Question 14 Marks
A stationary charge produces only an electrostatic field while a charge in uniform motion produces a magnetic field, that does not change with time. An oscillating charge is an example of accelerating charge. It produces an oscillating magnetic field, which in turn produces an oscillating electric fields and so on. The oscillating electric and magnetic fields regenerate each other as a wave which propagates through space.
$(i).$ Magnetic field in a plane electromagnetic wave is given by $\vec{B}= B _0 \sin ( kx +\omega t ) \hat{j} T$Expression for corresponding electric field will be (Where c is speed of light.)
$(a) \ \vec{E}= B _0 c \sin ( kx +\omega t ) \hat{k} V / m$
$(b) \ \vec{E}=- B _0 c \sin ( kx -\omega t ) \hat{k} V / m$
$(c) \ \vec{E}=- B _0 c \sin ( kx +\omega t ) \hat{k} V / m$
$(d) \ \vec{E}=\frac{B_0}{c} \sin ( kx +\omega t ) \hat{k} V / m$
$(ii)$ The electric field component of a monochromatic radiation is given by $\vec{E}=2 E _0 \hat{i} \cos kz \cos \omega t$. Its magnetic field $\vec{B}$ is then given by
$(a) \ -\frac{2 E_0}{c} \hat{j} \sin kz \sin \omega t$
$(b) \ \frac{2 E_0}{c} \hat{j} \sin kz \sin \omega t$
$(c) \ \frac{2 E_0}{c} \hat{j} \sin kz \cos \omega t$
$(d) \ \frac{2 E_0}{c} \hat{j} \cos kz \cos \omega t$
$(iii)$ A plane em wave of frequency $25 MHz$ travels in a free space along $x -$direction. At a particular point in space and time, $E =(6.3 \hat{j}) V / m$. What is magnetic field at that time?
$(a) \ 0.089 \mu T$
$(b) \ 0.124 \mu T$
$(c) \ 0.021 \mu T$
$(d) \ 0.095 \mu T$
OR
A plane electromagnetic wave travels in free space along $x-$axis. At a particular point in space, the electric field along $y-$axis is $9.3 V m ^{-1}$. The magnetic induction $(B)$ along $z -$axis is
$(a) \ 3.1 \times 10^{-8} T$
$(b) \ 3 \times 10^{-5} T$
$(c) \ 3 \times 10^{-6} T$
$(d) \ 9.3 \times 10^{-6} T$
$(iv)$ A plane electromagnetic wave travelling along the $x-$direction has a wavelength of $3 \ mm$ . The variation in the electric field occurs in the $y-$direction with an amptitude $66 V m ^{-1}$. The equations for the electric and magnetic fields as a function of $x$ and $t$ are respectively
$ \text { a) } E_y =11 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right), \text { b) } E_y =66 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right),$
$B_y =11 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right) B_z =2.2 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$\text { c) } E_x =33 \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right), \text { d) } E_y =33 \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right),$
$B_x =11 \times 10^{-7} \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right) B_z =1.1 \times 10^{-7} \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$(i).$ Magnetic field in a plane electromagnetic wave is given by $\vec{B}= B _0 \sin ( kx +\omega t ) \hat{j} T$Expression for corresponding electric field will be (Where c is speed of light.)
$(a) \ \vec{E}= B _0 c \sin ( kx +\omega t ) \hat{k} V / m$
$(b) \ \vec{E}=- B _0 c \sin ( kx -\omega t ) \hat{k} V / m$
$(c) \ \vec{E}=- B _0 c \sin ( kx +\omega t ) \hat{k} V / m$
$(d) \ \vec{E}=\frac{B_0}{c} \sin ( kx +\omega t ) \hat{k} V / m$
$(ii)$ The electric field component of a monochromatic radiation is given by $\vec{E}=2 E _0 \hat{i} \cos kz \cos \omega t$. Its magnetic field $\vec{B}$ is then given by
$(a) \ -\frac{2 E_0}{c} \hat{j} \sin kz \sin \omega t$
$(b) \ \frac{2 E_0}{c} \hat{j} \sin kz \sin \omega t$
$(c) \ \frac{2 E_0}{c} \hat{j} \sin kz \cos \omega t$
$(d) \ \frac{2 E_0}{c} \hat{j} \cos kz \cos \omega t$
$(iii)$ A plane em wave of frequency $25 MHz$ travels in a free space along $x -$direction. At a particular point in space and time, $E =(6.3 \hat{j}) V / m$. What is magnetic field at that time?
