10110001001

(ii) 110100012 = 209

(iii) 3𝐸1𝐶16 = 1590010

(iv) 100101000111𝐵𝐶𝐷 = 94710

(v) 186510 = 0001100001100101𝐵𝐶𝐷

3. (i) 𝑍 = 𝑋( + 𝑌)( + )

𝑍 = (𝑋 + 𝑋𝑌)( + 𝑋)

𝑍 = 0 + 𝑋𝑌( + 𝑋) = 𝑋𝑌 + 𝑋𝑌𝑋

𝑍 = 𝑋𝑌

(ii) 𝑍 = ( + ) + 𝑋𝑌

𝑍 = ( + ) + ( + )

𝑍 = ( + ) + ( + ). 1

𝑍 = ( + ) + ( + 1)

𝑍 = +

(iii) 𝑍 = (𝑋𝑊 + 𝑌)( + )

𝑍 = (𝑋𝑊 + 𝑌)(. )

𝑍=(𝑋𝑊 + 𝑌)(𝑋𝑊) 𝑍 = (𝑋𝑊 )(𝑋𝑊) + 𝑌(𝑋𝑊) 𝑍 = 0 + 𝑌𝑋𝑊

𝑍 = 𝑋𝑌𝑊 5. Answer: A(AB) +B(AB) = AAB + ABB = AB + AB = AB

7. The logic circuit will produce a HIGH output whenever the binary number is >0010 (i.e. 2

decimal ) and <1000 (i.e. 8 decimal). i.e.

A3 A2 A1 A0 X

0 0 0 0 0 0

1 0 0 0 1 0

2 0 0 1 0 0

3 0 0 1 1 1 OUTPUT HIGH

4 0 1 0 0 1 OUTPUT HIGH

5 0 1 0 1 1 OUTPUT HIGH

6 0 1 1 0 1 OUTPUT HIGH

7 0 1 1 1 1 OUTPUT HIGH

8 1 0 0 0 0

3 𝟑𝟐𝑨𝟏𝑨𝟎

4 𝟑𝑨𝟐𝟏𝟎

5 𝟑𝑨𝟐𝟏𝑨𝟎

6 𝟑𝑨𝟐𝑨𝟏𝟎

7 𝟑𝑨𝟐𝑨𝟏𝑨𝟎 (5 marks)

By ORing the outputs:

𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐𝟏𝟎 + 𝟑𝑨𝟐𝟏𝑨𝟎 + 𝟑𝑨𝟐𝑨𝟏𝟎 + 𝟑𝑨𝟐𝑨𝟏𝑨𝟎

58

𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐𝟏𝟎 + 𝟑𝑨𝟐𝟏𝑨𝟎 + 𝟑𝑨𝟐𝑨𝟏(𝟎 + 𝑨𝟎)

𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐𝟏𝟎 + 𝟑𝑨𝟐𝟏𝑨𝟎 + 𝟑𝑨𝟐𝑨𝟏

𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐𝟏(𝟎 + 𝑨𝟎) + 𝟑𝑨𝟐𝑨𝟏

𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐𝟏 + 𝟑𝑨𝟐𝑨𝟏 = 𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐(𝟏 + 𝑨𝟏)

𝟑𝟐𝑨𝟏𝑨𝟎 + 𝟑𝑨𝟐 = 𝟑(𝟐𝑨𝟏𝑨𝟎 + 𝑨𝟐)

9. 𝑍 = [𝑄 + 𝑌(𝑄 + 𝑊)] + 𝑋𝑌𝑊

𝑍 = 𝑄 + 𝑌𝑄 + 𝑌𝑊 + 𝑋𝑌𝑊 i.e.

