Total energy of earth in its circular orbit around the sun

Question :
Find out the total energy of earth in its circular orbit around the sun in terms of gravitational constant
Answer:
Let R be the total distance between the earth and the sun. If \({M_e}\) and \({M_s}\) are the mass of earth and sun respectively, the gravitational force of motion of earth and sun is given by

\(F =  - \frac{{G{M_e}{M_s}}}{{{R^2}}}\)

where G is the gravitational constant. Since the centripetal force balances the gravitational force of attraction, we have

\({F_c} = |{F_G}|\),

where

\({F_c} = \frac{{{M_e}{v^2}}}{R}\)

v being the velocity with which earth is moving. Hence we have

\(\frac{{{M_e}{v^2}}}{R} = \frac{{G{M_e}{M_s}}}{{{R^2}}}\)

or

\({M_e}{v^2} = \frac{{G{M_e}{M_s}}}{R}\)

Therefore kinetic energy of earth in motion is

\(T = \frac{1}{2}{M_e}{v^2} = \frac{1}{2}\frac{{G{M_e}{M_s}}}{R}\)

As we know that , force in terms of potential energy is

\({F_G} = - \frac{{\partial V}}{{\partial R}}\)
\(V = - \int {{F_G}dR} = \int {\frac{{G{M_e}{M_s}}}{{{R^2}}}dR = - } \frac{{G{M_e}{M_s}}}{R}\)

Now total energy of the earth in the orbit around the sun is

\(E = T + V\)
\(E = \frac{1}{2}\frac{{G{M_e}{M_s}}}{R} - \frac{{G{M_e}{M_s}}}{R}\)

\(E = - \frac{1}{2}\frac{{G{M_e}{M_s}}}{R}\)

This is the required expression.

Brachistochrone Problem

Question

If a particle falls from rest under the influence of gravity from higher to lower point in the minimum time, what is the curve that the particle will follow?

Solution

Suppose v is the speed of the particle along the curve, then in traversing ds portion of the curve time spent would be \(\frac{{ds}}{v}\) so that total time taken by the particle in moving from highest point 1 to lowest point 2 will be

\[{t_{12}} = \int\limits_1^2 {\frac{{ds}}{v}} \]

Suppose vertical distance of fall upto point 2 be x, then from the principle of conservation of energy of the particle we find that

\(\begin{array}{l}\frac{1}{2}m{v^2} = mgx\\or\\v = \sqrt {2gx} \\then\\{t_{12}} = \int\limits_1^2 {\frac{{\sqrt {d{x^2} + d{y^2}} }}{{\sqrt {2gx} }}} dx\\{\rm{      = }}\int\limits_1^2 {\frac{{\sqrt {1 + {{\dot y}^2}} }}{{\sqrt {2gx} }}} dx\\{\rm{      = }}\int\limits_1^2 {fdx} \end{array}\)

Where,

\( f=\left ( \frac{1+\dot{y}^{2}}{2gx} \right )^{\frac{1}{2}} \)

For \({t_{12}}\) to be minimum equation

\(\frac{d}{{dx}}\left( {\frac{{\partial f}}{{\partial \dot y}}} \right) - \frac{{\partial f}}{{\partial y}} = 0\)

must be satisfied. From expression for \(f\) we find that

\(\begin{array}{l}\frac{{\partial f}}{{\partial y}} = 0\\\frac{{\partial f}}{{\partial \dot y}} = \frac{{\dot y}}{{\sqrt {2gx} \sqrt {1 + {{\dot y}^2}} }}\\\frac{d}{{dx}}\left( {\frac{{\dot y}}{{\sqrt {2gx} \sqrt {1 + {{\dot y}^2}} }}} \right) = 0\\or\\\frac{{{{\dot y}^2}}}{{2gx(1 + {{\dot y}^2})}} = c'\end{array}\)

Where c’ is the constant of integration.

Since c’ is a constant we can also write

\(\frac{{{{\dot y}^2}}}{{x(1 + {{\dot y}^2})}} = c\)

where c is also a constant. On integrating above equation we find

\(\begin{array}{l}\frac{{{{\dot y}^2}}}{c} = x(1 + {{\dot y}^2})\\or,\\{{\dot y}^2}\left( {\frac{1}{c} - x} \right) = x\\{{\dot y}^2}\left( {\frac{x}{c} - {x^2}} \right) = {x^2}\\\dot y = \frac{x}{{\sqrt {\frac{x}{c} - {x^2}} }}\end{array}\)

Putting \(\frac{1}{c} = 2a\) , and on integration we get

\(\int dy=\int \frac{x}{\sqrt{2ax-x^{2}}}dx\)

\( y=acos^{-1}(1-\frac{x}{a})-(2ax-x^{2})^{1/2}+c''\)

where c'' is new constant of integration.
In case c'' is zero then y will be zero for x=0.

As such the equation

\( y=acos^{-1}(1-\frac{x}{a})-(2ax-x^{2})^{1/2}\)

And this y represents an inverted cycloid with its base along y axis and cusp at the origin and is the curve that particle will follow.

Double pendulum

Question 

In case of a double pendulum find the expression for the kinetic energy of the system

Solution

We take a simple case where lengths and masses are same.see below in the figure

Here on being displaced the co-ordinates of two pendulums are

\(\begin{array}{l}{x_1} = l\sin {\theta _1}\\{y_1} = l\cos {\theta _1}\end{array}\)

For the first pendulum where \({\theta _1}\) is the angle through which the first pendulum have been displaced.

