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Problem/EM-05004

Use Green’ reciprocity theorem to prove “mean value theorem”: the average of \(\phi(\vec{r})\) over a spherical surface \(S\) that encloses a charge-free volume is equal to the potential at the center of the sphere:\[ \frac{1}{4\pi R^2} \iint \phi(\vec{r}) dS = \phi(0) \]

Hint:

Apply recprocity theorem considering  two systems
(i) an empty sphere with charge spread uniformlyover its surface and
(ii) the given potential \(\phi(\vec{r})\) and its source \(\rho(\vec{r})\).

kapoor kapoor's picture 2 days 11 hours ago

Problem/EM-05002

Two closed equipotentials \(\phi_1\) and \(\phi_2\)are such that \(\phi_1\) contains \(\phi_2\);Let \(\phi\) be the potential at any point \(P\) between them. If a charge \(q\) is now put at point \(P\) and the equipotentials are replaced by grounded conducting surfaces, show that thecharges \(q_1, q_2\),  induced on the two conductors satisfy the relation \[\frac{q_1}{(\phi_2 - \phi_P)} = \frac{q_2}{\phi_p -\phi_1)} = \frac{q}{(\phi_2 -\phi_1)}\].

Hint

Use Green's reciprocity theorem

 

kapoor kapoor's picture 2 days 11 hours ago

Solved Problem/EM-02002

Summary (or abstract) not available for this node.
kapoor kapoor's picture 2 days 22 hours ago

Solved Problem/EM-02001

Summary (or abstract) not available for this node.
kapoor kapoor's picture 2 days 22 hours ago

Problem/EM-05005

Consider a point charge \(q\)  embedded in a semi-infinite dielectric medium of dielectric ) constant  \(\epsilon_1\), and located a distance  from a plane interface that separates the first medium from another semi-infinite dielectric medium of dielectric constant  \(\epsilon_2\). Suppose that the interface coincides with the plane. Find the electric field everywhere if the distance of the point charge from the interface is \(d\).

kapoor kapoor's picture 3 days 12 hours ago

Problem/EM-05003/Point charge and a dielectric plate

A point charge and a dielectric plate of permittivity  \(\epsilon_2\) are embedded in a dielectric medium of permittivity  \(\epsilon_1\). as shown in figure 2. Use method of images and find electric field everywhere.

kapoor kapoor's picture 3 days 12 hours ago

\(\maltese\) Problem/QM-05031 Temperature for Fusion Process in the Sun

Consider an ionised plasma of protons in thermal equilibrium. Assuming a Maxwell-Boltzmann distribution, estimate the temperature required for two protons to overcome the Coulomb barrier for fusion assuming an approach distance of 1 fm. Compare this with the temperature when quantum effects come into play. For this, use the distance between the two protons as the de-Broglie wavelength at which Coulomb energy becomes equal to the kinetic energy of the protons.   Click for Solution


       Source:Sarita Vig

kapoor kapoor's picture 4 days 19 hours ago

\(\maltese\) Solved Problem/QM-05031-SOL

Consider an ionised plasma of protons in thermal equilibrium. Assuming a Maxwell-Boltzmann distribution, estimate the temperature required for two protons to overcome the Coulomb barrier for fusion assuming an approach distance of 1 fm. Compare this with the temperature when quantum effects come into play. For this, use the distance between the two protons as the de-Broglie wavelength at which Coulomb energy becomes equal to the kinetic energy of the protons.  

See attached document for solution.

kapoor kapoor's picture 5 days 19 hours ago

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kapoor kapoor's picture 5 days 20 hours ago

\(\ddagger\) Problem/QM-05001 Uncertanty principle

Summary (or abstract) not available for this node.
kapoor kapoor's picture 5 days 20 hours ago

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