Answer the following questions:
Question: Answer the following questions: (a) When a low flying aircraft passes overhead, we sometimes notice a slight shaking of the picture on our TV screen. Suggest apossible explanation. (b) As you have learnt in the text, the principle of linearsuperposition of wave displacement is basic to understandingintensity distributions in diffraction and interference patterns.What is the justification of this principle? Solution: (a) The weak radar signals from the aircraft interfere with the TV sig...
Read More →A parallel beam of light of wavelength 500 nm
Question: A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Find the width of the slit. Solution: Wavelength of the beam of light, = 500 nm Distance between the slit and the screen, D= 1 m Distance of the first minimum from the centre of the screen, x = 2.5 mm = 2.5 x 10-3m First minima, n = 1 Consider the equat...
Read More →Two towers on top of two hills are 40 km apart.
Question: Two towers on top of two hills are 40 km apart. The line joining them passes 50 m above a hill halfway between the towers. What is the longest wavelength of radio waves, which can be sent between the towers without appreciable diffraction effects? Solution: Distance between the towers = 40 km Height of the line joining the hills, d = 50 m The radial spread of the radio waves must not exceed 50 m Aperture a = d = 50 m The hill is located halfway between the towers. Therefore, Fresnels d...
Read More →Answer the following questions:
Question: Answer the following questions: (a) In a single slit diffraction experiment, the width of the slit ismade double the original width. How does this affect the sizeand intensity of the central diffraction band? (b) In what way is diffraction from each slit related to theinterference pattern in a double-slit experiment?(c) When a tiny circular obstacle is placed in the path of light froma distant source, a bright spot is seen at the centre of the shadowof the obstacle. Explain why? (d) Tw...
Read More →In a double-slit experiment using the light of wavelength 600 nm,
Question: In a double-slit experiment using the light of wavelength 600 nm, the angular width of a fringe formed on a distant screen is 0.1. What is the spacing between the two slits? Solution: Wavelength of the light, = 600 nm The angular width of the fringe formed, = 0.10= 0.1/180 Spacing between the slits, d =/ = (600 x 10-9x 180)/(0.1 x 3.14) = 108000 x 10-9/0.314 d =3.44 x 10-4m...
Read More →In a double-slit experiment using the light of wavelength 600 nm,
Question: In a double-slit experiment using the light of wavelength 600 nm, the angular width of a fringe formed on a distant screen is 0.1. What is the spacing between the two slits? Solution: Wavelength of the light, = 600 nm The angular width of the fringe formed, = 0.10= 0.1/180 Spacing between the slits, d =/ = (600 x 10-9x 180)/(0.1 x 3.14) = 108000 x 10-9/0.314 d =3.44 x 10-4m...
Read More →The mean and standard deviations of a group of 100 observations were found to be 20 and 3 respectively.
Question: The mean and standard deviations of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations 21, 12 and 18 were incorrect. Find the mean and standard deviation if the incorrect observations were omitted. Solution: Given that number of observations (n) = 100 Incorrect Mean $(\bar{x})=20$ and Incorrect Standard deviation, $(\sigma)=3$ We know that, $\overline{\mathrm{x}}=\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm...
Read More →For sound waves, the Doppler formula for frequency
Question: For sound waves, the Doppler formula for frequency shift differs slightly between the two situations: (i) source at rest; observer moving, and (ii) source moving; observer at rest. The exact Doppler formulas for the case of light waves in vacuum are, however, strictly identical for these situations. Explain why this should be so. Wouldyou expect the formulas to be strictly identical for the two situations in the case of light travelling in a medium? Solution: The Doppler formuladiffers...
Read More →Let us list some of the factors, which could possibly influence the speed of wave propagation:
Question: Let us list some of the factors, which could possibly influence thespeed of wave propagation: (i) nature of the source. (ii) the direction of propagation. (iii) the motion of the source and/or observer. (iv) wavelength. (v) the intensity of the wave.On which of these factors, if any, does(a) the speed of light in a vacuum, (b) the speed of light in a medium (say, glass or water),depend? Solution: (a) The speed of light in the vacuum does not depend on any of the factors listed. (b) The...