𝑄 𝑊𝑄 𝑊

0 1 1 0

𝑌 0 1 1 1

𝑋𝑌 0 0 0 1

𝑋 0 0 0 0

MSP equation: 𝑍 = 𝑄 + 𝑌𝑊

11. X Y W Q Z

0 0 0 0 0

0 0 0 1 0

0 0 1 0 0

0 0 1 1 0

0 1 0 0 1 OUTPUT Z HIGH

0 1 0 1 1 OUTPUT Z HIGH

0 1 1 0 0

0 1 1 1 0

1 0 0 0 0

1 0 0 1 0

1 0 1 0 1 OUTPUT Z HIGH

1 0 1 1 1 OUTPUT Z HIGH

1 1 0 0 1 OUTPUT Z HIGH

1 1 0 1 1 OUTPUT Z HIGH

1 1 1 0 1 OUTPUT Z HIGH

1 1 1 1 1 OUTPUT Z HIGH

(5 marks)

𝑄 𝑊𝑄 𝑊

0 0 0 0

𝑌 1 1 0 0

𝑋𝑌 1 1 1 1

𝑋 0 0 1 1 (2 marks)

𝑍 = 𝑌 + 𝑋𝑊 (2 marks)

(ii) Complementing the function Z, i.e. ,

𝑄 𝑊𝑄 𝑊

1 1 1 1

𝑌 0 0 1 1

59

𝑋𝑌 0 0 0 0

𝑋 1 1 0 0 (2 marks)

𝑍 = + 𝑊 (2 marks)

(iii) Verification

From 𝑍 = 𝑌 + 𝑋𝑊

𝑍 = 𝑌 + 𝑋𝑊 = (𝑌) + (𝑋𝑊 )

𝑍 = ( + )( + ) = ( + 𝑊)( + )

𝑍 = + + 𝑊 + 𝑊 = + + 𝑊

𝑍 = (𝑊 + ) + + 𝑊 =

𝑍 = 𝑊( + 1) + ( + 1)

𝑍 = + 𝑊

13. 𝑄 𝑊𝑄 𝑊

1 0 0 1

𝑌 1 1 1 0

𝑋𝑌 1 1 1 0

𝑋 1 0 0 1

Minimal network:

𝑍 = + 𝑌𝑄 +

𝑍 = ( + ) + 𝑌𝑄 = (𝑌𝑊 ) + 𝑌𝑄

𝑍 = (𝑌𝑊 𝑌𝑄 = (𝑌𝑊 ) (𝑌𝑄 ) 𝑁𝐴𝑁𝐷 𝑓𝑜𝑟𝑚

19. The pumping action must not commence unless:

(i) the pH has reached a specified value AND

(ii) the temperature (T) of the contents is correct AND

(iii) the vat is full (VF) AND

(iv) the start button (SB) is pressed

The pumping action must stop

(v) immediately the emergency button (EB) is pressed OR

(vi) if the temperature (T) drops too low a value OR

(vii) if the pH becomes too low (or too high) OR

(viii) 5 seconds after the level of the liquid in the vat has fallen below a specified level (5 sec.

Delay, DY)

The commencement of the pumping action is governed by AND function while the cessation is

governed by OR function, i.e.

60

ph AND T AND VF AND SB

EB OR OR 𝑝𝐻 OR DY

2

3

4

5

1

U1:A

4012

2

3

4

5

1

U2:A

4002

1

2

3

U3:A

4011

5

6

4

U3:B

4011

pHT

VFSB

EB

DY

MOTOR PUMPMEMORY DRIVER

61

COURSE CODE: PHY 332 (TUTORIAL)

COURSE TITLE: Mathematical Methods in Physics II

1.

2. find the fourier transform of the exponenetial decay functionf(t)=0 for t<0 and f(t)= Ae-λt for t ≥ 0 (λ>0)

3.

.

4.

5.

62

.

6.

.

7.

8.

63

9.

9.

10

.

11.

64

12.

13.

14

15.

65

16.

17.

18.

19

20.

66

67

COVENANT UNIVERSITY

MARKING GUIDE

COURSE CODE: PHY334

COURSE TITLE: Thin Films Technology and Solar Energy Laboratory SEMESTER: ALPHA

1A (i) DC Sputtering

DC Sputtering is the simplest model among other sputtering systems. The dc sputtering system

is composed of a pair of planar electrodes. One of the electrodes is a cold cathode and the other

is the anode. The front surface of the cathode is covered with target materials to be deposited.

