July 2013

PHE-14

ASSIGNMENT BOOKLET Bachelor's Degree Programme (B.Sc.) MATHEMATICAL METHODS IN PHYSICS-III Valid from July 1, 2013 to March 31, 2014

It is compulsory to submit the Assignment before filling in the Term-End Examination Form.

Please Note • You can take electives (56 to 64 credits) from a minimum of TWO and a maximum of FOUR science disciplines, viz. Physics, Chemistry, Life Sciences and Mathematics. • You can opt for elective courses worth a MINIMUM OF 8 CREDITS and a MAXIMUM OF 48 CREDITS from any of these four disciplines. • At least 25% of the total credits that you register for in the elective courses from Life Sciences, Chemistry and Physics disciplines must be from the laboratory courses. For example, if you opt for a total of 64 credits of electives in these 3 disciplines, at least 16 credits should be from lab courses. • You cannot appear in the Term-End Examination of any course without registering for the course. Otherwise, your result will not be declared and the onus will be on you.

School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi-110068 (For July, 2013 Cycle)

Dear Student, We hope you are familiar with the system of evaluation to be followed for the Bachelor’s Degree Programme. At this stage you may probably like to re-read the section on assignments in the Programme Guide for Elective Courses that the University sent you after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous evaluation which would consist of one tutor-marked assignment for this course. Instructions for Formating Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your TMA answer sheet, please write the details exactly in the following format:

ENROLMENT NO. : ……………………………………… NAME : ……………………………………… ADDRESS : ……………………………………… ……………………………………… ……………………………………… COURSE CODE

: ……………………………..

COURSE TITLE

: ……………………….…….

ASSIGNMENT NO. : ………………………….… STUDY CENTRE

: ………………………..…….

DATE : ….……………………………...……

PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. 2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise and in your own words. Do not copy answers from study material. 5) While solving problems, clearly indicate the question number along with the part being solved. Write units at each step of your calculations as done in the text because marks will be deducted for such mistakes. Take care of significant digits in your work. Recheck your work before submitting it. 6) This assignment will remain valid from July 1, 2013 to March 31, 2014. However, you are advised to submit it within 12 weeks of receiving this booklet to accomplish its purpose as a teaching-tool. Answer sheets received after the due date shall not be accepted. We strongly feel that you should retain a copy of your assignment response to avoid any unforeseen situation and append, if possible, a photocopy of this booklet with your response. We wish you good luck.

2

Tutor Marked Assignment PHE-14: Mathematical Methods in Physics-III Course Code: PHE-14 Assignment Code: PHE-14/TMA/2013-14 Max. Marks: 100

Note: Attempt all questions. Symbols have their usual meanings. The marks for each question are indicated against it. 1. a) Determine the eigenvalues and eigenvectors of the following matrix

(5)

 2 3 0  3 2 0   0 0 1 b) Verify that the matrix is hermitian. Calculate its eigenvalues and eigenvectors. Construct the diagonalising matrix. Is it unitary. Verify your result. (10)

 3 1 − i 1 + i 2    c) What is a cyclic group? Show that the roots of the equation z 4 − 1 = 0 form a cyclic group of order 4. (5) d) Show that the contravariant and covariant components of a vector are identical in a Cartesian coordinate system. (5) 2. a) Show that u =

1 log ( x 2 + y 2 ) is harmonic and find its harmonic conjugate. 2

b) Find the type of singularity of the function

f ( z) =

( z − 2) z

2

(10) (5)

 1  sin    z −1

c) Show that

(10) ∞

dx

π

∫ x2 + 4 = 4 0

3. a)

Represent the function:

(10)

1, − 1 < x < 1 f ( x) =  0, | x | > 1

as a Fourier integral. Use the result to show that: ∞

sin α π dα = α 2 0

∫

3

b)

Calculate the inverse Laplace transform of F (s) =

c)

(5)

2s − 1 2

s − 2s + 10

Solve the initial value problem using the method of Laplace transforms:

(10)

y′′ + 16 y = 8 cos 4t ; y (0) = 0; y′(0) = 8

4. a)

Show that

(5) J 0 ( x) − J 2 ( x) = 2

d [ J1 ( x)] dx

b)

Expand the function f ( x) = x3 − x in a Legendre series.

c)

Show that

(5) Pn′ (1) =

d)

(10)

d 1 Pn ( x = 1) = n (n + 1) dx 2

Show that

(5) 2

d    2 x −  1 = H 2 ( x) dx   ******

4

TENTATIVE SCHEDULE FOR PHYSICS PROGRAMMES BROADCAST ON GYAN DARSHAN 1 July 2013 – May 2014 This is the tentative schedule for Physics students. The students may approach their cable operator to avail this facility.

