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  {\bf {\underline{INDIAN INSTITUTE OF INFORMATION TECHNOLOGY, ALLAHABAD}}}\\
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  {\small{\bf Back-Paper Examination, July-2018}}
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\rightline{{\bf{\small Date of Examination: 13. 07. 2018 }}}
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{\small{\bf Program Code \& Semester: B. Tech. (IT/EC/Integrated)/ 1st Semester}
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{\small
\begin{center}
 {\bf Paper Title: Engineering Physics\\
  Paper Code: SEGP132C\\
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 Paper Setter: Dr. Akhilesh Tiwari, Dr. Sanjai Singh, Dr. Srijit Bhattacharjee, \\Dr. Pramod Kumar}
 %(Sec.A- AT/ Sec.B- SB/Sec.C- SS)
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{\bf\leftline{Max Marks: 75 \hspace{10cm} Duration: 3 hours}}
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{\bf Note: Use of non-programmable calculator is allowed. $ ~n_i^{Si}= 10^{10} cm^{-3}; m_e=9.1 \times 10^{-31} kg.; \\ KT/e= .026 ~eV. $}
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\begin{enumerate}
\item	(a) A particle is executing a projectile motion in $x-z$ plane. Find out it's Lagrangian and deduce equations of motion. Find out the constant(s) of motion. \\

(b) Consider a mass $M$ is constrained to move in a straight line along a frictionless surface with a simple  pendulum of mass $m$, length $l$ attached to it. Find out the Lagrangian of the system. Find the equations of motion. \hfill[7+5]

\item	(a) Consider the physical system of a battery of voltage $V$ is connected in series with an inductance $L$ and a resistance $R$. Construct the Lagrangian of the system and write down the Lagrange's equation of motions. Find out the current flowing at any instant through the resistance. What is the steady-state current of the system? \hfill[7]\\

(b) Find out the shortest curve joining two points on a plane.  \hfill[5]
 
\item	(a) Using Heisenberg's uncertainty principle show that an electron can not reside on a nucleus. \hfill[7]\\

(b) Write down the time dependent Schr\"{o}dinger equation and find out the time independent equation from it. \hfill[5]

\item Let us consider a particle restricted to move along $x-axis$ between $x=0$ and $x=L$, by ideally reflecting, infinitely high walls of infinite potential. Suppose the potential energy of the particle is zero inside the box and rises to infinity outside. Find out the energy eigenfunctions for the particle. What is the expectation value of the position of the particle in its first excited state? \hfill[12]
 

\item What is mean free time and how it is related with mobility of carriers? What is the hole diffusion constant in a piece of silicon doped with $3 \times 10^{15} cm^{-3}$ of donors and $7 \times 10^{15} cm^{-3}$ of acceptors at 300 K? at 400 K? What is built in potential in a p-n junction device?
Given, $\mu_p=420~cm^2/V.s$ at $300 K$ and $200~ cm^2/V.s$ at $400 K$. \hfill [12] 


\item (a) Find the built-in potential for a p-n Si junction at room temperature if the bulk resistivity of Si is $1 \Omega $cm. Electron mobility in Si at room temperature is $1400~ cm^2 V^{-1} s^{-1}; \mu_n/\mu_p = 3.1; n_i = 1.05 \times 10^{10} cm^{-3}$.\\

(b) Establish a relation between depletion layer width and built in potential. For the p-n Si junction from the previous problem calculate the width of the space charge region for the applied voltages $V =-10, 0, and +0.3 V. $ $\epsilon_{Si} = 11.9$. \hfill[6+9]
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