The substrates are placed on the anode. The sputtering chamber is filled with sputtering gas,

typically argon gas at 5 Pa (4 × 10-2 torr). The glow discharge is maintained under the application

of dc voltage between the electrodes. The Ar+ ions generated in the glow discharge are

accelerated at the cathode fall (sheath) and sputter the target, resulting in the deposition of the

thin films on the substrates. The target is composed of metal since the glow discharge (current

flow) is maintained between the metallic electrodes.

4 Marks

1A (ii) RF Sputtering

In the rf-sputtering system, the thin films of the insulator are sputtered directly from the insulator

target.

By simple substitution of an insulator for the metal target in the dc sputtering discharge system,

the sputtering discharge cannot be sustained because of the immediate build up of a surface

charge of positive ions on the front side of the insulator. To sustain the glow discharge with the

insulator target, rf voltage is supplied to the target. This system is called rf-sputtering.

4 Marks

68

1A (iii) Magnetron Sputtering

In magnetron sputtering, a magnetic field is superposed on the cathode and glow discharge,

which is parallel to the cathode surface. The electrons in the glow discharge show cycloidal

motion, and the centre of the orbit drifts in the direction of E×B with the drift velocity of E/B,

where E and B denote the electric field in the discharge and the superposed transverse magnetic

field, respectively. The magnetic field is oriented such that these drift paths for electrons form a

closed loop. This electron trapping effect increases the collision rate between the electrons and

the sputtering gas molecules. This enables one to lower the sputtering gas pressure as low as 10-5

torr, but more typically 10-2 torr. In the magnetron sputtering system, the magnetic field

increases the plasma density which leads to increases in the current density at the cathode target,

effectively increasing the sputtering rate at the target. Due to the gas’s low working pressure, the

sputtered particles traverse the discharge space without collisions, which results in a high

deposition rate.

3 Marks

1A (iv)

(i) Resistive heating 1

2 Mark

(ii) Electron beam 1

2 Mark

Resistive heating is most commonly used for the deposition of thin films. The source materials

are evaporated by a resistively heated filament or boat, generally made of refractory metals such

as W, Mo, or Ta, with or without ceramic coatings. Crucibles of quartz, graphite, alumina,

beryllia, boron-nitride, or zirconia are used with indirect heating. The refractory metals are

evaporated by electron-beam deposition since simple resistive heating cannot evaporate high

melting point materials.

69

31

2 Marks

E-beam evaporation is a process similar to thermal evaporation that is, a source material is

heated above its boiling/sublimation temperature and evaporated to form a film on the surfaces

that is stroke by the evaporated atoms. The noticeable advantages of e-beam evaporation over

thermal evaporation are: precise control of low or high deposition rates; excellent material

utilization; co-deposition and sequential deposition systems and, a uniform low temperature

4 Marks

1B (1) Transmission sputtering occurs when a thin foil is bombarded with energetic particles,

and some of the scattered atoms transmit through the foil. 1 Mark

1B (ii) Cathode sputtering is the destruction of a solid through bombardment by charged or

neutral particles. It shortens the life of electronic devices. It is of practical use for cleaning

70

surfaces, revealing the structure of a substance (ionic etching), depositing thin films, and

producing directional molecular beams.

The bombarding ions that penetrate the target produce displacement of its atoms.

These displaced atoms in turn canproduce new displacements. Some of the atoms

reach the surface and pass out of it. Under certain conditions particles may leave the

target’s surface in the form of ions. In single crystals conditions are most favourable

for particle emission in directions in which the atomic packing density is greatest.

1 Mark

1B (iii) Hot wall reactor uses a heating system that heats up not only the wafer, but the walls of

the reactor itself, an example of which is radiant heating from resistance-heated coils.