Date/Day July 2013 07-07-13 (Sun) 09-07-13 (Tues) 14-07-13 (Sun) 21-07-13 (Sun) 23-07-13 (Tues) August 2013 06-08-13 (Tues) 13-08-13 (Tues) 20-08-13 (Tues) 27-08-13 (Tues)

Time Slot

Course Code

Video Programme Title

9.00-9.30 pm

PHE-04

Vector Calculus Part-I

8.30-9.00 am

BPHE-101

9.00-9.30 pm

PHE-11

Exploring Physics: Experiments with Low cost Materials Matter Waves and the Uncertainty Principle

9.00-9.30 pm

PHE-11

Wave Particle Duality

8.30-9.00 am

BPHE-101

The Bicycle: A Vehicle for Teaching Physics

8.30-9.00 am

BPHE-101

The Physics of Dance

8.30-9.00 am

PHE-15

The Milky Way

8.30-9.00 am

PHE-04

Vector Calculus Part-II

8.30-9.00 am

PHE-15

Stellar Spectra and Classification

PHE-10

Digital Modulation

PHE-10

Amplitude Modulation

PHE-06

First Law of Thermodynamics

PHE-06

Applications of the First Law of Thermodynamics

PHE-15

Stellar Structure

PHE-10

Transistor Biasing

PHE-06

Thermodynamics in Action

PHE-06

Ushmagatiki ke Anuprayog

PHE-10

Electronic Devices

September 2013 03-09-13 8.30-9.00 am (Tues) 08-09-13 9.00-9.30 am (Sun) 10-09-13 8.30-9.00 am (Tues) October 2013 06-10-13 9.00–9.30 pm (Sun) 13-10-13 9.00–9.30 pm (Sun) November 2013 10-11-13 9.00–9.30 pm (Sun) December 2013 15-12-13 9.00–9.30 pm (Sun) 17-12-13 8.30-9.00 am (Tues) 24-12-13 8.30-9.00 am (Tues)

5

Date/Day

Time Slot

Course Code

Video Programme Title

January 2014 05-01-14 (Sun)

9.00–9.30 pm

PHE-15

Glimpses of the Cosmos

12-01-14 (Sun)

9.00–9.30 pm

PHE-02

Simple Harmonic Motion

19-01-14 (Sun)

9.00–9.30 pm

PHE-15

Exploring the Night Sky

21-01-14 (Tues)

8.30-9.00 am

PHE-02

Dolan

February 2014 02-02-14 (Sun)

9.00–9.30 pm

PHE-15

On the Trail of Stars

04-02-14 (Tues)

8.30-9.00 am

PHE-06

Second Law of Thermodynamics

11-02-14 (Tues)

8.30-9.00 am

PHE-02

Coupled Oscillations

04-03-14 (Tues)

8.30-9.00 am

PHE-13

Introduction to Crystal Structure

11-03-14 (Tues)

8.30-9.00 am

PHE-15

Astronomical Coordinates

16-03-14 (Sun)

9.00–9.30 pm

PHE-10

Operational Amplifier

18-03-14 (Tues)

8.30-9.00 am

PHE-06

Carnot Cycle

06-04-14 (Sun)

9-00-9.30 pm

PHE-15

Astronomical Measurements (Measurement of Distance and Brightness)

13-04-14 (Sun)

9-00-9.30 pm

BPHE-101

Rotating Frames of Reference

11-05-14 (Sun)

9.00-9.30 pm

PHE-15

Astronomical Measurements (Measurement of Mass, Temperature and Time)

13-05-14 (Tues)

8.30-9.00 am

PHE-10

Logic Gates

March 2014

April 2014

May 2014

6

TENTATIVE SCHEDULE FOR PHYSICS STUDENTS ON DD1 (National Network) JULY 2013 TO MAY 2014 Time: 6.00 AM to 6.30 AM July 2013 08-07-13

PHE-10

Operational Amplifier

August 2013 12-08-13

PHE-02

Coupled Oscillations

September 2013 09-09-13

PHE-10

Transistor Configuration

October 2013 14-10-13

PHE-06

Thermodynamics in Action

November 2013 11-11-13

PHE-15

Exploring the Night Sky

January 2014 13-01-14

PHE-13

Introduction to Crystal Structure

February 2014 10-02-14

BPHE-101

Ghurni Nirdesh Tantra (Hindi)

March 2014 10-03-14

PHE-15

The Milky Way

April 2014 14-04-14

PHE-10

Semiconductor Diode

May 2014 12-05-14

PHE-15

On the Trail of Stars

* Second Monday of every month

7