Problem set 91

1. Which of the following equations signifies the conservative nature of the electric field $\vec E$?
1. $\vec\nabla\cdot\vec E(\vec r)=\frac{\rho(\vec r)}{\epsilon_0}$
2. $\vec\nabla\times\vec E(\vec r)=\vec 0$
3. $\vec\nabla\times\vec E(\vec r,t)=\frac{-\partial\vec B(\vec r,t)}{\partial t}$
4. $\epsilon_0\mu_0\frac{\partial\vec E(\vec r,t)}{\partial t}=\vec\nabla\times\vec B(\vec r,t)-\mu_0\vec J(\vec r,t)$

A field is said to be conservative if it can be expressed as a gradient of scalar potential $V$. The equation $\vec\nabla\times\vec E(\vec r)=\vec 0$ implies that $\vec E(\vec r)$ can be expressed as $\vec E(\vec r)=-\vec\nabla V$.

Hence, answer is (B)

2. Plane electromagnetic wave is propagating through a perfect dielectric material of refractive index $\frac{3}{2}$. The phase difference between the fields $\vec E$ and $\vec B$ associated with the wave passing through the material is
1. Zero
2. $\pi$
3. $\frac{3}{2}\pi$
4. any non-zero value between $-\pi$ and $\pi$

For a perfect dielectric the phase difference between $\vec E$ and $\vec B$ is zero.

Hence, answer is (A)

3. An electromagnetic wave is propagating in a dielectric medium of permittivity $\epsilon$ and permeability $\mu$ having an electric field vector $\vec E$ associated with the wave. The associated magnetic field $\vec H$ is
1. Parallel to $\vec E$ with magnitude $E\sqrt{\mu/\epsilon}$
2. Parallel to $\vec E$ with magnitude $E\sqrt{\epsilon/\mu}$
3. Perpendicular to $\vec E$ with magnitude $E\sqrt{\mu/\epsilon}$
4. Perpendicular to $\vec E$ with magnitude $E\sqrt{\epsilon/\mu}$

Perpendicular to $\vec E$ with magnitude $E\sqrt{\epsilon/\mu}$

Hence, answer is (D)

4. Power radiated by a point charge moving with constant acceleration of magnitude $\alpha$ is proportional to
1. $\alpha$
2. $\alpha^2$
3. $\alpha^{-1}$
4. $\alpha^{-2}$

The power radiated by a point charge is given by Larmor formula as $$P=\frac{\mu_0q^2\alpha^2}{6\pi c}$$

Hence, answer is (B)

5. The output of a laser has a bandwidth of $1.2\times10^{14}$ Hz. The coherence length $l_c$ of the output radiation is
1. 3.6 mm
2. 50 $\mu$m
3. 2.5 $\mu$m
4. 1.5 cm

Coherence length of laser is given by $$l_c=\frac{c}{\Delta\nu}$$ $$l_c=\frac{3\times10^8}{1.2\times10^{14}}=2.5 \mu m$$

Hence, answer is (C)