A plane EM wave travelling along z-direction
Question: 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 $I_{a v}=\frac{1}{2} c \epsilon_{0} E_{0}^{2}$ Solution: (i) The energy density due to electric field E is uE = 1/2 0E2 The energy density due ...
Read More →Find the mean, variance and standard deviation for the numbers 4, 6, 10, 12,
Question: Find the mean, variance and standard deviation for the numbers 4, 6, 10, 12, 7, 8, 13, 12. Solution: Given data: 4, 6, 10, 12, 7, 8, 13, 12 To find: MEAN We know that Mean $(\overline{\mathrm{x}})=\frac{\text { Sum of observations }}{\text { Total number of observations }}$ $=\frac{4+6+10+12+7+8+13+12}{8}$ $=\frac{72}{8}$ $\overline{\mathrm{x}}=9$ To find: VARIANCE Variance, $\sigma^{2}=\frac{\sum\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}}{\mathrm{n}}$ $=\frac{74}{8...
Read More →A long straight cable of length l is placed symmetrically
Question: A long straight cable of length l 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) = I0sin (2vt) 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 E(s,t) = 0I0v coz (2vt). In $\left(\frac{s}{a}\right) \hat{k}$a) calculate the displacement current density inside the cable (b) in...
Read More →Seawater at frequency v = 4 108 Hz has permittivity ε = 80 ε0,
Question: Seawater at frequency v = 4 108 Hz has permittivity = 80 0, permeability = 0and resistivity = 0.25 Ωm. Imagine a parallel plate capacitor immersed in seawater and driven by an alternating voltage source V(t) = V0sin (2vt). What fraction of the conduction current density is the displacement current density? Solution: The separation between the plates of the capacitor is V(t) = V0sin (2vt) Ohms law for the conduction of current density J0c= V0/d The displacement current density is Jod= 2...
Read More →An infinitely long thin wire carrying a uniform linear
Question: An infinitely long thin wire carrying a uniform linear static charge density 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 $S=1 / \mu_{0}$ (ExB). Solution: The electric field in an infinitely long thin wire is $\vec{E}=\frac{\lambda \hat{e}_{s}}{2 \pi \epsilon_{0} a} \hat{j}$ Magnetic field due to the wire is $\vec{B}=\frac{\mu_{0} i}{2 \pi a} \hat{i}$ The equivalent current flow...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{x-1}{3 x^{2}-4 x+3} d x$ Solution: $I=\int \frac{x-1}{3 x^{2}-4 x+3} d x$ As we can see that there is a term of $x$ in numerator and derivative of $x^{2}$ is also $2 x$. So there is a chance that we can make substitution for $3 x^{2}-4 x+3$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(3 \mathrm{x}^{2}-4 \mathrm{x}+3\right)=6 \mathrm{x}-4$ $\therefore$ Let, $x-1=A(6 x-4)+B$ $\Rightarrow x-1=6 A x-4 A+B$ On...
Read More →Even though an electric field E exerts a force qE
Question: Even though an electric field E exerts a force qE on a charged particle yet the electric field of an EM wave does not contribute to the radiation pressure. Explain. Solution: Even though an electric field E exerts a force qE on a charged particle yet the electric field of an EM wave does not contribute to the radiation pressure because radiation pressure is the result of the action of the magnetic field of the wave on the electric currents which is induced by the electric field....
Read More →What happens to the intensity of light from a bulb
Question: 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? Solution: When the distance is doubled, the intensity of light becomes one-fourth. Following are the geometrical characteristics of the LASER: (a) unidire...
Read More →Find the mean, variance and standard deviation for the numbers 4, 6, 10, 12,
Question: Find the mean, variance and standard deviation for the numbers 4, 6, 10, 12, 7, 8, 13, 12. Solution: Given data: 4, 6, 10, 12, 7, 8, 13, 12br data-mce-bogus="1"/ppTo find: MEANbr data-mce-bogus="1"/ppWe know that,br data-mce-bogus="1"/ppMean $(\overline{\mathrm{x}})=\frac{\text { Sum of observations }}{\text { Total number of observations }}$br data-mce-bogus="1"/pp$=\frac{4+6+10+12+7+8+13+12}{8}$br data-mce-bogus="1"/pp$=\frac{72}{8}$/pp$\overline{\mathrm{x}}=9$/ppTo find: VARIANCEbr ...
Read More →Show that the radiation pressure exerted
Question: Show that the radiation pressure exerted by an EM wave of intensity I on a surface kept in vacuum is I/c. Solution: Energy received by the surface per second = E = IA No.of photons received by the surface per second = N The perfect absorbing is = h/ Pressure, P = F/A = I/c...
Read More →You are given a 2μF parallel plate capacitor.
Question: You are given a 2F parallel plate capacitor. How would you establish an instantaneous displacement current of 1 mA in the space between its plates? Solution: The capacitance of the capacitor C = 2F Displacement current Id = 1 mA Charge in capacitor, q = CV dV/dt = 500 V/s...
Read More →Show that the average value of radiant flux
Question: Show that the average value of radiant flux density S over a single period T is given by S = 1/2c0E02. Solution: Radiant flux density is given as $\vec{S}=\frac{1}{\mu_{0}}(\vec{E} \times \vec{B})=c^{2} \epsilon_{0}(\vec{E} \times \vec{B})$ E = E0cos (kx ꞷt) B = B0cos (kx ꞷt) EB = c20(E0B0) cos2(kx ꞷt) Average value of the radiant flux density is Sav = E02/20c...
