The magnetic field of a beam emerging from a filter facing a floodlight is given by
$$B_0=12 \times 10^{-8} \sin \left(1.20 \times 10^7 z-3.60 \times 10^{15} t\right) \mathrm{T}$$
What is the average intensity of the beam?
Magnetic field $\mathbf{B}=B_0$ sin $\omega t$
Given, equation $B=12 \times 10^{-8} \sin \left(1.20 \times 10^7 z-3.60 \times 10^{15} t\right) \mathrm{T}$.
On comparing this equation with standard equation, we get
$$B_0=12 \times 10^{-8}$$
The average intensity of the beam $I_{\mathrm{av}}=\frac{1}{2} \frac{B_0^2}{\alpha_0} \cdot c=\frac{1}{2} \times \frac{\left(12 \times 10^{-8}\right)^2 \times 3 \times 10^8}{4 \pi \times 10^{-7}}$
$$=1.71 \mathrm{~W} / \mathrm{m}^2$$
Poynting vectors $\mathbf{S}$ is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propogation. Mathematically, it is given by $\mathbf{S}=\frac{1}{\propto_0} \mathbf{E} \times \mathbf{B}$. Show the nature of $\mathbf{S}$ versus $t$ graph.
Consider and electromagnetic waves, let $\mathbf{E}$ be varying along $y$-axis, $\mathbf{B}$ is along $z$-axis and propagation of wave be along $x$-axis. Then $\mathbf{E} \times \mathbf{B}$ will tell the direction of propagation of energy flow in electromegnetic wave, along $x$-axis.
Let
$$\begin{aligned} \mathbf{E} & =E_0 \sin (\omega t-k x) \hat{\mathbf{j}} \\ \mathbf{B} & =B_0 \sin (\omega t-k x) \hat{\mathbf{k}} \\ \mathbf{S} & =\frac{1}{\alpha_0}(\mathbf{E} \times \mathbf{B})=\frac{1}{\alpha_0} E_0 B_0 \sin ^2(\omega t-k x)[\hat{\mathbf{j}} \times \hat{\mathbf{k}}] \\ & =\frac{E_0 B_0}{\alpha_0} \sin ^2(\omega t-k x) \hat{\mathbf{i}} \end{aligned}$$
The variation of $|\mathbf{S}|$ with time $t$ will be as given in the figure below
Professor CV Raman surprised his students by suspending freely a tiny light ball in a transparent vacuum chamber by shining a laser beam on it. Which property of EM waves was he exhibiting? Give one more example of this property.
An electromagnetic wave carries energy and momentum like other waves. Since, it carries momentum, an electromagnetic wave also exerts pressure called radiation pressure. This property of electromagnetic waves helped professor CV Raman surprised his students by suspending freely a tiny light ball in a transparent vacuum chamber by shining a laser beam on it. The tails of the camets are also due to radiation pressure.
Show that the magnetic field $B$ at a point in between the plates of a parallel plate capacitor during charging is $\frac{\propto_0 \varepsilon_0 r}{2} \frac{d E}{d t}$ (symbols having usual meaning).
Consider the figure ginen below to prove that the magneti field $B$ at a point in between the plater of a paravel- plate copocior during charging is $\frac{\varepsilon_0 \propto_0 r}{2} \frac{d E}{d t}$
Let $I_d$ be the displacement current in the region between two plates of parallel plate capacitor, in the figure.
The magnetic field induction at a point in a region between two plates of capacitor at a perpendicular distance $r$ from the axis of plates is
$$\begin{array}{rlr} B & =\frac{\propto_0 2 I_d}{4 \pi r}=\frac{\propto_0}{2 \pi r} I_d=\frac{\propto_0}{2 \pi r} \times \varepsilon_0 \frac{d \phi_E}{d t} & {\left[\because I_d=\frac{E_0 d \phi_E}{d t}\right]} \\ & =\frac{\propto_0 \varepsilon_0}{2 \pi r} \frac{d}{d t}\left(E \pi r^2\right)=\frac{\propto_0 \varepsilon_0}{2 \pi r} \pi r^2 \frac{d E}{d t} & {\left[\because \phi_E=E \pi r^2\right]} \\ B & =\frac{\propto_0 \varepsilon_0 r}{2} \frac{d E}{d t} & \end{array}$$
Electromagnetic waves with wavelength
(i) $\lambda_1$ is used in satellite communication.
(ii) $\lambda_2$ is used to kill germs in water purifies.
(iii) $\lambda_3$ is used to detect leakage of oil in underground pipelines.
(iv) $\lambda_4$ is used to improve visibility in runways during fog and mist conditions.
(a) Identify and name the part of electromagnetic spectrum to which these radiations belong.
(b) Arrange these wavelengths in ascending order of their magnitude.
(c) Write one more application of each.
(a) (i) Microwave is used in satellite communications.
So, $\lambda_1$ is the wavelength of microwave.
(ii) Ultraviolet rays are used to kill germs in water purifier. So, $\lambda_2$ is the wavelength of UV rays.
(iii) $X$-rays are used to detect leakage of oil in underground pipelines. So, $\lambda_3$ is the wavelength of $X$-rays.
(iv) Infrared is used to improve visibility on runways during fog and mist conditions. So, it is the wavelength of infrared waves.
(b) Wavelength of X-rays < wavelength of UV $<$ wavelength of infrared $<$ wavelength of microwave.
$$\Rightarrow \quad \lambda_3<\lambda_2<\lambda_4<\lambda_1 $$
(c) (i) Microwave is used in radar.
(ii) UV is used in LASIK eye surgery.
(iii) X-ray is used to detect a fracture in bones.
(iv) Infrared is used in optical communication.