1 Marks

(i) Cold-wall reactors use heating systems that minimize the heating up of the reactor

walls while the wafer is being heated up, an example of which is heating via infrared

lamps inside the reactor. 1Marks

2A (i) Reactive sputtering occurs when a reactive gas species such as oxygen or nitrogen is

introduced into the chamber, and the thin films of compounds (i.e., oxides or nitrides)

are deposited by the sputtering of the appropriate metal targets. 1 Mark

2A (ii)

(a) Low pressure chemical vapour deposition 1

2 Mark

(b) Atmospheric pressure chemical deposition 1

2 Mark

(c) Plasma enhanced chemical vapour deposition 1

2 Mark

2A (iii)

71

(a) 3

1

2 Marks

2A (iv) Chemical vapor deposition is the formation of a non-volatile solid film on a substrate by

the reaction of vapor phase chemicals (reactants) that contain the required constituents.

1Mark

2 B

(i) Low-pressure CVD reactors operate at medium vacuum (30-250 Pa) and higher temperature

than APCVD reactors. 1 Mark

(ii) Plasma Enhanced CVD reactors also operate under low pressure, but do not depend

completely on thermal energy to accelerate the reaction processes. 1 Mark

(iii) Atmospheric pressure CVD (APCVD) reactors operate at atmospheric pressure, and are

therefore the simplest in design. 1 Mark

CVD

Process Advantages Disadvantages

LPCVD

Excellent purity,

Excellent uniformity,

Good step coverage,

Large wafer capacity

High temperature,

Slow deposition

Arrival Flow Rate

Substrate

Input Flow Rate

r = Growth Rate of Film g

r g Surface Reaction Rate

Gro

wth

Ra

te

Fil

m

Chemical Vapor Deposition

CVD Process

Surface Reaction

72

9 Marks

2C A basic CVD process consists of the following steps:

(i) A predefined mix of reactant gases and diluents inert gases are introduced at a

specified flow rate into the reaction chamber;

(ii) The gas species move to the substrate

(iii) The reactants get adsorbed on the surface of the substrate

(iv) The reactants undergo chemical reactions with the substrate to form the film

(v) The gaseous by-products of the reactions are desorbed and evacuated from the

reaction chamber.

(vi) Reactions that take place in the gas phase are known as homogeneous reactions.

(vii) Homogeneous reactions form gas phase aggregates of the depositing material, which

adhere to the surface poorly and at the same time form low-density films with lots of

defects.

(viii) Heterogeneous reactions are much more desirable than homogeneous reactions

during chemical vapor deposition. 41

2 Marks

3(i) Transmission electron microscopy (TEM) is a microscopy technique in which a beam

of electrons is transmitted through an ultra-thin specimen, interacting with the specimen as it

passes through. An image is formed from the interaction of the electrons transmitted through the

specimen; the image is magnified and focused onto an imaging device, such as

a fluorescent screen, on a layer of photographic film. TEMs are capable of imaging at a

PECVD

Low temperature,

Good step coverage

Chemical and particle

contamination

APCVD

Simple,

Fast deposition,

Low temperature

Poor step coverage,

contamination

73

significantly higher resolution than light microscopes, owing to the small de Broglie

wavelength of electrons. This enables the instrument's user to examine fine detail even as small

as a single column of atoms, which is thousands of times smaller than the smallest resolvable

object in a light microscope. TEM forms a major analysis method in a range of scientific fields,

in both physical and biological sciences. TEMs find application in cancer

research, virology, materials science as well as pollution, nanotechnology,

and semiconductor research. At smaller magnifications TEM image contrast is due to absorption

of electrons in the material, due to the thickness and composition of the material. At higher

magnifications complex wave interactions modulate the intensity of the image, requiring expert

analysis of observed images. Alternate modes of use allow for the TEM to observe modulations

in chemical identity, crystal orientation, electronic structure and sample induced electron phase

shift as well as the regular absorption based imaging.

6 Marks

Scanning electron microscope (SEM) is a type of electron microscope that produces images of

a sample by scanning it with a focused beam of electrons. The electrons interact with atoms in

the sample, producing various signals that can be detected and that contain information about the

sample's surface topography and composition. The electron beam is generally scanned in a raster

scan pattern, and the beam's position is combined with the detected signal to produce an image.

