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CBSE Class 12 NCERT Physics Electromagnetic Waves
Electromagnetic Waves

A plane EM wave travelling along z-direction is described by $E=E_{0} \sin (k z-\omega t) \hat{i}$ and $B=B_{0} \sin (k z-\omega t) \hat{j}$ Show that
i) the average energy density of the wave is given by $u_{a v}=\frac{1}{4} \epsilon_{0} E_{0}^{2}+\frac{1}{4} \frac{B_{0}^{2}}{\mu_{0}}$
ii) the time-averaged intensity of the wave is given by $I_{a v}=\frac{1}{2} c \epsilon_{0} E_{0}^{2}$

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i) The energy density due to electric field $E$ is given as $\mathrm{uE}=1 / 2 \varepsilon_{0} \mathrm{E}^{2}$ The energy density due to magnetic field $B$ is givena as $\mathrm{uB}=1 / 2...

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A plane EM wave travelling in vacuum along z-direction is given by $E=E_{0} \sin (k z-\omega t) \hat{i} \text { and } B=B_{0} \sin (k z-\omega t) \hat{j}$
a) Use equation $\oint E . d l=\frac{-d \phi_{B}}{d t}$ to prove $E_ 0 / \mathbf B_ 0=\mathrm{c}$
b) by using a similar process and the equation $\oint B . d l=\mu_{0} I \epsilon_{0} \frac{-d \phi_{E}}{d t}$, prove that $c=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}$

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a) Substituting the above equations in the following equation we get ${c} \oint E . d l=-\frac{d \phi_{B}}{d t}=-\frac{d}{d t} \oint B \cdot d s$ So, $E_{0} / B_{0}=0$ b) We get $c=1 /...

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A plane EM wave travelling in vacuum along z-direction is given by $E=E_{0} \sin (k z-\omega t) \hat{i} \text { and } B=B_{0} \sin (k z-\omega t) \hat{j}$
a) evaluate $\oint E . d l \quad$ over the rectangular loop 1234 shown in the figure
b) evaluate $\int B \cdot d s$ over the surface bounded by loop 1234

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Solution: (a) $\oint_{\vec{E}} \cdot \overrightarrow{d l}=E_{0} h\left[\sin \left(k z_{2}-\omega t\right)-\sin \left(k z_{1}-\omega t\right)\right]$ (b) $\int \vec{B}.{\overrightarrow{d s}} =...

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A long straight cable of length $I$ is placed symmetrically along the z-axis and has radius $a$. The cable consists of a thin wire and a co-axial conducting tube. An alternating current $I(t)=I_{0}$ sin $(2\pi vt)$ flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance $s$ from the wire inside the cable is $\mathbf{E}(\mathrm{s}, \mathrm{t})=\mu_{0} \mathrm{l}_{0} \mathrm{~V} cos (2\pi vt)$. In $\left(\frac{s}{a}\right) \hat{k}$,
compare the conduction current 10 with the displacement current $I_{0}^{\mathrm{d}}$

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The displacement will be, $I_{0}^{\mathrm{d}} / \mathrm{I}_{0}=(\mathrm{am} / \lambda)^{2}$

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A long straight cable of length $I$ is placed symmetrically along the z-axis and has radius $a$. The cable consists of a thin wire and a co-axial conducting tube. An alternating current $I(t)=I_{0}$ sin $(2\pi vt)$ flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance $s$ from the wire inside the cable is $\mathbf{E}(\mathrm{s}, \mathrm{t})=\mu_{0} \mathrm{l}_{0} \mathrm{~V} cos (2\pi vt)$. In $\left(\frac{s}{a}\right) \hat{k}$,
a) calculate the displacement current density inside the cable
b) integrate the displacement current density across the cross-section of the cable to find the total displacement current I

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a) The displacement current density is given as $\vec{J}_{d}=\frac{2 \pi I_{0}}{\lambda^{2}} \ln \frac{a}{s} \sin 2 \pi v t \hat{k}$ b) Total displacement current will be, $I^{d}=\int J_{d} 2 \pi s...