$(a) \ 0.089 \mu T$
$(b) \ 0.124 \mu T$
$(c) \ 0.021 \mu T$
$(d) \ 0.095 \mu T$
OR
A plane electromagnetic wave travels in free space along $x-$axis. At a particular point in space, the electric field along $y-$axis is $9.3 V m ^{-1}$. The magnetic induction $(B)$ along $z -$axis is
$(a) \ 3.1 \times 10^{-8} T$
$(b) \ 3 \times 10^{-5} T$
$(c) \ 3 \times 10^{-6} T$
$(d) \ 9.3 \times 10^{-6} T$
$(iv)$ A plane electromagnetic wave travelling along the $x-$direction has a wavelength of $3 \ mm$ . The variation in the electric field occurs in the $y-$direction with an amptitude $66 V m ^{-1}$. The equations for the electric and magnetic fields as a function of $x$ and $t$ are respectively
$ \text { a) } E_y =11 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right), \text { b) } E_y =66 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right),$
$B_y =11 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right) B_z =2.2 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$\text { c) } E_x =33 \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right), \text { d) } E_y =33 \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right),$
$B_x =11 \times 10^{-7} \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right) B_z =1.1 \times 10^{-7} \cos \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
Answer
View full question & answer→A stationary charge produces only an electrostatic field while a charge in uniform motion produces a magnetic field, that does not change with time. An oscillating charge is an example of accelerating charge. It produces an oscillating magnetic field, which in turn produces an oscillating electric fields and so on. The oscillating electric and magnetic fields regenerate each other as a wave which propagates through space.
$(i) \ (a) \ \vec{E}= B _0 c \sin ( kx +\omega t ) \hat{k} V / m$
Explanation: Given : $\vec{B}= B _0 \sin ( kx +\omega t ) \hat{j} T$
The relation between electric and magnetic field is, $c =\frac{E}{B}$ or $E = cB$
The electric field component is perpendicular to the direction of propagation and the direction of magnetic field. Therefore, the electric field component along z-axis is obtained as $\vec{E}= cB _0 \sin ( kx +\omega t ) \hat{k} V / m$
$(ii) \ (b) \ \frac{2 E_0}{c} \hat{j} \sin kz \sin \omega t$
Explanation: $\frac{d E}{d z}=-\frac{d B}{d t}$
$\frac{d E}{d z}=-2 E _0 k \sin kz \cos \omega t =-\frac{d B}{d t}$
$dB =+2 E _0 k \sin kz \cos \omega tdt$
$B =+2 E _0 k \sin kz \int \cos \omega tdt =+2 E _0 \frac{k}{\omega} \sin kz \sin \omega t$
$\frac{E_0}{B_0}=\frac{\omega}{k}=c$
$B =\frac{2 E_0}{c} \sin kz \sin \omega t \therefore \vec{B}=\frac{2 E_0}{c} \sin kz \sin \omega t \hat{j}$
$E$ is along $y-$direction and the wave propagates along $x-$axis.
$\therefore B$ should be in a direction perpendicular to both $x$-and $y-$axis.