SEM can achieve resolution better than 1 nanometre. Specimens can be observed in high

vacuum, in low vacuum, in wet conditions (in environmental SEM), and at a wide range of

cryogenic or elevated temperatures. The most common mode of detection is by secondary

electrons emitted by atoms excited by the electron beam. On a flat surface, the plume of

secondary electrons is mostly contained by the sample, but on a tilted surface, the plume is

partially exposed and more electrons are emitted. By scanning the sample and detecting the

secondary electrons, an image displaying the topography of the surface is created.

6 Marks

74

51

2

Marks

3 (iii) Molecular beam epitaxy (MBE) for a typical MBE-deposition process the material that

needs to be deposited is heated in ultra high vacuum (UHV) and forms a molecular beam. The

atoms of the beam are then adsorbed by the sample surface (adatoms). During the deposition of

the atoms the adatoms interact with the atoms of the surface. This interaction depends on the

type of adatoms, the substrate, and the temperature of the substrate. To achieve good-quality film

growth, the growth rate must be small (typical growth rate: 0.05 ... 1 Å/s) and therefore the

vacuum pressure in the ultra-high vacuum regime is typically only a few 10−11 mbar.

Transmission electron microscope Scanning electron microscope

It has Broad and static beams

Beam focused to fine point; sample is

scanned line by line

Its voltage ranges from 60-300,000

volts

Accelerating voltage much lower; not

necessary to penetrate the specimen

Its specimen must be very thin

Wide range of specimens allowed;

simplifies sample preparation

Its electrons must pass through and be

transmitted by the specimen

Information needed is collected near

the surface of the specimen

Transmitted electrons are collectively

focused by the objective lens and

magnified to create a real image

Beam is scanned along the surface of

the sample to build up the image

75

6 Marks

4 (i)

(1) Amorphous silicon 1

2 Mark

(2) Cadmium telluride 1

2 Marks

(3) Copper indium gallium Diselenide 1

2 Marks

4 (ii)

Advantages

(1) It requires little semiconductor material

(2) The glass is cheap to get

(3) Easy to handle

(4) More flexible solar cells

(5) Available as thin wafer sheets

Disadvantages

1) Less efficiency (20 to 30% of light converted into electricity)

2) Complex structure

3) Need to be very careful in handling

76

4) Difficult to manufacture good films

4 (iii)

(a) Optical properties 1Mark

(b) Electrical properties 1Mark

(c) Magnetic properties 1Mark

(d) Mechanical properties 1Mark

4 (B)

(i) Transmissivity: Is defined a fraction of incident radiation that is transmitted.

𝑇 =𝐼𝑇

𝐼𝑂 2 Marks

(ii) Absorptivity: Is defined as a property of a material that determines the fraction of incident

radiation that is absorbed.

𝐴 =𝐼𝐴

𝐼𝑂 2 Marks

(ii) Reflectivity: Is the function of incident radiation reflected by a surface.

𝑅 =𝐼𝑅

𝐼𝑂

𝐼𝑂 = 𝐼𝑇 + 𝐼𝐴 + 𝐼𝑅

T + A + R = 1 2 marks

4C (i) Optics

(a) Antireflection coating on lenses 1

2 Mark

(b) Reflection coating for mirrors 1

2 Mark

(c) Coatings to produce decorations 1

2 Mark

(d) CD’s, DVD’S and upcoming D’s 1

2 Mark

4C (ii) Chemistry

(a) Diffusion barriers

(b) Sensors for liquid 11

2 Marks

77

4C (iii) Mechanics

(a) Hard layer (drill bits)

(b) Adhesions providers 11

2 Marks

4C (iv) Magnetism

(a) Hard disc

(b) Video/audio tape 11

2 Marks

4C (v) Electricity

(a) Insulating film

(b) Conducting film 11

2 Marks

5 (i) Thin film is defined as a low-dimensional material created by condensing, one-by one,

atomic/molecular/ionic species of matter. 11

2 marks

The thickness is typically less than several microns, while thick film is defined as a low-

dimensional material created by thinning a three-dimensional material of atomic species.