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Seawater at frequency $v=4 \times 10^8 \mathrm{~Hz}$ has permittivity $\varepsilon=80 \varepsilon_{0}$, permeability $\mu=\mu_{0}$ and resistivity $\rho=$ $0.25 \Omega \mathrm{m}$. Imagine a parallel plate capacitor immersed in seawater and driven by an alternating voltage source $V(t)=V_{0} \sin (2 \pi v t) .$ What fraction of the conduction current density is the displacement current density?

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The separation between the plates of the capacitor is given as $V(t)=V_{0} \sin (2 \pi v t)$ Ohm's law for the conduction of current density is given as $\mathrm{J}_{0}{...

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An infinitely long thin wire carrying a uniform linear static charge density $\Lambda$ is placed along the z-axis. The wire is set into motion along its length with a uniform velocity $v=v \hat{k}_{z} \quad .$ Calculate the pointing vectors $s=1 / \mu_{0}(E \times B)$

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The electric field in an infinitely long thin wire is given by the expression, $\vec{E}=\frac{\lambda \hat{e}_{s}}{2 \pi \epsilon_{0} a} \hat{j}$ Magnetic field due to the wire is given by the...

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What happens to the intensity of light from a bulb if the distance from the bulb is doubled? As a laser beam travels across the length of a room, its intensity essentially remains constant. What geometrical characteristics of the LASER beam is responsible for the constant intensity which is missing in the case of light from the bulb?

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When the distance between two points is doubled, the intensity of light is reduced by one-fourth. Geometrical characteristics of the LASER are: a) unidirectional b) monochromatic c) coherent...

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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 runaways during fog and mist conditions
a) identify and name the part of the electromagnetic spectrum to which these radiations belong
b) arrange these wavelengths in ascending order of their magnitude
c) write one more application of each

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a) i) $\lambda_{1}$ is a microwave, used in satellite communication. ii) $\lambda_{2}$ is UV rays, used in a water purifier for killing germs. iii) $\lambda_{3}$ is X-rays, used in improving the...

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The magnetic field of a beam emerging from a filter facing a floodlight is given by $B_{0}=12 \times 10^{-8} \sin$ $\left(1.20 \times 10^{7} \mathrm{z}-3.60 \times 10^{15} \mathrm{t}\right) \mathrm{T}$. What is the average intensity of the beam?

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$\begin{array}{l} B_{0}=12 \times 10^{-8} \sin \left(1.20 \times 10^{7} z-3.60 \times 10^{15} \mathrm{t}\right) \mathrm{T} \\ \mathrm{B} 0=12 \times 10^{-8} \mathrm{~T} \\ \mathrm{lav}=1.71...

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An EM wave of intensity I falls on a surface kept in vacuum and exerts radiation pressure $p$ on it. Which of the following are true?
a) radiation pressure is $\mathrm{I} / \mathrm{c}$ if the wave is totally absorbed
b) radiation pressure is $I / c$ if the wave is totally reflected
c) radiation pressure is $2 \mathrm{I} / \mathrm{c}$ if the wave is totally reflected
d) radiation pressure is in the range $I / c

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The correct options are: a) radiation pressure is $\mathrm{l} / \mathrm{c}$ if the wave is totally absorbed c) radiation pressure is $2 \mathrm{l} / \mathrm{c}$ if the wave is totally reflected d)...

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A charged particle oscillates about its mean equilibrium position with a frequency of $10^{9} \mathrm{~Hz}$. The electromagnetic waves produced:
a) will have a frequency of $10^{9} \mathrm{~Hz}$
b) will have a frequency of $2 \times 10^{9} \mathrm{~Hz}$
c) will have a wavelength of $0.3 \mathrm{~m}$
d) fall in the region of radiowaves

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The correct options are: a) will have a frequency of $10^{9} \mathrm{~Hz}$ c) will have a wavelength of $0.3 \mathrm{~m}$ d) fall in the region of radiowaves

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An electromagnetic wave travelling along the $z$-axis is given as: $E=E_{0} \cos (k z-\omega t)$. Choose the correct options from the following a) the associated magnetic field is given as $B=\frac{1}{c} \hat{k} \times E=\frac{1}{\omega}(\hat{k} \times E)$
b) the electromagnetic field can be written in terms of the associated magnetic field as$E=c(B \times \hat{k})$
c)$\hat{k} \cdot E=0, \hat{k} \cdot B=0$
d)$\hat{k} \times E=0, \hat{k} \times B=0$

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a) the associated magnetic field is given as $B=\frac{1}{c} \hat{k} \times E=\frac{1}{\omega}(\hat{k} \times E)$ b) the electromagnetic field can be written in terms of the associated magnetic field...