For second pendulum

\(\begin{array}{l}{x_2} = l\sin {\theta _1} + l\sin {\theta _2}\\{y_2} = l\cos {\theta _1} + l\cos {\theta _2}\end{array}\)

Where \({\theta _2}\) is the angle through which second pendulum has been displaced.

The total kinetic energy of the system is given by the expression

\(T = \frac{1}{2}m(\dot x_1^2 + \dot y_1^2) + \frac{1}{2}m(\dot x_2^2 + \dot y_2^2)\)

Now

And

Which is the required result

How to Find Kinetic energy in terms of spherical co-ordinates

Question

How to Find Kinetic energy in terms of \((r,\theta ,\phi )\)

Solution

We have to find the kinetic energy in terms of \((r,\theta ,\phi )\) that is in terms of spherical c0-ordinates. Kinetic energy in terms of Cartesian co-ordinates is

\(T = \frac{1}{2}m\left( {{{\dot x}^2} + {{\dot y}^2} + {{\dot z}^2}} \right)\)                                                                (1)

Where \(\dot x,\dot y{\rm{ and }}\dot z\) are derivatives of z, y and z with respect to time.

Cartesian co-ordinates x, y, z in terms of \(r,\theta \) and \(\phi \) are

\(\begin{array}{l}x = r\sin \theta \cos \phi \\y = r\sin \theta \sin \phi \\z = r\cos \theta \end{array}\)

Now derivatives of x, y and z w.r.t. t are

\(\begin{array}{l}\dot x = \frac{{dx}}{{dt}} = \dot r\sin \theta \cos \phi  + r\cos \theta \cos \phi \dot \theta  - r\sin \theta \sin \phi \dot \phi \\\dot y = \frac{{dy}}{{dt}} = \dot r\sin \theta \sin \phi  + r\cos \theta \sin \phi \dot \theta  + r\sin \theta \cos \phi \dot \phi \\\dot z = \frac{{dz}}{{dt}} = \dot r\cos \theta  - r\sin \theta \dot \theta \end{array}\)  

Where

\(\dot r = \frac{{dr}}{{dt}},\dot \theta  = \frac{{d\theta }}{{dt}},\dot \phi  = \frac{{d\phi }}{{dt}}\)  means all \(r,\theta \) and \(\phi \) changes with time as the particle moves or changes its position with time.

Now calculate for \({\dot x^2},{\dot y^2},{\dot z^2}\) and add them. After adding them we get

\({(\dot x)^2} + {(\dot y)^2} + {(\dot z)^2} = {\dot r^2} + {r^2}{\dot \theta ^2} + {r^2}{\sin ^2}\theta {\dot \phi ^2}\)

Putting this value of \({(\dot x)^2} + {(\dot y)^2} + {(\dot z)^2}\)in equation 1 we get kinetic energy of particle or system in terms of \(r,\theta \) and \(\phi \).

Hence

\(T = \frac{1}{2}m({\dot r^2} + {r^2}{\dot \theta ^2} + {r^2}{\sin ^2}\theta {\dot \phi ^2})\)

Equations of motion of coupled pendulum using the lagrangian method

Question

Obtain the equations of motion of coupled pendulum using the lagrangian method.

Solution

Consider a system of coupled pendulums as shown below in the figure

The displacement of A is \({x_1}\) and B is\({x_2}\) , condition being \({x_1}\) < \({x_2}\). In such state the spring gets stretched. The lengths of the strings of both the pendulums are same (say l). The angular displacement of A is \({\theta _1}\) and that of B is \({\theta _2}({\theta _2} > {\theta _1})\).

Therefore

\(\begin{array}{l}{x_1} = l{\theta _1} \Rightarrow {\theta _1} = \frac{{{x_1}}}{l}{\rm{                      (1)}}\\{x_2} = l{\theta _2} \Rightarrow {\theta _2} = \frac{{{x_2}}}{l}{\rm{                     (2)}}\end{array}\)

As the spring gets stretched, it is clear from the figure that restoring force works along the direction of displacement \({\theta _1}\) and opposite to the direction of displacement\({\theta _2}\) . Now A and B at zero potential level, the total potential energy of the system is given as

\(V = mgl(1 - \cos {\theta _1}) + mgl(1 - \cos {\theta _2}) + \frac{1}{2}k{({x_2} - {x_1})^2}\)

Where m is the mass of each one of the bob and k is the spring constant.

Since \({\theta _1}\)and \({\theta _2}\)are small so,

\(\begin{array}{l}\cos {\theta _1} = 1 - \frac{{\theta _1^2}}{2} + \frac{{\theta _1^4}}{4} + ......\\\cos {\theta _2} = 1 - \frac{{\theta _2^2}}{2} + \frac{{\theta _2^4}}{4} + ......\end{array}\)

Neglecting the higher powers other than squares of \({\theta _1}\)and \({\theta _2}\)the expression of potential energy can be written as

\(\begin{array}{l}V = mgl\frac{{\theta _1^2}}{2} + mgl\frac{{\theta _2^2}}{2} + \frac{1}{2}k{({x_2} - {x_1})^2}\\{\rm{    = }}\frac{{mgx_1^2}}{{2l}} + \frac{{mgx_2^2}}{{2l}} + \frac{1}{2}k{({x_2} - {x_1})^2}\end{array}\)

