Problem set 12

1. A plane wave of electric and magnetic fields $\vec E_0$ , $\vec B_0$ and frequency $\omega$ enters in a conducting bar of conductivity $\sigma$ along z-axis. Which of the following pairs of equations best represents the propagating wave? ($k\longrightarrow$ wave number)
1. $\vec E(z,t)=E_0e^{-ikz}e^{i(kz-\omega t)}\hat x$ and $\vec B(z,t)=\frac{k}{\omega} E_0e^{-ikz}e^{i(kz-\omega t+\phi)}\hat y$
2. $\vec E(z,t)=E_0e^{-kz}e^{i(kz-\omega t)}\hat x$ and $\vec B(z,t)=\frac{k}{\omega} E_0e^{-kz}e^{i(kz-\omega t+\phi)}\hat y$
3. $\vec E(z,t)=\frac{k}{\omega}E_0e^{-kz}e^{i(kz-\omega t)}\hat x$ and $\vec B(z,t)=\frac{k}{\omega} E_0e^{-ikz}e^{i(kz-\omega t+\phi)}\hat y$
4. $\vec E(z,t)=\frac{k}{\omega}E_0e^{-kz}e^{i(kz-\omega t)}\hat x$ and $\vec B(z,t)= E_0e^{-kz}e^{i(kz-\omega t+\phi)}\hat y$
In electromagnetic wave the electric and magnetic fields and propagation direction are mutually perpendicular to each other. Hence as wave travels in z-direction, electric and magnetic fields must be in x and y-directions. Let electric field is in x-direction. The wave equation for electric field in electromagnetic wave is given by $$\vec E(z,t)=E_0e^{-kz}e^{i(kz-\omega t)}\hat x$$ The term $e^{i(kz-\omega t)}$ is a plane equation and the term $e^{-kz}$ gives exponential decay of wave.
Magnetic field in plane electromagnetic wave in a conducting medium is related to the electric field by $$\begin{align*} \vec B&=\frac{\vec k}{\omega}\times\vec E\\ &=\frac{k}{\omega}\left|\vec E\right|\hat z\times\hat x\\ &=\frac{k}{\omega}\left|\vec E\right|\hat y\\ &=\frac{k}{\omega}E_0e^{-kz}e^{i(kz-\omega t)}\hat y \end{align*}$$ Also in conducting media there is a phase difference of $\phi$ between $\vec E$ and $\vec B$ $$\therefore\vec B=\frac{k}{\omega}E_0e^{-kz}e^{i(kz-\omega t+\phi)}\hat y$$
2. A plane electromagnetic wave travelling in vacuum is incident normally on a non-magnetic, non-absorbing medium of refractive index $n$. The incident $(E_i)$, reflected $(E_r)$ and transmitted $(E_t)$, electric fields are given as $E_i=E_{0i}exp[i(kz-\omega t)]$, $E_r=E_{0r}exp[i(k_rz-\omega t)]$, $E_t=E_{0t}exp[i(k_tz-\omega t)]$. If $E_{0i}=2 V/m$ and $n=1.5$ then the application of appropriate boundary conditions leads to
1. $E_{0r}=-\frac{3}{5} V/m$, $E_{0t}=\frac{7}{5} V/m$
2. $E_{0r}=-\frac{1}{5} V/m$, $E_{0t}=\frac{9}{5} V/m$
3. $E_{0r}=-\frac{2}{5} V/m$, $E_{0t}=\frac{8}{5} V/m$
4. $E_{0r}=\frac{4}{5} V/m$, $E_{0t}=\frac{6}{5} V/m$
Wave is travelling in vacuum, hence $n_1=1$, for the medium $n_2=1.5$. $$\begin{align*} E_{0r}&=\frac{n_1-n_2}{n_1+n_2}E_{0i}\\ &=\frac{1-1.5}{1+1.5}\times 2\\ &=-\frac{1}{2.5}=-\frac{2}{5} \end{align*}$$ $$\begin{align*} E_{0t}&=\frac{2n_1}{n_1+n_2}E_{0i}\\ &=\frac{2}{1.5+1}\times 2\\ &=\frac{4}{2.5}=\frac{8}{5} \end{align*}$$
3. The magnetic field due to the $TE_{11}$ mode in a rectangular wave guide aligned along Z-axis is given by $H_z=H_1\cos{(0.5\:\pi x)}\cos{(0.6\:\pi y)}$, where $x$ and $y$ are in cm. Then dimensions of the rectangular wave guide $a$ and $b$, respectively, are
1. 2.00 cm and 1.66 cm
2. 1.66 cm and 2.66 cm
3. 2.54 cm and 1.66 cm
4. 1.66 cm and 1.25 cm
In a rectangular wave guide the longitudinal magnetic field is given by $$H_z(x,y)=H_0\cos{(k_xx)}\cos{(k_yy)}$$ where $k_x=\frac{n\pi}{a}$ and $k_y=\frac{m\pi}{b}$. For $TE_{11}$ mode $n=m=1$.
Hence, $k_x=\frac{\pi}{a}$ and $k_y=\frac{\pi}{b}$
Hence, $a=\frac{\pi}{k_x}=\frac{1}{0.5}=2$ cm and $b=\frac{\pi}{k_y}=\frac{1}{0.6}=1.66$ cm
Hence, answer is (A) 2.00 cm and 1.66 cm.
4. The Boolean expression $B\cdot(A+B)+A\cdot(\bar B+A)$ can be realized using minimum number of
1. 1 AND gate
2. 2 AND gates
3. 1 OR gate
4. 2 OR gates
$$\begin{align*} {}&{}B\cdot(A+B)+A\cdot(\bar B+A)\\ &=B\cdot A+B\cdot B+A\cdot\bar B+A\cdot A \end{align*}$$ But $B\cdot B=B$ and $A\cdot A=A$ $$\begin{align*} {}&{}B\cdot(A+B)+A\cdot(\bar B+A)\\ &=B\cdot A+B+A\cdot\bar B+A\\ &=B\cdot(A+1)+A\cdot(\bar B+1) \end{align*}$$ But $A+1=1$ and $(\bar B+1)=1$ $$B\cdot(A+B)+A\cdot(\bar B+A)=B+A$$ Hence answer is (C) 1 OR gate
5. For a diatomic molecule with the vibrational quantum number $n$ and rotational quantum number $J$, the vibrational level spacing $\Delta E_n=E_n-E_{n-1}$ and the rotational level spacing $\Delta E_J=E_J-E_{J-1}$ are approximately
1. $\Delta E_n=$ constant, $\Delta E_J=$ constant
2. $\Delta E_n=$ constant, $\Delta E_J\propto J$
3. $\Delta E_n\propto n$, $\Delta E_J\propto J$
4. $\Delta E_n\propto n$, $\Delta E_J\propto J^2$
(B)$\Delta E_n=$ constant, $\Delta E_J\propto J$