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An electromagnetic wave travels in vacuum along z-direction: $E=\left(E_{1} \hat{i}+E_{2} \hat{j}\right) \cos (k z-\omega t) \cdot$ Choose the correct options from the following:
a) the associated magnetic field is given as $B=\frac{1}{c}\left(E_{1} \hat{i}-E_{2} \hat{j}\right) \cos (k z-\omega t)$
b) the associated magnetic field is given as $B=\frac{1}{c}\left(E_{1} \hat{i}-E_{2} \hat{j}\right) \cos (k z-\omega t)$
c) the given electromagnetic field is circularly polarised
d) the given electromagnetic waves is plane polarised

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a) the associated magnetic field is given as $B=\frac{1}{c}\left(E_{1} \hat{i}-E_{2} \hat{j}\right) \cos (k z-\omega t)$ d) the given electromagnetic waves is plane polarised

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Answer the following questions:
(a) If the earth did not have an atmosphere, would its average surface temperature be higher or lower than what it is now?
(b) Some scientists have predicted that a global nuclear war on the earth would be followed by a severe ‘nuclear winter’ with a devastating effect on life on earth. What might be the basis of this prediction?

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(a) There would be no greenhouse effect if there was no atmosphere. As a result, the earth's temperature would plummet. (b) Smoke clouds from a worldwide nuclear war might potentially cover large...

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Answer the following questions:
(a) Optical and radio telescopes are built on the ground but X-ray astronomy is possible only from satellites orbiting the earth. Why?
(b) The small ozone layer on top of the stratosphere is crucial for human survival. Why?

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(a) X-rays are absorbed by the atmosphere, although visible and radio waves can get through. (b) The ozone layer absorbs ultraviolet energy from the sun, preventing it from reaching the earth's...

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About $5 \%$ of the power of a $100 \mathrm{~W}$ light bulb is converted to visible radiation. What is the average intensity of visible radiation.
(a) at a distance of $1 \mathrm{~m}$ from the bulb?
(b) at a distance of $10 \mathrm{~m}$ ? Assume that the radiation is emitted isotropically and neglect reflection.

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(a) Average intensity of the visible radiation is given by the expression, $I=P^{\prime} / 4 \pi d^{2}$ So, the power of the visible radiation will be $P^{\prime}=(5 / 100) \times 100=5 \mathrm{~W}$...

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Suppose that the electric field part of an electromagnetic wave in vacuum is $E=\{(3.1$ $\left.\mathrm{N} / \mathrm{C}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) \mathrm{y}+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) \mathrm{t}\right]\right\}^{\wedge} \mathrm{i}$. Write an expression for the magnetic field part of the wave.

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Magnetic wave is directed along the negative z-direction. Thus, B = Bo​cos(ky+ωt)k $B_{z}=B_{0} \cos (k y+\omega t)^{2} k=\left{(10.3 \mathrm{nT}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m})...

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Suppose that the electric field part of an electromagnetic wave in vacuum is $E=\{(3.1$ $\left.\mathrm{N} / \mathrm{C}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) \mathrm{y}+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) \mathrm{t}\right]\right\}^{\wedge} \mathrm{i}$
(a) What is the frequency v?
(b) What is the amplitude of the magnetic field part of the wave?

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(a) Frequency is calculated as $v=\omega / 2 \pi=5.4 \times 10^{6} /(2 \times 3.14)=0.859 \times 10^{6} \mathrm{~Hz}$ (b) Amplitude of the magnetic field can be calculated...

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Suppose that the electric field part of an electromagnetic wave in vacuum is $E=\{(3.1$ $\left.\mathrm{N} / \mathrm{C}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) \mathrm{y}+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) \mathrm{t}\right]\right\}^{\wedge} \mathrm{i}$
(a) What is the direction of propagation?
(b) What is the wavelength $\lambda$ ?