Also the kinetic energy of whole system is

\(T = \frac{1}{2}m\dot x_1^2 + \frac{1}{2}m\dot x_2^2 = \frac{1}{2}m(\dot x_1^2 + \dot x_2^2)\)

Hence Lagrangian L would be

\(\begin{array}{l}L = T - V\\L = \frac{1}{2}m(\dot x_1^2 + \dot x_2^2) - \frac{{mgx_1^2}}{{2l}} - \frac{{mgx_2^2}}{{2l}} - \frac{1}{2}k{({x_2} - {x_1})^2}\end{array}\)

Now

 \(\begin{array}{l}  \frac{\partial L}{\partial {{x}_{1}}}=-\frac{mg{{x}_{1}}}{l}+k({{x}_{2}}-{{x}_{1}}) \\\end{array}\)

\(\begin{array}{l}\frac{\partial L}{\partial {{{\dot{x}}}_{1}}}=m{{{\dot{x}}}_{1}} \\\end{array}\)

\(\begin{array}{l}\therefore \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{x}}}_{1}}} \right)=\frac{d}{dt}(m{{{\dot{x}}}_{1}})=m{{{\ddot{x}}}_{1}} \\\end{array}\)

Hence Lagrangian equation in terms of \({x_1}\)is

\(\begin{array}{l}\frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial {{\dot x}_1}}}} \right) - \frac{{\partial L}}{{\partial {x_1}}} = 0\\or,\\m{{\ddot x}_1} + \frac{{mg{x_1}}}{l} - k({x_2} - {x_1}) = 0\\or,\\m{{\ddot x}_1} =  - \frac{{mg{x_1}}}{l} + k({x_2} - {x_1})\end{array}\)

Also,

\(\begin{array}{l}\frac{{\partial L}}{{\partial {x_2}}} =  - \frac{{mg{x_2}}}{l} - k({x_2} - {x_1})\\\frac{{\partial L}}{{\partial {{\dot x}_2}}} = m{{\dot x}_2}\\and\\\frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial {{\dot x}_2}}}} \right) = m{{\ddot x}_2}\end{array}\)

Hence Lagrangian equation in terms of \({x_2}\)is

\(\begin{array}{l}\frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial {{\dot x}_2}}}} \right) - \frac{{\partial L}}{{\partial {x_2}}} = 0\\or,\\m{{\ddot x}_2} =  - \frac{{mg{x_2}}}{l} - k({x_2} - {x_1})\end{array}\)

The equation of motion for given system are

\(\begin{array}{l}m{{\ddot x}_1} =  - \frac{{mg{x_1}}}{l} + k({x_2} - {x_1})\\m{{\ddot x}_2} =  - \frac{{mg{x_2}}}{l} - k({x_2} - {x_1})\end{array}\)

IISER pune Notice for Admission to Ph. D. Program 2012

Notice for Admission to Ph. D. Program

Applications are invited for the Ph. D. program scheduled to commence from August  2012  in the following areas of basic sciences:

1.      Biological/Ecological  Sciences :   Application Form

2.      Chemical Sciences :  Application Form

3.      Mathematical Sciences :  Write to mathphdselection@iiserpune.ac.in

4.      Physical Sciences :  Application Form

ONLINE APPLICATION FORM available UNTIL April 30,  2012

Test/Interview Dates: (New!!)

Biological/Ecological Sciences:

Written Test: June 20, 2012

Interviews: June 21-22, 2012

Chemical Sciences:

Written Test and Interviews: May 24-25, 2012

Physical Sciences:

Written Test and Interviews: May 30-31, 2012

Note: For Biological/Ecological Sciences, candidates should come prepared to stay foratleast 3 days. For Chemical and Physical Sciences, candidates should come prepared to stay for atleast 2 days.

Eligibility

General Criteria:

Nationality: Indian

Age: Not more than 28 years on August 01, 2012.  Age relaxation of 5 years for SC/ST/OBC/PwD/Female applicants

Minimum Educational Qualifications/Selection Procedure

Biological Sciences

       60%  marks (55% for SC/ST)  in:

 ·  M.Sc in any branch of Science – Biological Sciences/Ecology & EnvironmentalSciences/Biodiversity/Biochemical/Biophysical/ Biomathematical or equivalent. Including Physics, Chemistry & Mathematics.

·   B.E/B.Tech/M.Tech or MBBS degree holders may also apply.

·   The candidate MUST qualify National Eligibility Test (NET) with valid CSIR-JRF/UGC-JRFor DBT-JRF-A or ICMR-JRF or ICAR-JRF or CSIR-LS or INSPIRE-PhD or DAE-JEST or NBHM or have a valid GATE qualification.

·   Additional criteria may be used for shortlisting of candidates.

---

Chemical Sciences60% marks (55 for SC/ST)  in:

M. Sc. or equivalent degree in Chemistry/Physics/Biochemistry/Material Science/Bioinformatics/Pharmacy

The candidate should have qualified National Eligibility Test (NET) with CSIR-JRF/UGC-JRF, CSIR-LS (only inorganic and physical chemistry), JEST or INSPIRE-Ph.D or GATE.

The following ranks for GATE will be eligible.

1. Organic  chemistry applicants: 1-500 ranks
2. Inorganic and Physical chemistry applicants:  All ranks

Additional criteria may be used for shortlisting of candidates.

