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[NOTES/EM-02008]-Maxwell's Equations for Electrostatics-IINode id: 5580pageThe Gauss law of electrostatics follows from the Coulomb's law for a point charge and superposition principle. The Gauss law along with the Gauss divergence theorem of vector calculus imply Maxwell's first equation \[ \nabla\cdot\bar{E}=\frac{\rho}{\epsilon_0}. \] for electrostatics.
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Quantum Field Theory --- Notes for Lectures and Problems --- [QFT-MIXED-LOT]]Node id: 4685collection |
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[NOTES/EM-03007]-Work done in field of a point chargeNode id: 5644pageWe discuss the path independence of the work done by static electric field. This leads to, as in mechanics, introduction of the electric potential. An expression of the electric potential is derived by an explicit computation of work done by on a unit positive charge by the electric field of a point charge \(q\). For an arbitrary distribution of charges, the electric potential is obtained by making use of the superposition principle.
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[NOTES/QM-17007] Summary of main results on addition of angular momentaNode id: 4820page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$
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[NOTES/ME-14011]-Rotation of a Rigid Body with One Point FixedNode id: 5704page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$
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[NOTES/EM-12003]-Fields of a Moving Point ChargeNode id: 5761page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$
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The electromagnetic field tensor \(F_{\mu\nu}\) is a second rank Lorentz tensor. It transformation properties are used to derive the transformation properties of electric and magnetic fields under Lorentz transformations. The equation of motion of a relativistic charged particle are written down in a covariant notation.
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[QUE/SM-08001] SM-PROBLEMNode id: 5073pageFor a black body of temperature \(T\), Planck's law can be used to show that the
wavelength for which the emissivity \(e(\lambda, T)\)
is maximum is given by
\[ \beta h c = 4.965 \lambda_\text{max}\]
Solar radiation has a maximum intensity near
\(\lambda = 5\times 10^{-5}\)cm. Assume
that the the sun's surface is in thermal equilibrium. Solve the above
equation numerically and determine its temperature.
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[QUE/TH-01005] TH-PROBLEMNode id: 5190pageA new scale is to defined in terms of ideal gas scale of
temperature. Let the difference of ice points and the steam points
be fixed at $t_s-t_i=180$ deg. If
$\displaystyle{\lim_{P_{TP}\to0}\left({P_s\over P_i}\right)}$ is
given to be 1.3661 as the best experimental value, find $t_s$ and
$t_i$ on the new scale.
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[2008EM/HMW-03]Node id: 5400pageIMSc-IV Physics-IV : Electricity and Magnetism Jan-Apr 2008
Set-IV : Tutorial-III
Electric Potential
- Four charges $q,2q,-q,-2q$ are placed at the corners of a square of side $a$. Find the electric field at a point distance $z$ above the center of the square. \input{\QBankDir/em-que-01016}
- The electric field due to a line segment of length $2L$, and carrying a uniform line charge $\lambda$ at a distance $d$ above the mid point is given by $$ E = \frac{1}{4\pi\epsilon_0} \, \frac{2\lambda L}{d \sqrt{d^2 +L^2}}$$ Using the above result find the electric field at a distance $z$ above the center of a square loop ( side $2L$) carrying a uniform line charge $\lambda$.
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[1998TH/LNP-02]--Lecture -2 Mechanics vs ThermodynamicsNode id: 5544page |
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[NOTES/ME-06002a]-General Properties of Motion in One DimensionNode id: 5676page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$
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[NOTES/EM-10002]-Poynting Theorem--ExamplesNode id: 5732pageBy means of several examples it is shown that the energy density of electromagnetic fields equals the well known expressions of energy of solenoid and a charged capacitor. By means of example of a capacitor being charged, the presence of the displacement current leads to a consistent Ampere's law.
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[QUE/TH-06001] TH-PROBLEMNode id: 5160pageTwo heat reservoirs, each of fixed volume $V$ are at temperatures $T_1$ and $T_2$ ( $T_1\,>\,T_2$) initially. An engine operates between the two till both of them reach the same temperature $T_3$. The specific heat $c_V$ of the reservoirs can be assumed to constant through out. The engine is back to its inital state.
- Show that $$ T_3^2\,\geq\,T_1T_2 $$
- Calculate the maximum work that can be extracted from the two reservoirs. ( Answer in terms of $T_1$ and $T_2$)
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[QUE/TH-08007] TH-PROBLEMNode id: 5218pageFor a two component system with coordinates $P,V,X,Y$
($dw=-PdV+YdX$) show that the maximum workout put at constant $V$ &
$T$ is less than or equal to the decrease in free energy
($F_1-F_2$).
