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Title+Summary	Name	Date
[Solved/EM-05001] Electromagnetic Theory Node id: 2363page Question: A dielectric sphere contains a dipole \(\vec{p}_0\). Find the net dipole moment of the system. Solution: The dielectric gets polarized and the dipole and acquires a polarization \(\vec{P}\). The dipole moment of the dielectric is \begin{eqnarray} \vec{p} &=& \int_V \vec{P} d^3r = \int_V \vec{D}-\epsilon_0 \vec{E} d^3 r\\ &=& \int_V\epsilon_0(\kappa-1) \vec{E} d^3r = \epsilon_0(\kappa-1) \int_V\vec{E} d^3r \end{eqnarray} For arbitrary charge distribution the average of electric field, \( \int_V\vec{E} d^3r\), over sphere is \(-\dfrac{\vec{p}_\text{tot}}{3\epsilon_0}\). Hence we get \[ \vec{p}= - \frac{(\kappa-1)}{3} \vec{p}_\text{tot}\] using \(\vec{p}_\text{tot}= \vec{p}_0 + \vec{p}\), we get \begin{eqnarray}\vec{p}_\text{tot} &=& \vec{p}_0 -\frac{(\kappa-1)}{3} \vec{p}_\text{tot}\\\text{or } \vec{p}_\text{tot} &=&\frac{3}{\kappa+2} \vec{p}_0\end{eqnarray} Zangwill	kapoor	22-02-27 17:02:33	n
[QUE/CM-01005#Solu] Periodic Motion $V(x) = V_0 \left( e^{-2x/\alpha} - 2 e^{-x/\alpha}\right) $ Node id: 1912page Question Plot the Morse potential $$V(x) = V_0 \left( \exp(-2x/\alpha) - 2 \exp(-x/\alpha)\right) $$ and find the period of small oscillations for a particle having energy $E$ and moving in fore field described by the Morse potential.}	kapoor	22-02-27 17:02:21	n
[Solved/QM-20002] Quantum Mechanics --- Solved Problem Node id: 2264page	kapoor	22-02-27 17:02:01	n
[Solved/QM-20003] Quantum Mechanics --- Solved Problem Node id: 2261page	kapoor	22-02-27 17:02:46	n
[Solved/QM-20002 ] Quantum Mechanics --- Solved Problem Node id: 2260page	kapoor	22-02-27 17:02:12	n
[SolvedQM-20001] Quantum Mechanics --- Solved Problem Node id: 2259page	kapoor	22-02-27 17:02:04	n
[Solved/QM-19001] Quantum Mechanics Solved Problem Node id: 2256page	kapoor	22-02-27 16:02:02	n
[Solved/QM-17003] Quantum Mechanics ---- Solved Problem Node id: 2255page	kapoor	22-02-27 16:02:19	n
[Solved/QM-17002] Quantum Mechanics Solved Problem Node id: 2254page	kapoor	22-02-27 16:02:10	n
[Solved/QM-17001] Quantum Mechanics --- Solved Problem Node id: 2253page	kapoor	22-02-27 16:02:40	n
[Solved/EM-04004] A point charge and an insulated conducting sphere at potential \(\phi_0\) Node id: 2231page	kapoor	22-02-27 16:02:00	n
[Solved/EM-04003] Charged sphere and a point charge Node id: 2230page An insulated conducting sphere carries total charge \(Q\) and a charge \(q\) is placed at a distance \(a\) from the centre of the sphere. Find the potential at an arbitrary point outside the sphere.	kapoor	22-02-27 16:02:32	n
[Solved/EM-02001] Two unifromaly charged overlapping spheres Node id: 2222page	kapoor	22-02-27 16:02:58	n
[QUE/CM-01018#Solu] Sliding rope (Solved) Node id: 2430page Falling rope: A rope of length $L$ slides over the edge of a table. Initially a piece $x_0$ of it hangs without motion over the side of the table.Let $x$ be the length of the rope hanging vertically at time $t$. The rope is assumed to be perfectly flexible. Show that the principle of energy in the form of $T+V$ gives an integral of motion. Source::SOMMERFELD	kapoor	22-02-27 16:02:17	n
[QUE/CM-01012#Solu] -- Motion on Curves (Solved) Node id: 2424page A particle of mass $m$ moves on a cycloid under influence of uniform gravitational field. The parametric equations of the cycloid are given by $$ x= R( \phi + \sin\phi) , \qquad y=R(1-\cos\phi).$$ Find a suitable transformation to show that the equations of motion is identical to that of a simple harmonic motion with frequency $\omega=g/4R$. Source{Sommerfeld}	kapoor	22-02-27 16:02:16	n
[QUE/CM-01009] Periodic Motion $V(x) = V_0 \left( \exp(-2x/\alpha) - 2 \exp(-x/\alpha)\right) $ (Solved) Node id: 2416page Find the equilibrium position and the frequency of small oscillations about the equilibrium position for the potential $c$ [HSM 42(a)]	kapoor	22-02-27 16:02:51	n
[QUE/CM-01003#Solu] Periodic Motion $\displaystyle V(x) = V_0 \tan^2(\alpha x) $ Node id: 1910page Question : For a particle in one dimensional potential well $\displaystyle V(x) = V_0 \tan^2(\alpha x) $ find the time period as function of the energy of the particle. Answer : $\displaystyle ({\pi\over\alpha})\sqrt{2m\over (E+V_0)} $	kapoor	22-02-27 16:02:27	n
[QUE/CM-01017#Solu] Falling Chain (solved) Node id: 2429page A chain lies pushed together at the edge of a table, except for a piece which hangs over it, initially at rest. The links of the chain start moving one at a time; neglect friction. The energy written in the usual form is no longer an integarl of motion. Instead impulsive ( Carnot) energy loss must be taken into accountb in writing the balance of energy. Source::Sommerfeld	kapoor	22-02-27 16:02:37	n
[Solved/QM-13001#Example] Node id: 1384page	kapoor	22-02-27 16:02:42	n
[QUE/CM-02001#Solu] Classical Mechanics Solved Problem Node id: 1390page	kapoor	22-02-27 16:02:14	y

Question:
A dielectric sphere contains a dipole $\vec{p}_0$. Find the net dipole moment of the system.