---

Mathematical Sciences.

60 marks (55%  for SC/ST)  in:

M.Sc/M.Stat/M.Tech/M.A in any field,

or in:

B.Sc/B.Stat/B.Tech/B.A in any field.

The candidates must have appeared for the NBHM Ph.D. screening test for the year 2012. Candidates with Undergraduate degree will be admitted to the Integrated  Ph.D  program. Shortlisted candidates from the NBHM exam will be contacted directly.

---

Physical Sciences 60% marks (55 for SC/ST)  in:

(i) M. Sc. or equivalent degree in Physics/Material Science OR

(ii) BE / BTech / MTech

The candidate should have qualified at least one of the following tests/fellowship in Physics valid for the current academic year:

(i)                 CSIR-JRF/UGC-JRF

(ii)              JEST/GATE with ranks less than or equal to 500

(iii)            INSPIRE-Ph.D fellowship.

Additional criteria may be used for shortlisting of candidates.

---

Selection Procedure

• Only Eligible candidates that meet the minimum educational qualifications and satisfy the additional criteria set by the respective departments will be called for a written test.  The additional criteria may  include cut-off marks at different levels of examinations, rank in the national examinations etc. No correspondence/inquiries will be entertained in this regard.
• Candidates who are shortlisted by the selection committee will be informed the dates and details of the written test/interview by e-mail.
•  Only candidates who qualify the written test will be interviewed.  The interview may be conducted in 1 or 2 rounds.
• Based on their academic record and the performance of the candidates in the written test and interview, the departmental selection committee will recommend candidates to the Chairman for admission to the Ph. D. program.

E-mail will be the primary means of communication. All applicants should specify their e-mail ID clearly and legibly in their application forms.

How to Apply/ Application Deadline

Application form has to be filled ONLINE before  April 30, 2012. Candidates intending to apply to more than one program/Department should apply separately application forms foreach program/ Department.

• Fill the online application form completely. Incomplete application forms may be rejected.
• Attested copies of all supporting documents (mark sheets, certificates, age proof, caste certificate, etc.) are to be brought in at the time of the interview.

Please carry one photograph. In addition, applicants for Biological/Ecological  Sciences should bring along one recommendation letter at the time of interview The last date for receipt of completed applications ONLINE is April 30, 2012.

Contact e-mail

Please contact Coordinator for each Department at the following address in case you have questions:

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Chemistry (chemphdselection@iiserpune.ac.in)

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Physics (physicsphdselection@iiserpune.ac.in)

Test/Interview Dates (New!!)

Biological/Ecological Sciences:

Written Test: June 20, 2012

Interviews: June 21-22, 2012

Chemical Sciences:

Written Test and Interviews: May 24-25, 2012

Physical Sciences:

Written Test and Interviews: May 30-31, 2012

Note: For Biological/Ecological Sciences, candidates should come prepared to stay for atleast 3 days. For Chemical and Physical Sciences, candidates should come prepared to stay for atleast 2 days.

LAST DATE FOR RECEIPT OF COMPLETED APPLICATION FORMS, FILLED ONLINE is April 30, 2012

Fourier Series

Fourier series is an expansion of a periodic function of period $2\pi$ which is representation of a function in a series of sine or cosine such as

$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)+\sum_{n=1}^{\infty }b_{n}sin(nx)$
 
   where $a_{0}$ , $a_{n}$ and $b_{n}$ are constants and are known as fourier coefficients.
   In applying fourier theorem for analysis of an complex periodic function , given function must satisfy following condition
  (i) It should be single valued
  (ii) It should be continuous.
 
  Drichlet's Conditions(sufficient but not necessary)

  When a function $f(x)$ is to be expanded in the interval (a,b)
  (a) $f(a)$ is continous in interval (a,b) except for finite number of finite discontinuties.
  (b) $f(x)$ has finite number of maxima and minima in this interval.

Orthogonal property of sine and cosine functions
  $\int_{-\pi}^{\pi}sin(mx)cos(mx)dx=0$
  $\int_{-\pi}^{\pi}sin(mx)sin(nx)dx= \int_{-\pi}^{\pi}sin(mx)sin(nx)dx=\begin{bmatrix}
\pi\delta_{mn} &m\neq 0 \\
 0& m=0
\end{bmatrix} $
 $\int_{-\pi}^{\pi}cos(mx)cos(nx)dx=\begin{bmatrix}
\pi\delta_{mn} &m\neq 0 \\
 2\pi& m=0
\end{bmatrix} $
 

Fourier Constants
 $a_{0}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx$
 $a_{0} is the average value of function $f(x)$ over the interval
 $a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx$
 $b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx$
  

For even functions
  $f(-x)=f(x)$ and fourier series becomes
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)$
 

 For odd functions
  $f(-x)=-f(x)$ and fourier series becomes
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}sin(nx)$
  

Complex form of fourier series
  putting $c_{0}=c_{0}$
  $c_{n}=\frac{a_{n}-ib_{n}}{2}$
  and
  $c_{-n}=\frac{a_{n}+ib_{n}}{2}$
  $f(x)=\sum_{-\infty }^{\infty }C_{n}e^{inx}$
  coefficent
  $C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$
  