$(iii) \ (c) \ 0.021 \mu T$
Explanation: Here, $E =6.3 \hat{j} ; c =3 \times 10^8 m / s$
The magnitude of $B$ is
$B_{Z}=\frac{E}{c}=\frac{6.3}{3 \times 10^8}=2.1 \times 10^{-8} T=0.021 \mu T$
$OR$
$(a) 3.1 \times 10^{-8} T$
Explanation: At a particular point, $E =9.3 V m ^{-1}$
$\therefore$ Magnetic field at the same point $=\frac{9.3}{3 \times 10^8}$
$=3.1 \times 10^{-8} T$
$(iv) \ (b) \ E_y=66 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right), B_z=2.2 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
Explanation: Here : $E _0=66 Vm ^{-1}, E _y=66 \cos \omega\left(t-\frac{x}{c}\right)$,
$\lambda=3 mm=3 \times 10^{-3} m, k=\frac{2 \pi}{\lambda}$
$\frac{\omega}{k}=c \Rightarrow \omega=ck=3 \times 10^8 \times \frac{2 \pi}{3 \times 10^{-3}}$
$\text { or } \omega=2 \pi \times 10^{11}$
$\therefore E _{ y }=66 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$B _{ z }=\frac{E_y}{c}=\left(\frac{66}{3 \times 10^8}\right) \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$=2.2 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$(i) \ (a) \ \vec{E}= B _0 c \sin ( kx +\omega t ) \hat{k} V / m$
Explanation: Given : $\vec{B}= B _0 \sin ( kx +\omega t ) \hat{j} T$
The relation between electric and magnetic field is, $c =\frac{E}{B}$ or $E = cB$
The electric field component is perpendicular to the direction of propagation and the direction of magnetic field. Therefore, the electric field component along z-axis is obtained as $\vec{E}= cB _0 \sin ( kx +\omega t ) \hat{k} V / m$
$(ii) \ (b) \ \frac{2 E_0}{c} \hat{j} \sin kz \sin \omega t$
Explanation: $\frac{d E}{d z}=-\frac{d B}{d t}$
$\frac{d E}{d z}=-2 E _0 k \sin kz \cos \omega t =-\frac{d B}{d t}$
$dB =+2 E _0 k \sin kz \cos \omega tdt$
$B =+2 E _0 k \sin kz \int \cos \omega tdt =+2 E _0 \frac{k}{\omega} \sin kz \sin \omega t$
$\frac{E_0}{B_0}=\frac{\omega}{k}=c$
$B =\frac{2 E_0}{c} \sin kz \sin \omega t \therefore \vec{B}=\frac{2 E_0}{c} \sin kz \sin \omega t \hat{j}$
$E$ is along $y-$direction and the wave propagates along $x-$axis.
$\therefore B$ should be in a direction perpendicular to both $x$-and $y-$axis.
$(iii) \ (c) \ 0.021 \mu T$
Explanation: Here, $E =6.3 \hat{j} ; c =3 \times 10^8 m / s$
The magnitude of $B$ is
$B_{Z}=\frac{E}{c}=\frac{6.3}{3 \times 10^8}=2.1 \times 10^{-8} T=0.021 \mu T$
$OR$
$(a) 3.1 \times 10^{-8} T$
Explanation: At a particular point, $E =9.3 V m ^{-1}$
$\therefore$ Magnetic field at the same point $=\frac{9.3}{3 \times 10^8}$
$=3.1 \times 10^{-8} T$
$(iv) \ (b) \ E_y=66 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right), B_z=2.2 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
Explanation: Here : $E _0=66 Vm ^{-1}, E _y=66 \cos \omega\left(t-\frac{x}{c}\right)$,
$\lambda=3 mm=3 \times 10^{-3} m, k=\frac{2 \pi}{\lambda}$
$\frac{\omega}{k}=c \Rightarrow \omega=ck=3 \times 10^8 \times \frac{2 \pi}{3 \times 10^{-3}}$
$\text { or } \omega=2 \pi \times 10^{11}$
$\therefore E _{ y }=66 \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$B _{ z }=\frac{E_y}{c}=\left(\frac{66}{3 \times 10^8}\right) \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
$=2.2 \times 10^{-7} \cos 2 \pi \times 10^{11}\left(t-\frac{x}{c}\right)$