Solution:
The dielectric gets polarized and the dipole and acquires a polarization $\vec{P}$. The dipole moment of the dielectric is
\begin{eqnarray}
\vec{p} &=& \int_V \vec{P} d^3r = \int_V \vec{D}-\epsilon_0 \vec{E} d^3 r\\ &=& \int_V\epsilon_0(\kappa-1) \vec{E} d^3r = \epsilon_0(\kappa-1) \int_V\vec{E} d^3r \end{eqnarray}
For arbitrary charge distribution the average of electric field, $ \int_V\vec{E} d^3r$, over sphere is $-\dfrac{\vec{p}_\text{tot}}{3\epsilon_0}$. Hence we get

\[ \vec{p}= - \frac{(\kappa-1)}{3} \vec{p}_\text{tot}\]

using $\vec{p}_\text{tot}= \vec{p}_0 + \vec{p}$, we get

\begin{eqnarray}\vec{p}_\text{tot} &=& \vec{p}_0 -\frac{(\kappa-1)}{3} \vec{p}_\text{tot}\\\text{or } \vec{p}_\text{tot} &=&\frac{3}{\kappa+2} \vec{p}_0\end{eqnarray}

Zangwill

Question Plot the Morse potential
$$V(x) = V_0 \left( \exp(-2x/\alpha) - 2 \exp(-x/\alpha)\right) $$
and find the period of small oscillations for a particle having energy
$E$ and moving in fore field described by the Morse potential.}

An insulated conducting sphere carries total charge $Q$ and a charge $q$ is placed at a distance $a$ from the centre of the sphere. Find the potential at an arbitrary point outside the sphere.

Falling rope: A rope of length $L$ slides over the edge of a table. Initially a piece $x_0$ of it hangs without motion over the side of the table.Let $x$ be the length of the rope hanging vertically at time $t$. The rope is assumed to be perfectly flexible. Show that the principle of energy in the form of $T+V$ gives an integral of motion.

Source::SOMMERFELD

A particle of mass $m$ moves on a cycloid under influence of uniform gravitational field. The parametric equations of the cycloid are given by $$ x= R( \phi + \sin\phi) , \qquad y=R(1-\cos\phi).$$ Find a suitable transformation to show that the equations of motion is identical to that of a simple harmonic motion with frequency $\omega=g/4R$.

Source{Sommerfeld}

Find the equilibrium position and the frequency of small oscillations about the equilibrium position for the potential $c$

[HSM 42(a)]

Question : For a particle in one dimensional potential well $\displaystyle V(x) = V_0 \tan^2(\alpha x) $ find the time period as function of the energy of the particle.
Answer : $\displaystyle ({\pi\over\alpha})\sqrt{2m\over (E+V_0)} $

A chain lies pushed together at the edge of a table, except for a piece which hangs over it, initially at rest. The links of the chain start moving one at a time; neglect friction. The energy written in the usual form is no longer an integarl of motion. Instead impulsive ( Carnot) energy loss must be taken into accountb in writing the balance of energy.

Source::Sommerfeld

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[Solved/EM-05001] Electromagnetic Theory

[QUE/CM-01005#Solu] Periodic Motion $V(x) = V_0 \left( e^{-2x/\alpha} - 2 e^{-x/\alpha}\right) $

[Solved/QM-20002] Quantum Mechanics --- Solved Problem

[Solved/QM-20003] Quantum Mechanics --- Solved Problem

[Solved/QM-20002 ] Quantum Mechanics --- Solved Problem

[SolvedQM-20001] Quantum Mechanics --- Solved Problem

[Solved/QM-19001] Quantum Mechanics Solved Problem

[Solved/QM-17003] Quantum Mechanics ---- Solved Problem

[Solved/QM-17002] Quantum Mechanics Solved Problem

[Solved/QM-17001] Quantum Mechanics --- Solved Problem

[Solved/EM-04004] A point charge and an insulated conducting sphere at potential \(\phi_0\)

[Solved/EM-04003] Charged sphere and a point charge

[Solved/EM-02001] Two unifromaly charged overlapping spheres

[QUE/CM-01018#Solu] Sliding rope (Solved)

[QUE/CM-01012#Solu] -- Motion on Curves (Solved)

[QUE/CM-01009] Periodic Motion $V(x) = V_0 \left( \exp(-2x/\alpha) - 2 \exp(-x/\alpha)\right) $ (Solved)

[QUE/CM-01003#Solu] Periodic Motion $\displaystyle V(x) = V_0 \tan^2(\alpha x) $

[QUE/CM-01017#Solu] Falling Chain (solved)

[Solved/QM-13001#Example]

[QUE/CM-02001#Solu] Classical Mechanics Solved Problem

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