Fourier series in interval (0,T)
  General fourier series of a periodic piecewise continous function $f(T)$ having period $T=\frac{2\pi}{\omega}$ is
  $f(t)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)+\sum_{n=1}^{\infty }b_{n}sin(nx)$
  where
  $a_{0}=\frac{1}{T}\int_{0}^{T}f(t)dt$
$a_{n}=\frac{2}{T}\int_{0}^{T}f(t)cos(n\omega T)dt$
$b_{n}=\frac{2}{T}\int_{0}^{T}f(t)sin(n\omega T)dt$
 

 Complex Form of Fourier Series
  $f(x)=\sum_{n=-\infty }^{\infty }C_{n}e^{-i\omega t}$
  where
  $c_{n}=\frac{1}{T}\int_{0}^{T}f(t)e^{-i\omega t}dx$
 

 Advantages of Fourier series
  1. It can also represent discontinous functions
  2.  Even and odd functions are conveniently represented as cosine and sine series.
  3.  Fourier expansion gives no assurance of its validity outside the interval.

  Change of interval from $(-\pi,\pi)$ to $(-l,l)$
  Series will be
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(\frac{nx\pi}{l})+\sum_{n=1}^{\infty }b_{n}sin(\frac{nx\pi}{l})$
  with
  $a_{0}=\frac{1}{2l}\int_{-l}^{l}f(x)dx$
  $a_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)cos(\frac{n\pi x}{l})dx$
  $b_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)sin(\frac{n\pi x}{l})dx$
  

Fourier Series in interval $(0,l)$
  Cosine series when function $f(x)$ is even
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(\frac{n\pi x}{l})$
  $a_{0}=\frac{1}{l}\int_{0}^{l}f(x)dx$
  $a_{n}=\frac{2}{l}\int_{0}^{l}f(x)cos(\frac{n\pi x}{l})dx$
  Sine series when function $f(x)$ is odd
  $f(x)=\sum_{n=1}^{\infty }a_{n}sin(\frac{n\pi x}{l})$
  $b_{n}=\frac{2}{l}\int_{0}^{l}f(x)sin(\frac{n\pi x}{l})dx$

Complex Analysis Part 2

Liouville's Theorem 
If a function $f(z)$ is analytic for all finite values of z, and is bounded then it is a constant.
Note:- $e^{z+2\pi i} = e^z$

Taylor's Theorem
If a function $f(z)$ is analytic at all points inside a circle C, with its centre at point a and radius R then at each point z inside C
$f(z)=f(a)+(z-a)f'(a)+\frac{1}{2!}(z-a)^2f''(a)+.......+\frac{1}{n!}(z-a)^nf^n(a)$
Taylor's theorem is applicable when function is analytic at all points inside a circle.

 Laurent Series
If $f(z)$ is analytic on $C_{1}$ and $C_{2}$ and in the annular region R bounded by the two concentric circles $C_{1}$ and $C_{2}$ of radii $r_{1}$and $r_{2}$ ($r_{1} > r_{2}$) with their centre at a then for all z inside R
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.........$
where,
$a_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(n+1)}}$
$b_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(-n+1)}}$

Singular points
If a function $f(z)$ is not analytic at point z=a then z=a is known as a singular point or there is a singularity of $f(z)$ at z=a for example
$f(z)=\frac{1}{z-2}$
z=2 is a singularity of $f(z)$

Pole of order m
If $f(z)$ has singularity at z=a then from laurent series expansion
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.....+\frac{b_{m}}{(z-a)^m}+\frac{b_{m+1}}{(z-a)^{m+1}}$
if
$b_{m+1}=b_{m+2}=0$
then
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.....+\frac{b_{m}}{(z-a)^m}$
and we say that function $f(z)$ is having a pole of order m at z=a. If m=1 then point z=a is a simple pole.

Residue
The constant $b_{1}$ , the coefficent of $(z-z_{0})^{-1}$ , in the Laurent series expansion is called the residue of $f(z)$ at singularity $z=z_{0}$
$b_{1}=Res_{z=z_{0}}f(z) = \frac{1}{2\pi i}\int_{C_{1}}f(z)dz$

Methods of finding residues
1.  Residue at a simple pole
if $f(z)$ has a simple pole at z=a then
$Res f(a) =\lim_{z\rightarrow a}(z-a)f(z) $
2. If $f(z)=\frac{\Phi(z)}{\Psi (z)}$
and $\Psi(a)=0$ then
$Res f(a)=\frac{\Phi(z)}{\Psi^{'} (z)}$
3. Residue at pole of order m
If $f(z)$ is a pole of order m at z=a then
$Res f(a)= \frac{1}{(m-1)!}\left \{ \frac{d^{m-1}}{dz^{m-1}}(z-a)^{m}f(z) \right \}_{z=a}$

 Residue Theorem

If $f(z)$ is analytic in closed contour C excapt at finite number of points (poles) within C, then
$\int_{C}f(z)dz = 2\pi i \textsl { [sum of the residues at poles within C]}$

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Complex analysis part 2

Complex Analysis Part 1

Complex Variables

•  A function is said to be analytic in a domain D if it is single valued and differentiable at every point in the domain D.
• Points in a domain at which function is not differentiable are singularities of the function in domain D.
• Cauchy Riemann conditions for a function $\textit{f(z)=u(x,y)+iv(x,y)}$ to be analytic at point z
  $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
  $\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$
•  Cauchy Riemann equations in polar form are
  $\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta}$
  $\frac{1}{r}\frac{\partial u}{\partial \theta}=-\frac{\partial v}{\partial r}$

