Pointing vectors $\vec S$ is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propagation. Mathematically, it is given by $\vec S = \frac{1}{{{\mu _0}}}(\vec E \times \vec B)$. Show the nature of $\vec S$ vs $t$ graph.
Pointing vectors $\vec S$ is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propagation. Mathematically, it is given by $\vec S = \frac{1}{{{\mu _0}}}(\vec E \times \vec B)$. Show the nature of $\vec S$ vs $t$ graph.
In electromagnetic wave let $\overrightarrow{\mathrm{E}}$ is in $y$-direction, $\overrightarrow{\mathrm{B}}$ is in $z$-direction and electromagnetic wave propagate in $x$-direction energy propagating will be in direction of $\overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{B}}$ (in $x$-direction).
$\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \sin (\omega t-k x) \hat{j}$
$\overrightarrow{\mathrm{B}}=\mathrm{B}_{0} \sin (\omega t-k x) \hat{k}$
$\therefore \overrightarrow{\mathrm{S}}=\frac{1}{\mu_{0}}(\overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{B}})=\frac{1}{\mu_{0}} \mathrm{E}_{0} \mathrm{~B}_{0} \sin ^{2}(\omega t-k x)(\hat{j} \times \hat{k})$
$\therefore \overrightarrow{\mathrm{S}}=\frac{\mathrm{E}_{0} \mathrm{~B}_{0}}{\mu_{0}} \sin ^{2}(\omega t-k x) \hat{i}[\because \hat{j} \times \hat{k}=\hat{i}]$
Variation in magnitude of $|\overrightarrow{\mathrm{S}}|$ with time is shown in figure shown below.
Similar Questions
Given below are two statements:
Statement $I$ : Electromagnetic waves are not deflected by electric and magnetic field.
Statement $II$ : The amplitude of electric field and the magnetic field in electromagnetic waves are related to each other as $E _0=\sqrt{\frac{\mu_0}{\varepsilon_0}} B_0$
In the light of the above statements, choose the correct answer from the options given below:
Given below are two statements:
Statement $I$ : Electromagnetic waves are not deflected by electric and magnetic field.
Statement $II$ : The amplitude of electric field and the magnetic field in electromagnetic waves are related to each other as $E _0=\sqrt{\frac{\mu_0}{\varepsilon_0}} B_0$
In the light of the above statements, choose the correct answer from the options given below:
- [JEE MAIN 2023]
For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric $\left( U _{ e }\right)$ and magnetic $\left( U _{ m }\right)$ fields is
For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric $\left( U _{ e }\right)$ and magnetic $\left( U _{ m }\right)$ fields is
- [JEE MAIN 2021]
In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10}\; Hz$ and amplitude $48\; Vm ^{-1}$
$(a)$ What is the wavelength of the wave?
$(b)$ What is the amplitude of the oscillating magnetic field?
$(c)$ Show that the average energy density of the $E$ field equals the average energy density of the $B$ field. $\left[c=3 \times 10^{8} \;m s ^{-1} .\right]$
In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10}\; Hz$ and amplitude $48\; Vm ^{-1}$
$(a)$ What is the wavelength of the wave?
$(b)$ What is the amplitude of the oscillating magnetic field?
$(c)$ Show that the average energy density of the $E$ field equals the average energy density of the $B$ field. $\left[c=3 \times 10^{8} \;m s ^{-1} .\right]$
The electric field in an electromagnetic wave is given by $\overrightarrow{\mathrm{E}}=\hat{\mathrm{i}} 40 \cos \omega\left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right) N \mathrm{NC}^{-1}$. The magnetic field induction of this wave is (in SI unit):
The electric field in an electromagnetic wave is given by $\overrightarrow{\mathrm{E}}=\hat{\mathrm{i}} 40 \cos \omega\left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right) N \mathrm{NC}^{-1}$. The magnetic field induction of this wave is (in SI unit):
- [JEE MAIN 2024]
The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}=30(2 \hat{x}+\hat{y}) \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{V} \mathrm{m}^{-1}$. Which of the following option($s$) is(are) correct?
[Given: The speed of light in vacuum, $c=3 \times 10^8 \mathrm{~ms}^{-1}$ ]
($A$) $B_x=-2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{Wbm}^{-2}$.
($B$) $B_y=2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{Wbm}^{-2}$
($C$) The wave is polarized in the $x y$-plane with polarization angle $30^{\circ}$ with respect to the $x$-axis.
($D$) The refractive index of the medium is $2$ .
The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}=30(2 \hat{x}+\hat{y}) \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{V} \mathrm{m}^{-1}$. Which of the following option($s$) is(are) correct?
[Given: The speed of light in vacuum, $c=3 \times 10^8 \mathrm{~ms}^{-1}$ ]
($A$) $B_x=-2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{Wbm}^{-2}$.
($B$) $B_y=2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{Wbm}^{-2}$
($C$) The wave is polarized in the $x y$-plane with polarization angle $30^{\circ}$ with respect to the $x$-axis.
($D$) The refractive index of the medium is $2$ .
- [IIT 2023]