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:\[ \frac{1}{4\pi R^2} \iint \phi(\vec{r}) dS = \phi(0) \]
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 \(\rho(\vec{r})\).
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 \[\frac{q_1}{(\phi_2 - \phi_P)} = \frac{q_2}{\phi_p -\phi_1)} = \frac{q}{(\phi_2 -\phi_1)}\].
Use Green's reciprocity theorem
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 \(d\).
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.
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
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.
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