Cauchy' Theorem

If  $\textit{f(z)}$ is an analytic function of z and  $\textit{f'(z)}$ is continuous at each point within and on a closed contour C then

$\oint_C{f(z)dz}=0$

Green's Theorem

If $\textit{M(x,y)}$ and $\textit{N(x,y)}$ are two functions of x and y and have continous derivatives

$\oint_C{(Mdx+Ndy)}=\iint_{S}\left ( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right )\delta x\delta y$

Theorem:-

If function $\textit{f(z)}$ is not analytic in the whole region enclosed by a closed contour C but it is analytic in the region bounded between two contours $C_{1}$ and $C_{2}$ then

$\int_Cf(z)dz=\int_{C_{1}}f(z)dz+\int_{C_{2}}f(z)dz$

Cauchy's Integral Formula

If  $\textit{f(z)}$ is an analytic function on and within the closed contour C the value of $\textit{f(z)}$ at any point z=a inside C is given by the following contour integral

$f(a)=\frac{1}{2\pi i}\oint _{C}\frac{f(z)}{z-a}dz$

Cauchy's Integral Formula for derivative of an analytic function

If  $\textit{f(z)}$ is an analytic function in a region R , then its derivative at any point z=a is given by

$f'(a)=\frac{1}{2\pi i}\oint _{C}\frac{f(z)}{(z-a)^{2}}dz$

generalizing it we get

$f^{n}(a)=\frac{n!}{2\pi i}\oint _{C}\frac{f(z)}{(z-a)^{n+1}}dz$

Morera Theorem

It is inverse of Cauchy's theorem. If $\textit{f(z)}$ is continuous in a region R and if $\oint f(z)dz$ taken around a simple closed contour in region R is zero then $\textit{f(z)}$ is an analytic function.

Cauchy's inequality

If  $\textit{f(z)}$ is an analytic function within a circle C i.e., $\left | z-a \right |=R$ and if $\left | f(z) \right |\leq M$ then

$\left | f^{n}(a) \right |\leq \frac{Mn!}{R^{n}}$

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Complex Analysis Part 1

Complex Numbers

Complex numbers are the numbers of the form a+ib where i=√(-1) and a and b are real numbers.

Definition:- Complex numbers are defined as an ordered pair of real numbers like (x,y) where

z=(x,y)=x+iy

and both x and y are real numbers and x is known as real part of complex number and y is known as imaginary part of the complex number.

Addition of complex numbers

Let z₁=x₁+iy₁ and z₂=x₂+iy₂ then

z₁+z₂=(x₁+x₂)+i(y₁+y₂)

Subtraction

z₁-z₂=(x₁-x₂)+i(y₁-y₂)

Multiplication

(z₁.z₂)=(x₁+iy₁).(x₂+iy₂)

Division

To divide complex number by another , first write quotient as a fraction. Then reduce the fraction to rectangular form by multiplying the numerator and denominator by the complex conjugate of the denominatormaking the denominator real.

Points to remember

1. Complex conjugate of z=x+iy is $\overline{z}=x-\mathrm{iy}$

2. Modulus of the absolute value of z is denoted by |z| and is defined by √(x²+y²)

3. Let r be any non negative number and θ any real number. If we take x=rcosθ and y=rsinθ then,

$r=\sqrt{x^2+y^2}$ which is the modulus of z and $\theta ={\tan }^{-1}\frac{y}{x}$ which is the argument or amplitude of z and is denoted by arg.z

we also have x+iy=r(cosθ+isinθ)=r[cos(2nπ+θ)+isin(2nπ+θ)] , where n=0, ±1, ±2, ....

4. Argument of a complex number is not unique since if θ is the value of argument then 2nπ+θ (n=0, ±1, ±2, ....) are also values of the argument. Thus, argument of complex number can have infinite number of values which differ from each other by any multiple of 2π.

5. Arg(0) is not defined.

6. argument of positive real number is zero.

7. argument of negative real number is ±π

8. Properties of moduli:-

• |z₁+z₂|≤|z₁|+|z₂|
• |z₁-z₂|≥|z₁|-|z₂|
• |z₁•z₂|=|z₁|•|z₂|
• |z₁/z₂|=|z₁|/|z₂|

Physics books for CSIR-NET/JRF and GATE

This is the list of few books recommended for those preparing for net/jrf in physics . It is hard for a person to collect these many books so i would advise you to go through these books while doing your masters in physics and make yourself appropriate notes for further reference and study.

Classical mechanics

1.      Classical Mechanics by Herbert Goldstein

2.      Classical mechanics by Gupta and Kumar

Mathematical physics

1.      Mathematical Methods in Physical sciences by Mary L. Boas

2.      Mathematical Physics Including Classical Mechanics by Satya Prakash

Electrodynamics

1. Introduction to electrodynamics by Griffith
2. Classical electrodynamics by J.D. Jackson
3. Feynman Lectures Vol. 2

Thermodynamics and statistical physics

1. Statistical Physics Vol. 5, Berkeriey, Physics Courses.
2. A Treatise on Heat-MN. Saha and B.N. Srivastava.
3. Thermodynamics and Statistical Physics by F. Reif.
4. Thermodynamics and Statistical Physics-Silokanathan and D.P. Khandelwal.
5. Thermodynamics, Kinetic, Theory of gases and Statistical Mechanics-Sears.