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(a) The motion is going in the opposite direction of the y-axis. To put it another way, along -j. (b) The given equation is compared with the equation and we get, $E=E_{0} \cos (k y+\omega t)$...

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In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10} \mathrm{~Hz}$ and amplitude $48 \mathrm{~V} \mathrm{~m}^{-1}$. Show that the average energy density of the $E$ field equals the average energy density of the B field. $\left[\mathrm{c}=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\right]$

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Frequency of the electromagnetic wave is given as $\mathrm{v}=2 \times 10^{10} \mathrm{~Hz}$ Electric field amplitude is given as $E_{0}=48 \vee \mathrm{m}^{-1}$ Speed of light is known as $c=3...

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In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10} \mathrm{~Hz}$ and amplitude $48 \mathrm{~V} \mathrm{~m}^{-1}$.
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field?

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Frequency of the electromagnetic wave is given as $\mathrm{v}=2 \times 10^{10} \mathrm{~Hz}$ Electric field amplitude is given as $E_{0}=48 \vee \mathrm{m}^{-1}$ Speed of light is known as $c=3...

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The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formula $E=h v$ (for the energy of a quantum of radiation: photon) and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?

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The energy of a photon is represented by the expression, $\mathrm{E}=\mathrm{hv}=\frac{h c}{\lambda}$ Where, $\mathrm{h}$ is the planck's constant with a value of$6.6 \times 10^{-34} J_{S}$...

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Suppose that the electric field amplitude of an electromagnetic wave is $E_{0}=120 \mathrm{~N} / C$ and that its frequency is $\mathbf{v}=50 \mathrm{MHz}$.(a) Determine $B_{0}, \omega, k$ and $\lambda$ (b) Find expressions for $\mathrm{E}$ and $\mathrm{B}$.

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Electric field amplitude is given as $E_{0}=120 \mathrm{~N} / C$ Frequency of source is given as $v=50 \mathrm{MHz}=50 \times 10^{6} \mathrm{~Hz}$ Speed of light is known as $c=3 \times 10^{8}...

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A parallel plate capacitor made of circular plates each of radius $\mathbf{R}=\mathbf{6 . 0} \mathbf{~ c m}$ has a capacitance $C=100 \mathrm{pF}$. The capacitor is connected to a $230 \mathrm{~V}$ ac supply with an (angular) frequency of $300 \mathrm{rad} \mathrm{s}^{-1}$. Determine the amplitude of $\mathrm{B}$ at a point $3.0 \mathrm{~cm}$ from the axis between the plates.

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Solution: Radius of each circular plate is given as $0.06m$ Capacitance of a parallel plate capacitor is given as $\mathrm{C}=100 \mathrm{pF}=100 \times 10^{-12} \mathrm{~F}$ Supply voltage is given...

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A parallel plate capacitor made of circular plates each of radius $\mathbf{R}=\mathbf{6 . 0} \mathbf{~ c m}$ has a capacitance $C=100 \mathrm{pF}$. The capacitor is connected to a $230 \mathrm{~V}$ ac supply with an (angular) frequency of $300 \mathrm{rad} \mathrm{s}^{-1}$.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?

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Solution: Radius of each circular plate is given as $0.06m$ Capacitance of a parallel plate capacitor is given as $\mathrm{C}=100 \mathrm{pF}=100 \times 10^{-12} \mathrm{~F}$ Supply voltage is given...

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The Figure shows a capacitor made of two circular plates each of radius $12 \mathrm{~cm}$ and separated by $\mathbf{5 . 0} \mathbf{~ c m}$. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15A. Is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.

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Solution: Yes Kirchhoff's first rule holds true for each capacitor plate if we use the total of conduction and displacement to calculate current.

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The Figure shows a capacitor made of two circular plates each of radius $12 \mathrm{~cm}$ and separated by $\mathbf{5 . 0} \mathbf{~ c m}$. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15A.
(a) Calculate the capacitance and the rate of change of the potential difference between the plates.
(b) Obtain the displacement current across the plates.

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The radius of each circular plate $(r)$ is given as $0.12 \mathrm{~m}$ The distance between the plates $(d)$ is given as $0.05 \mathrm{~m}$ The charging current (I) is given as $0.15 \mathrm{~A}$...

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