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22-01-23 18:01:01 |
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[NOTE/EM-02014] Flux of Feld of a Point ChargeNode id: 5959article |
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[2019EM/HMW-03]Node id: 5365pageElectrodynamics March 8, 2019 Tutorial-III
- The potential takes the constant value \(\phi_0\) on the closed surface \(S\) which bounds the volume \(V\). The total charge inside V is \(Q\). There is no charge anywhere else. Show that the electrostatic energy contained in the space outside of S is \(\displaystyle U_\text{E}(out)=\frac{1}{2} Q\phi_0\)
- The line segment between \(A\) to \(B\) in the diagram below carries a uniform charge per unit length \(\lambda\). The vector \(\vec{a}\) is coincident with the segment. The vectors \(\vec{b}\) and \(\vec{c}\) point from the observation point \(\vec{r}\) to the beginning and end of \(\vec{a}\), respectively.
 Evaluate the integral for the potential in a coordinate-free manner by parameterizing the line source using a variable \(s\) which is zero at the point on the charged segment that is closest to \(\vec{r}\). Show thereby that \begin{eqnarray} \phi(\vec{r}) &=& \frac{\lambda}{4\pi\epsilon_0}\ln {\left|\frac{\vec{b}\cdot\vec{a}}{a} +\sqrt{\left(\frac{\vec{b}\cdot\vec{a}}{a}\right)^2 + \frac{|\vec{b}\times\vec{a}|^2}{a^2} } \right| }\nonumber\\ && -\frac{\lambda}{4\pi\epsilon_0} \ln {\left|\frac{\vec{c}\cdot\vec{a}}{a} +\sqrt{\left(\frac{\vec{c}\cdot\vec{a}}{a}\right)^2 + \frac{|\vec{b}\times\vec{a}|^2}{a^2} } \right| } \end{eqnarray}
- Suppose the electrostatic potential of a point charge were \[V (r) =\frac{q}{4\pi\epsilon_0}\frac{1}{r^{(1+\epsilon)}}\] rather than the usual Coulomb formula.
- Find the potential \(V(r)\) at a point at a distance \(r\) from the center of a spherical shell of radius \(R > r\) with uniform surface charge per unit area \(\sigma\). Check the Coulomb limit \(\epsilon = 0\).
- To first order in \(\epsilon\), show that \begin{equation} \frac{V(R)-V(r)}{V(R)} =\frac{\epsilon}{2}\left[\frac{R}{r}\ln\frac{R+r}{R-r} - \ln \frac{4R^2}{R^2-r^2}\right] \end{equation}
Since the time of Cavendish, formulas like this one have been used in experimental tests of the correctness of Coulomb’s law.
- Find the primitive, Cartesian monopole, dipole, and quadrupole moments for each of the following charge distributions. Use the geometrical center of each as the origin.
- Two charges \(+q\) at two diagonal corners of a square \((\pm a, \pm a, 0)\) and two minus charges \(−q\) at the two other diagonals of the square \((\pm a, \mp a, 0)\).
- A line segment with uniform charge per unit length λ which occupies the interval \(-\ell \le z \le \ell\).` .
- An origin-centered ring in the \(X-Y\) plane with uniform charge per unit length \(\lambda\) and radius \(R\).
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[2018EM/QUIZ-02]Node id: 5429page |
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[NOTES/EM-01006]--Hall EffectNode id: 5510pageThe Hall effect described here provided an early method to study the effect of magnetic field on a current. An expression for Hall voltage and Hall resistance is obtained in this section.
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[1998TH/LNP-11]-Work in ThermodynamicsNode id: 5582page |
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Statistical Mechanics --- Notes for Lectures and Problems [SM-MIXED-LOT]Node id: 4690collectionStatistical Mechanics
Mixed Lot of Problems in Statistical Mechanics
- SM-Mod01 Probability and Statistics
- SM-Mod02 Thermodynamics -- Review Problems
- SM-Mod03 Microcanonical Ensemble
- SM-Mod04 Canonical Partition Function
- SM-Mod05 Application to Nondegenerate Perfect Gases
- SM-Mod06 Grand Canonical Partition Function
- SM-Mod-08 Statistical Mechanics of Photons
- SM-Mod09 Statistical Mechanocs of Fermions
- SM-Mod10 Perfect Boson Gas; Bose Einstein Condensation
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