 Quantum Mechanics

     1.      Introduction to quantum mechanics by D.J. Griffiths

2. P.M. Methews and K. Venkatesan-A Textbook of Quantum Mechanics.
3. A.K. Ghatak and S. Lokanathan-Quantum Mechanics (Third Edition).

Electronics

1. Ap. Malvino : Electronics Principles
2. A.P. Malvino : Digital Computer Electronics
3. Van Valcumgurg : Network Analysis
4. J. Milliman and C.C. Halkias : Integrated Electronics
5. G.K. Mithal : Integrated Electronics.

Nuclear and particle physics

1. Nuclear Physics:Irving Kaplan
2. Concepts of Nuclear Physics:B.L. Cohen
3. Introduction Nuclear Physics:Kenneth S.Krane

Solid State physics

1. Materials Science and Engineering by V.Raghvan, Prentice-Hall Edition 1993.
2. Solid State Electronic Engineering Materials by S.O. Pillai, Wiley Eastern Ltd.
3. Solid state Physics by C. Kittel V.Edition
4. Introduction to Solid by L.Azroff.
5. Solid state physics by N.W. Ascheroft and N.W. Ascheroft and N.D. Mermin CBS Publishing Asia Ltd.

Atomic and molecular physics

1. G.Herzberg; "Atomic Spectra and atomic structure".
2. H. Kuhn:"Atomic Spectra".
3. Walker and Straugha, "spectroscopy, Vol. I, II, III."
4. H. Herzberg; "Molecular Spectra and Molecular structure."
5. H. Barrow: "Theory of Atomic Spectra."
6. R.C. Johnson:"Introduction to Molecular Spectra."
7. White;Atomic 'Spectra'.
8. B.K.Agrawal:"X-ray Spectroscopy." 

Important dates for CSIR-NET/JRF Exam June 2012

1. CSIR will hold the Joint CSIR-UGC Test on 17th June, 2012 for determining the eligibility of the Indian National candidates
for the award of Junior Research Fellowships (JRF) NET and for determining eligibility for appointment of Lecturers (NET) in
certain subject areas falling under the faculty of Science. The award of Junior Research Fellowship (NET) to the successful
eligible candidates will depend on their finding admission/placement in a university/ national laboratory/ institution of higher
learning and research, as applicable.
1.1 A candidate may apply either for ‘JRF + Lectureship’ or for ‘Lectureship (LS) only’ He/she may indicate his/her preference in
the O.M.R Application Form/Online Application, as the case may be. CSIR may consider candidates applying for ‘JRF + LS’ or
‘Lectureship only” depending on number of fellowships available & performance in the test subject to the condition that they
fulfill the laid down eligibility criterion . If a candidate is found to be over-age for JRF (NET) he/she will automatically be
considered for Lectureship (NET) only.
1.2 Two separate merit lists, one comprising the candidates qualifying for the award of Junior Research Fellowship (JRF - NET)
and the second, of those candidates qualifying the Eligibility Test for Lectureship (NET), will be made on the basis of their
performance in the above Test. Candidates qualifying for JRF (NET), will also be eligible for Lectureship (NET) subject to
fulfilling the eligible criteria laid down by UGC. The candidates qualifying for Lectureship will be eligible for recruitment as
Lecturers as well as for JRF-ship in a Scheme/Project, if otherwise suitable. However, they will not be eligible for Regular JRFNET
Fellowship. They will be eligible to pursue Ph.D. programme with or without any fellowship other than JRF-NET.
Candidates qualifying for the award of JRF (NET) will receive fellowship either from CSIR or UGC as per their assignment or
from the Schemes with which they may find association. The candidates declared eligible for Junior Research Fellowship under
UGC scheme will be governed by UGC rules/regulations in this regard.
1.3 The final result of this Single MCQ test may be declared sometime in the month of August, 2012 and fellowship to successful
candidates could be awarded from 01.01.2013.

CSIR UGC NET JUNE 2012 NOTIFICATION

Joint CSIR-UGC Test for Junior Research Fellowship and Eligibility for Lectureship (NET), June, 2012

CSIR will hold Joint CSIR-UGC Test for Junior Research Fellowship and Eligibility for Lectureship (NET) June, 2012 on 17th June, 2012, Sunday

CSIR - UGC NET 2012 - Educational Qualification and Extension in last date for sale of information bulletin/on-line submission of fee

As notified vide our advt No-10-2(5)/2012(i)-EU-II published in Employment News issue dtd 18th-24th February, 2012 and also placed on website www.csirhrdg.res.in that the educational qualification for the candidates applying for the Joint CSIR-UGC Test for Junior Research Fellowship and Eligibility for Lectureship (NET) June, 2012 in Chemical Sciences; Earth, Atmospheric, Ocean and Planetary Sciences; Life Sciences; Mathematical Sciences; Physical Sciences is M.Sc or equivalent degree with minimum 55% marks for General/OBC candidates; 50% for SC/ST candidates, Physically and Visually Handicapped candidates and Ph.D degree holders who had passed Mater’s degree prior to 19th September, 1991

Students enrolled in Integrated MS-Ph.D program are also eligible to apply for JRF in subject areas of NET

Their eligibility for Lectureship will be subject to fulfilling the criteria laid down by UGC

