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\noindent (a)~ Consider a PV system undergoing change of state from 1 to
2. If the system is in contact with a thermal reservoir at
temperature $T$, show that the maximum amount of work out put
$|W_0|$ is given by
\begin{eqnarray}
|W_0| \le& F_1-F_2&\hfill{[4]}
\end{eqnarray}
\noindent(b)~For a reversible process show that
$$ |W_0| = F_1-F_2 $$
where $F$ is the Helmholtz free energy \hfill{[4]}

In this section the concept of solid angle is defined as a generalization of angle in plane geometry

Electrodynamics Feb 21, 2019

Tut-02

A line charge carrying a charge $\lambda$ per unit length and extending from $-a,0,0$ to $+a,0,0$ lies along the $x$- axis. Find the potential at a point on the $X$- axis at point $(d,0,0), d>0$ and at a point $(0,d,0)$ on the $Y$-axis.

We use the Lorentz force on a unit positive charge to define the electric and magnetic fields.

NOTES FOR LECTURES ON CLASSICAL MECHANICS

NOTES FOR LECTURES

PROBLEMS

Maxwell's equation, $\nabla \times \vec{E}=0$, can be easily proved by direct computation of curl of electric field of a point charge and appealing to the superposition principle.

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$
$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$
$\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
$\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$
$\newcommand{\ket}[1]{\langle #1\rangle}$
qm-lec-17009

Using $ e(\nu, T) d\nu =\frac{2\pi h}{c^2})\frac{\nu^3}{e^{\beta h \nu}-1}\, d\nu$, where $e(\nu, T)$ is called the black-body emissivity, show that the energy radiated per unit area and time in the range $d\lambda$ of $\lambda$ (where $\lambda = c/\nu $ is the wavelength) is \[ \left(\frac{2\pi c^2h}{\lambda^5}\right)(e^{\frac{\beta h c }{\lambda}}-1)^{-1} d\lambda \equiv e(\lambda, T) d\lambda.\]
Show that the wavelength for which $e(\lambda, T)$ is a maximum is given by \[ \beta h c = 4.965 \lambda_\text{max}\] What does $\frac{\lambda_\text{max}\nu_{max}}{c}$ equal?
Solar radiation has a maximum intensity near $\lambda = 5\times 10^{-5}$cm. Assuming that the the sun's surface is in thermal equilibrium, determine its temperature.

In the table below, a number in the top row represents the
pressure of a gas in the bulb of a constant volume gas thermometer
when the bulb is immersed in the triplet cell. The bottom row
represents the corresponding readings of the pressure when the bulb
is surrounded by a material at constant unknown temperature.
Calculate the ideal gas temperature of this material (use five
significant figures) {Zemansky}

$P_{TP}$,~ mm Hg	1000.00	750.00	500.00	250.00
$P_{TP}$, mm Hg	1535.30	1151.60	767.82	383.95

Is it necessary to convert the

pressures from mm Hg to Pascal?

2019Classical Mechanics

2019CM/HMW-01
2019CM/HMW-02
2019CM/HMW-03
2019CM/HMW-04
2019CM/HMW-05
2019CM/HMW-07
2019CM/HMW-08
2019CM/QUIZ-01
2019CM/QUIZ-02
2019CM/QUIZ-03
2019CM/QUIZ-04
2019CM/TEST-01
2019CM/EXM-01

2018 Classical Mechanics

Consider a cycle $ABCD$ with perfect gas as the working substance. $AB$ is at constant volume $V_1$ and $CD$ is at constant volume at $V_2$ with $V_2\,>\,V_1$ The parts $BC$ and $DA$ are adiabatic. Calculate the efficiency of this engine in terms of $V_1$ and $V_2$. (Note this is different from Carnot's engine and so we can not draw similar conclusions about the efficiency being maximum)

For a system, as in $Q[2]$, undergoing a process from state 1
to state 2 at constant pressure and temperature, show that the
maximum ``non'' $PdV$ work out put is given by
\begin{eqnarray}
|A_{TP}| \le G_1-G_2 &\qquad G=\text{Gibbs function}&
\end{eqnarray}