11

2 Marks

5 (ii) Characterization is a measurement of thin-film properties such as chemical composition,

crystalline structure, and optical, electrical, and mechanical properties which are indispensable

for the study of thin-film materials and devices. 2 Marks

5 (iii)

Film Properties Evaluation Methods

Thickness

Ellipsometry

Cross-sectional SEM & TEM

Mechanical stylus

Surface roughness

Mechanical stylus and/or optical

microscope

Scanning electron microscope

Atomic force microscope

Chemical composition

Inductively coupled plasma

Optical emission

Spectroscopy(ICP)

78

Rutherford backscattering

spectroscopy(RBS)

Auger electron

Spectroscopy(AES)

Electron probe microanalysis(EPMA)

X-ray photoelectron spectroscopy

(SIMS)

Structure

Electron and / or X-ray diffraction

analysis

X-ray photoelectron spectroscopy

(XPS)

Electron energy loss spectroscopy

(EELS)

Optical absorption

Adhesion

Peeling method,

Scratching method

Pulling method

Stress

Disk method

Bending beam method

X-ray diffraction method

Hardness

Micro-Vickers hardness

measurement and

nano-identation measurement

Wear and friction

Wear test between film

coated ball and iron plate.

Sand blast method.

Electrical resistivity

Standard four terminals

resistive measurements

79

Dielectric constant

Dielectric measurements:

at sandwich structure;

evaporated electrode /

dielectric film / evaporated

electrode on substrate, or at

inter-digital electrodes (IDE)

on dielectric films.

Piezo-electricity Electro-mechanical coupling

kt: admittance

measurements at sandwich

structure; evaporated

electrode / piezoelectric

film / evaporated electrode

on fuzzed quartz substrate.

d33, at sandwich structure;

piezoelectric film /

conductive substrate, Si,

La-doped SrTiO3. d31, at

micro-cantilevers

181

2 Marks

80

COVENANT UNIVERSITY

COLLEGE OF SCIENCE AND TECHNOLOGY

DEPARTMENT: INDUSTRIAL PHYSICS

COURSE TITLE: Energy Conversion and Storage

COURSE CODE: PHY 335

COURSE UNIT: 2 UNITS

TIME: 2 hours

Instruction: Answer Question one and any other two questions.

1. a. Explain the concept of ‘total energy usage’; use a bar chart and any other useful resource to

complement your ideas. 15 marks

b. Distinguish between traditional energy production and alternate forms of energy. Give

examples of each type. 5 marks

c. Give a detailed account of the need for alternate forms of energy; with respect to the

following glossary terms:

. 10 marks

2. a. Review the law of thermodynamics with respect to energy generation and flow.

5 marks

b. Derive the Inverse Square law, relate it to Wien’s law (use the concept of solar

radiation, irradiance and iiradiation). 4 marks

c. Describe a simple experiment to illustrate the Inverse Square Law. 4 marks

d.(i) List the factors that determine the quality of light. 2 marks

(ii) Calculate the brightness of a bulb if its luminosity is given as 4000 lumens, the

radius sphere is 1 metre. 5 marks

3. a. Explain the following terms:

(i) p- type Semi conductor

(ii) electron-hole pair

(iii) band gap

(iv) solar cell 8 marks

(Illustrative diagrams are important)

b. Define the following short circuit current (Isc), open circuit voltage (Voc), Fill factor and

efficiency of a solar cell. Draw an I-V graph to illustrate (Isc), (Voc) and maximum power. 8

marks

81

c. the solar constant is given as 1360 W/m2

, area of solar cell is 100mm X 100mm, Isc is

0.45 A , Voc is 100mV. Given a maximum power output of 0.5W, determine the efficiency

of the solar cell. 4 marks

4. a. (i) Distinguish between windmills, heat engines and classical engines. 3 marks

(ii) list the types of wind turbines and heat engines, differentiate each type. The

merits, de-merits, efficiency. 2 marks

(iii) Illustrate the difference between lift and drag with a diagram, which is preferred

in turbines. Give reasons for your answer. 4 marks

b. (i) What are solar concentrators and what are they used for? 1 mark

(ii) Differentiate between the terms, ‘cut in’ speed and ‘cut out’ speed. Explain Betz

limit. 6marks

c. What is the Tip Speed Ratio (TSR) for a 90m diameter turbine rotating at 15rpm at

a wind speed of 10m/s?

