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With the help of a neat diagram, show that the potential due to a dipole at a point is given by V=\frac{1}{4\pi\epsilon_0}\,\frac{p\cos\theta}{r^2}, where p is the dipole moment of the charge distribution, \theta is the angle between the line joining the centre of the dipole to the point of interest and the axis of the dipole.
(i) Derive Planck's law of Black body radiation. (ii) Obtain its limiting forms for (i) very low frequency and (ii) very high frequency.
Write down Maxwell's equations for linear dielectrics and deduce the equation of continuity.
State and prove Poynting's theorem.
In free space, the electric field of electromagnetic wave is given as \vec{E}(x,t)=120\cos(\omega t-kx)\hat{y}\,\mathrm{V/m}. Find the average power crossing a circular area of radius one metre in the yz-plane.
Show that the displacement current between the plates of a parallel-plate capacitor is equal to the conduction current across the conductor.
If the magnetic field \vec{B}, at a point with position vector \vec{r} is uniform, show that the corresponding vector potential \vec{A}(\vec{r}) is given by \vec{A}(\vec{r}) = -\frac{1}{2} \left[ \vec{r} \times \vec{B} \right]
Show that for any electromagnetic wave propagating in free space, the total average energy per unit volume is \frac{1}{2}\epsilon_0 E_{\text{max}}^2; where E_{\text{max}} is the amplitude of the electric field associated with the electromagnetic wave.
Show that Maxwell's equations of electrodynamics are invariant under Lorentz transformations.
The spectral energy curve of the moon shows maxima at 470\,\mathrm{nm} and 14\,\mu\mathrm{m}. What inference can you draw from this data? Also calculate the energy density and radiation pressure in both cases. Given, Wien's constant b = 2.892 \times 10^{-3}\,\mathrm{m\,K}, Stefan's constant \sigma = 5.67 \times 10^{-8}\,\mathrm{J\,m^{-2}\,s^{-1}\,K^{-4}} and speed of light c = 3 \times 10^{8}\,\mathrm{m\,s^{-1}}.
Briefly explain Planck's law of blackbody radiation. Show that Planck's law reduces to Wien's law and Rayleigh-Jeans law at lower and higher wavelength limits respectively.
The electric potential of a grounded conducting sphere of radius a in a uniform electric field E = E_0\hat{z} is given as \phi(r, \theta) = -E_0 r \left[ 1 - \left( \frac{a}{r} \right)^3 \right] \cos\theta Find the expression for the surface charge density on the sphere.
In a simple AC circuit involving only a resistor R = 50\,\Omega and a voltage source V, find the linear frequency of the generator if V = 0 \cdot 5\,V_m (V_m is peak e.m.f.) at time t = \frac{1}{720}\text{ s} (assuming V = 0 at t = 0).
Verify whether the electric potential V = 15 x^2 y z - 5 y^3 z satisfies Laplace's equation or not.
In the circuit given below, find the values of currents I_1, I_2 and I.
A series RLC circuit has R=2\Omega. The energy stored in the circuit decreases by 1\% per period of oscillation. Its natural undamped frequency is 2\,\mathrm{kHz}. Determine the values of inductor L and the quality factor.
A series RLC circuit has a resistance of 100\Omega and an impedance of 210\Omega. If this circuit is connected to an a.c. source with an r.m.s. voltage of 220\ \mathrm{V}, how much is the average power dissipated in the circuit?
Viewing ionosphere as a dielectric medium of refractive index \mu = \sqrt{1 - \omega_p^2 / \omega^2}, \omega_p being known as the plasma frequency, determine the group velocity of a radio wave of frequency \omega = \sqrt{2}\omega_p.
A conducting sphere of radius 5\,\mathrm{cm} has a total charge of 12\,\mathrm{nC} uniformly distributed on its surface in free space. Determine the displacement vector \vec{D} on its surface and outside at a distance r from the centre of the sphere.
A plane electromagnetic wave propagating along +\hat{z} direction is incident normally on the boundary at z=0 between medium A(z<0) and medium B(z>0). Determine the reflection coefficient and transmission coefficient for the wave.