$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
Starting from Coulomb's law a proof is given that the electric field of a system of point charges obeys the Maxwell's equation.
\[\nabla \times \vec E =0\]

Electrodynamics March 26, 2019
Tutorial-IV

A uniform magnetic field $\vec{B}$ fills a cylindrical volume of radius $R$ and a metal rod of length $L$ is placed in it as shown in the figure.If the magnitude $B$ is changing at the rate $\frac{dB}{dt}$ show that the emf that is produced by the changing magnetic field and that acts between the ends of the rod is given by $${\cal E} = \frac{dB}{dt}\frac{L}{2}\sqrt{R^2-\frac{L^2}{2}}$$

Two identical coils each having radius $R$ and $n$- turns are kept parallel and with a distance $d$ between the two.
1. Find an expression for the magnetic field at a point on the common axis of the coils and at a distance $x$ from the mid-point between the coils.
2. Show that if the separation of the coils is equal to $R$, the first and the second derivatives of $B$ w.r.t. $x$ vanish at the mid point. This produces nearly constant magnetic field near the mid point, WHY?
3. For $R=5.0$cm, $I=50$amp,and 300 turn coils, plot the magnetic field as a function of $x$ in the range $x=-5$cm to $x=5$cm.
.
A wooden cylinder of mass $m=0.5$kg, radius $R=3$cm, length $\ell=10$cm, is placed on an inclined plane. It has 10 turns of wire wrapped around it longitudinally so that the plane of the wire contains the axis of the cylinder and is parallel to the inclined plane, see . Assuming no friction, what is the current that will prevent the cylinder from rolling down the inclined plane in presence of a uniform magnetic field of 0.5T?. Describe what happens if the block is a rectangular instead of a cylindrical one? What will be the current that will prevent the block from moving down the plane?
A square wire of length $L$, mass $m$, and resistance $R$ slides without friction down parallel rails of negligible resistance, as in \Figref{em-fig-015}. The rails are connected to each other at the bottom by a resistanceless rail parallel to the wire so that the wire and rails form a closed rectangular conducting loop. The plane of the rails makes an angle $\theta$ with the horizontal, and a uniform vertical magnetic field $\vec{B}$ exists in the region.
1. Show that the wire acquires a steady state velocity of magnitude $$v= \frac{mgR\sin\theta}{B^2 L^2\cos^2\theta}$$
2. Show that the above result is consistent with conservation of energy.
3. What changes will be necessary in the above results, if the direction of magnetic field is reversed?
A cylindrical shell of radius $R$, height $h$, and carrying a uniform surface charge density $\sigma$, rotates about its own axis with angular velocity $\omega$. Compute the magnetic field produced by the cylinder at a point on the axis

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Browse & Filter

[QUE/TH-08006] TH-PROBLEM

[NOTES/EM-02013] Solid Angle

[2019EM/HMW-02]

[2018EM/HMW-07]

[NOTES/EM-01005]--Defining the Electric and Magnetic Fields

[1998TH/LNP-22]-Pure Substances

Classical Mechanics --- Notes for Lectures and Problems [CM-MIXED-LOT]

NOTES FOR LECTURES

PROBLEMS

[NOTES/EM-03008]-Maxwell's Second Equations from Coulomb's Law

[NOTES/QM-17009] Addition of Angular Momenta Using Tables

[NOTES/ME-14013]-Angular velocity from rotation matrix

[QUE/SM-08002] SM-PROBLEM

[QUE/TH-01006] TH-PROBLEM

Contributed Problem Sets

[2008EM/HMW-04]

[1998TH/LNP-03]--Lecture -3 Microscopic vs macroscopic systems

[NOTES/ME-06002b]-Using graph of $V(x)$ to find motion

[QUE/TH-06002] TH-PROBLEM

[QUE/TH-08008] TH-PROBLEM

[NOTES/EM-02015] Proof of curl free nature of \(\vec E\)

[2019EM/HMW-04]

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