Given a wind turbine with the following parameters, calculate the annual energy

production: Cp = Cp max = 0.48 (where CP is the maximum value assumed constant for all

wind speeds).

Rated power= 5MW, wind speed =10m/s, Area = 7238.2𝑚2, 𝜌𝑎𝑖𝑟= 1.20Kg/m3

4 marks

5. a. Discuss the concept of energy storage you would recommend for nuclear energy in

reactors, is nuclear energy a renewable source of energy? Give reasons for and against this

notion. 5 marks

b. write short notes on the following:

(i) geothermal energy 5 marks

(ii) tidal energy 5 marks

(iii) ocean thermal energy 5 marks

How would you recommend them for the generation of electricity? Mention the operation

cost, handling cost. The advantages and disadvantages and their impact on the ecosystem.

82

COVENANT UNIVERSITY

CANAANLAND, KM 10, IDIROKO ROAD

P.M.B 1023, OTA, OGUN STATE, NIGERIA. TITLE OF EXAMINATION: B.Sc

COLLEGE: College of Science and Technology

SCHOOL: Natural and Applied Sciences

DEPARTMENT: Physics

SEMESTER: ALPHA

COURSE CODE: PHY 336 CREDIT UNIT: 2

COURSE TITLE: Electrical and Radiometric Methods

INSTRUCTION: Answer any 3 Questions TIME: 2 HOURS

1. (a)Describe the following arrays used in electrical resistivity survey.

(i) Wenner

(ii) Schlumberger

(iii)Dipole to Dipole

(iv) Pole to Dipole

(b)Derive the relationship between apparent resistivity, geometric factor and resistance. From your

derivation, deduce the relation between apparent resistivity, geometric factor and resistance when

using Wenner and Schlumberger arrays in electrical resistivity survey.

(c)Complete the table below

AB/2 (m) MN/2 (m) R(Ω) Geometric

Factor

Apparent

Resistivity

1 0.2 53.03

2 0.2 8.70

3 0.6 6.19

4 0.6 3.15

5 0.6 1.13

7 1.4 0.93

83

10 2.0 0.665

15 2.0 0.40

20 2.0 0.27

30 4.0 0.32

45 4.0 0.19

80 4.0 0.10

100 4.0 0.08

2. (a) (i) Describe electrical properties of rocks

(ii) Describe the following as relating to electrical resistivity method used in applied geophysics:

Electrolytic conduction and Dielectric conduction

(b) Describe the physicochemical model for self potential

(c)(i) Consider a continuous current flowing in an isotropic homogeneous media, show that

2

= 0

(ii) For a single point electrode at depth show that

𝜌 =4𝜋𝑟𝑣

𝐼

23mks

3. (a) Explain two (2) major field procedures used in electrical resistivity method

(b)(i) Describe Induced Polarization method

(ii) List and explain two classes of Induced Polarization

(iii) Explain two microscopic effects that causes ground to be chargeable

(c)Describe four types of commonly measured IP data

23mks

4. (a)(i) Describe theory of radioactive disintegration

(ii) Explain three (3) processes of radioactive disintegration

(iii) State Law of radioactive disintegration

(b)(i) Describe the instruments used for measuring Radioactivity.

(ii) List seven environmental applications of radiometric survey

(c)List 5(five) geological application of self potential (SP)

84

23mks

5. (a) Describe the occurrence of self potential

(b) Describe how self potential can be measured.

(c) Explain Potential Gradient Method.

23mks