Read More →Find the mean deviation about the median for the following data :
Question: Find the mean deviation about the median for the following data : Solution: The class interval containing $\frac{\mathrm{N}}{2}^{\mathrm{th}}$ or 25th item is 20-30. Therefore, 2030 is the median class. We know that Median $=1+\frac{\frac{\mathrm{N}}{2}-\mathrm{C}}{\mathrm{f}} \times \mathrm{h}$ Here, $I=20, C=14, f=11, h=10$ and $N=50$ Therefore, Median $=20+\frac{\frac{50}{2}-14}{11} \times 10=20+10=30$ Now We have, $\sum_{\mathrm{i}=1}^{6} \mathrm{f}_{\mathrm{i}}=50$ and $\sum_{\mat...
Read More →Electromagnetic waves with wavelength
Question: Electromagnetic waves with wavelength (i) 1 is used in satellite communication (ii) 2 is used to kill germs in water purifies (iii) 3 is used to detect leakage of oil in underground pipelines (iv) 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 Solution: (a) (...
Read More →Professor C.V.Raman surprised his students by suspending
Question: Professor C.V.Raman surprised his students by suspending freely a ting 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. Solution: The radiation pressure property of EM waves was exhibited by Professor CV Raman....
Read More →Find the mean deviation about the median for the following data :
Question: Find the mean deviation about the median for the following data : Solution: The class interval containing $\frac{\mathrm{N}^{\mathrm{th}}}{2}$ or 25th item is 20-30. Therefore, 2030 is the median class. We know that Median $=1+\frac{\frac{\mathrm{N}}{2}-\mathrm{C}}{\mathrm{f}} \times \mathrm{h}$ Here, $I=20, C=13, f=15, h=10$ and $N=50$ Therefore, $\quad$ Median $=20+\frac{\frac{50}{2}-13}{15} \times 10=20+8=28$ Now, We have, $\sum_{\mathrm{i}=1}^{6} \mathrm{f}_{\mathrm{i}}=50_{\text {...
Read More →The magnetic field of a beam emerging from a filter
Question: The magnetic field of a beam emerging from a filter facing a floodlight is given by B0= 12 10-8sin (1.20 107z 3.60 1015t)T. What is the average intensity of the beam? Solution: B0= 12 10-8sin (1.20 107z 3.60 1015t)T B0 = 12 10-8T Iav = 1.71 W/m2...
Read More →A variable frequency a.c source is connected
Question: A variable frequency a.c source is connected to a capacitor. How will the displacement current change with a decrease in frequency? Solution: Capacitive reaction Xc = 1/2fc As there is a decrease in frequency, Xc increases and the conduction current becomes inversely proportional to Xc....
Read More →The charge on a parallel plate capacitor varies as
Question: The charge on a parallel plate capacitor varies as q = q0cos 2vt. The plates are very large and close together. Neglecting the edge effects, find the displacement current through the capacitor? Solution: The displacement current through the capacitor is Id = Ic = dq/dt Given, q = q0cos 2vt Substituting the values, we get Id = Ic = -2vq0 sin 2 vt...
Read More →Why does a microwave oven heats up a food
Question: Why does a microwave oven heats up a food item containing water molecules most efficiently? Solution: The microwave oven heats up a food item containing water molecule most efficiently because the frequency of the microwave and the resonant frequency of the water molecules are the same....
Read More →Why is the orientation of the portable
Question: Why is the orientation of the portable radio with respect to broadcasting station important? Solution: The orientation of the portable radio with respect to the broadcasting station is important because the electromagnetic waves are plane polarised and the antenna should be placed parallel to the vibration of the electric or magnetic field of the wave....
Read More →An EM wave of intensity I falls on a surface kept in vacuum
Question: 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 I/c if the wave is totally absorbed (b) radiation pressure is I/c if the wave is totally reflected (c) radiation pressure is 2I/c if the wave is totally reflected (d) radiation pressure is in the range I/c p 2I/c for real surface Solution: (a) radiation pressure is I/c if the wave is totally absorbed (c) radiation pressure is 2I/c...
Read More →The source of electromagnetic waves can be a charge
Question: The source of electromagnetic waves can be a charge (a) moving with a constant velocity (b) moving in a circular orbit (c) at rest (d) falling in an electric field Solution: (b) moving in a circular orbit (d) falling in an electric field...
Read More →A charged particle oscillates about its mean equilibrium
Question: A charged particle oscillates about its mean equilibrium position with a frequency of 109 Hz. The electromagnetic waves produced: (a) will have a frequency of 109Hz (b) will have a frequency of 2 109Hz (c) will have a wavelength of 0.3 m (d) fall in the region of radiowaves Solution: (a) will have a frequency of 109Hz (c) will have a wavelength of 0.3 m (d) fall in the region of radiowaves...
Read More →Find the mean deviation about the median for the following data :
Question: Find the mean deviation about the median for the following data : Solution: we make the following table from the given data:br data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/02/22/image31372.png" alt="" The class interval containing $\frac{N^{t h}}{2}$ or 25th item is 20-30. Therefore, 2030 is the median class. We know that Median $=1+\frac{\frac{\mathrm{N}}{2}-\mathrm{C}}{\mathrm{f}} \times \mathrm{h}$ Here, l = 20, C = 13, f = 15, h = 10 and N = 50 Therefore, M...
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