Further to broaden the scope of NET and to attract meritorious students at early stages of their career, CSIR has revised the Educational Eligibility criteria for writing NET for JRF in the subject areas of NET

BS-4 year program/B.E./B.Tech/B.Pharm/MBBS/Integrated BS-MS/M.Sc or equivalent degree with at least 55% marks for General/OBC (50% for SC/ST candidates, Physically and Visually Handicapped candidates)

Candidates enrolled for M.Sc or having completed 10+2+3 years of the above qualifying examination are also eligible to apply in the above subject under the Result Awaited (RA) category on the condition that they complete the qualifying degree with requisite percentage of marks within the validity period of two years to avail the fellowship from the effective date of award

Such candidates will have to submit the attestation format (given at the reverse of the application form) duly certified by the Head of the Department/Institute from where the candidate is appearing or has appeared

B.Sc (Hons) or equivalent degree holders or students enrolled in Integrated MS-Ph.D program with at least 55% marks for General/OBC candidates; 50% marks for SC/ST candidates, physically and visually handicapped candidates are also eligible to apply

Candidates with bachelor’s degree, whether Science, Engineering or any other discipline, will be eligible for Fellowship only after getting registered/enrolled for Ph.D/Integrated Ph.D program within the validity period of two years

The eligibility for Lectureship of NET qualified candidates will be subject to fulfilling the criteria laid down by UGC

Ph.D degree holders who have passed Master’s degree prior to 19th September, 1991, with at least 50% marks are eligible to apply for Lectureship only

The candidates with the above qualification are advised to fill up their degree with percentage of marks in column number 20 and 21, as applicable (if they are applying through information bulletin) or column number 18 to 21, as applicable (for on-line applications)

There is no minimum age limit for writing the NET whereas other term and conditions/Syllabus/Scheme of examination etc will remain the same as notified vide our advertisement number 10-2(5)/2012(i)-EU-II published in Employment News dtd 18th-24th February, 2012 and placed on website www.csirhrdg.res.in

The last date for sale of information bulletin/online submission of fee and receipt of completed application forms have been extended

Date of close of sale of information bulletin by cash at all branches/stations: 27th March, 2012

Date of close of on-line deposit of fee: 27th March, 2012

Date of close of on-line submission of applications: 28th March, 2012

Last date of receipt of completed application forms (including duly completed hard copy of on-line application) in examination unit: 02nd April, 2012

Last date of receipt of completed application forms (including duly completed hard copy of on-line application) in examination unit from remote areas: 10th April, 2012

Introduction to Quantum Physics;Heisenberg''s uncertainty principle (Video Lecture)

I really like the style of the professor how he is teaching physics .....

Symmetries and conservation Laws: Part 2

Baryon and Lepton numbers

• The Baryon number B=1 is assigned to all baryons and B=-1 is assigned to all anti baryons: all other particles have B=0.
• The lepton number L_e=1 is assigned to electrons and electron neutrino, L_e=-1 to their anti particles ; all other particles have L_e=0
• L_μ=1 for muon and μ-neutrino and L_τ=1 for tau lepton and its neutrino.
• Significance of these numbers is that , in every process of whwtever kind , the total valuse of B, L_e,L_μ,L_τ separately remains constant.
• Conservation of leptons has a signefence for strond interactions.
• Another property that is conserved only in strong interactions is isospin.

Strangeness

• A number of particles were discovered that behsve so unexpectedly that they were called strange particles.
• They were created in pairs, and decay only in certain ways but not in others that were allowed by existing conservation laws.
• To clarify the observation Gell-Mann and Nishijina indepensently introduced the strangeness number.
• For photon , π⁰ and η⁰, B, L_e,L_μ, L_τ and S are zero . There is no way to distinguish between them and their antiparticles, and they are regarded as their own anti particles.
• Strangeness number is conserved in all processes mediated by strong and electromagnetic interactions.
• The multiple creation of particles with S≠0 is the result of this conservation principle
• S can change in an event mediated by the week interaction. Decays that proceed via a week interaction are relatively slow, a billion tomes slower than the interactions proceeded via strong interactions.
• Week interactions does not allow S to change by more than ±1 in a decay. For example,

  Ξ^- decays in two steps Ξ^- →Λ⁰+π^-→η⁰+π⁰

Isospin

• There are number of hadron families whose numbers have similar masses but different charges. Thes families are called multiplets. Member of multiplet represents different charged states of a single fundemental entity.
• Each multiplet according to number of charge states exhibits a number I such that the multiplicity of state is given by 2I+1.
• Isospin can be represented by vector I in an abstract iso space whose component in any specific direction is governed by the quantum number denoted by I₃.
• Possible values of I₃ varies from I, I-1 to -I. The charge of a baryon is related to its baryon numberB, its strangeness number S and the component I₃ of its isotopic spin by the formula

  $Q=e(I_3+\frac{B}{2}+\frac{S}{2})$

Conservation of statistics

• The interchange of identicle particles in a system is a type of symmetry operation which leads to the preservation of the wave
• Conservation of statistics signifies that no process occuring within an isolated system can change the statistical behaviour of the system.

Hypercharge

• Hypercharge is defined as Y=S+B
• Classification system for hadrons encompasses many short lived particles as well as relatively stable hadrons
• This scheme cillects isospin multiplets into submultiplets whose members have the same spin but different in isospin and a quantity called hypercharge.