Read More →Explain how Corpuscular theory predicts the speed of light in
Question: Explain how Corpuscular theory predicts the speed of light in a medium, say, water, to be greater than the speed of light in vacuum. Is the prediction confirmed by experimental determination of the speed of light in water? If not, which alternative picture of light is consistent with experiment? Solution: According to Newtons Corpuscular theory, velocity of light in the denser medium (water) is greater than the velocity of light in the rarer medium (vacuum). This was experimentally wro...
Read More →The 6563 Å Hα line emitted by hydrogen
Question: The 6563 Å H line emitted by hydrogen in a star is found to be red-shifted by 15 Å. Estimate the speed with which the star is receding from the Earth. Solution: =6563 Å Δ =15 Å Since the star is receding, the velocity (v) is negative. Δ = v/c v = cΔ/ = (3 x 108) x (15 Å/6563 Å) = 6.86 x 105m/s...
Read More →Estimate the distance for which ray optics
Question: Estimate the distance for which ray optics is a good approximation for an aperture of 4 mm and wavelength 400 nm. Solution: Fresnel's distance $\left(Z_{F}\right)$ is the distance which is used in ray optics for a good approximation. Following is the relation, $Z_{F}=\frac{a^{2}}{\lambda}$ Where, Aperture width, a = 4 mm= 4 10-3m Wavelength of light, $\lambda=400 \mathrm{~nm}=400 \times 10^{-9} \mathrm{~m}$ $Z_{F}=\frac{\left(4 \times 10^{-3}\right)^{2}}{400 \times 10^{-9}}=40 \mathrm{...
Read More →Light of wavelength 5000 Armstrong falls
Question: Light of wavelength 5000 Armstrong falls on a plane reflecting surface. What are the wavelength and frequency of the reflected light? For what angle of incidence is the reflected ray normal to the incident ray? Solution: Wavelength of incident light, $[\lambda]=5000$ Armstrong $=5000 \times 10^{-10} \mathrm{~m}$ Speed of light, $\mathrm{c}=3 \times 10^{8} \mathrm{~m}$ Following is the relation for the frequency of incident light: $\mathrm{v}=\frac{c}{\lambda}=\frac{3 \times 10^{8}}{500...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{x+5}{3 x^{2}+13 x-10} d x$ Solution: $I=\int \frac{x+5}{3 x^{2}+13 x-10} d x$ 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 $3 x^{2}+13 x-10$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(3 \mathrm{x}^{2}+13 \mathrm{x}-10\right)=6 \mathrm{x}+13$ $\therefore$ Let, $x+5=A(6 x+13)+B$ $\Rightarrow x+5=6 A x...
Read More →For a group of 200 candidates, the mean and standard deviations of scores were found to be 40 and 15 respectively.
Question: For a group of 200 candidates, the mean and standard deviations of scores were found to be 40 and 15 respectively. Later on it was discovered that the score of 43 was misread as 34. Find the correct mean and standard deviation. Solution: Given that number of observations (n) = 200 Incorrect Mean $(\overline{\mathrm{x}})=40$ and Incorrect Standard deviation, $(\sigma)=15$ We know that, $\overline{\mathrm{x}}=\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{x}_{\mathrm{i}}$ ...
Read More →What is the Brewster angle for
Question: What is the Brewster angle for air to glass transition? (Refractive index of glass=1.5.) Solution: Refractive index of glass, $\mu=1.5$ Consider Brewster angle $=\theta$' Following is the relation between the Brewster angle and the refractive index: $\tan \theta=\mu$ $\theta=\tan ^{-1}(1.5)=56.31^{\circ}$ Therefore, the Brewster angle for air to glass transition is 56.31...
Read More →In a double-slit experiment,
Question: In a double-slit experiment, 0.2 is found to be the angular width of a fringe on a screen placed 1 m away. The wavelength of light used is 600 nm. What will be the angular width of the fringe if the entire experimental apparatus is immersed in water? Take refractive index of water to be $\frac{4}{3}$. Solution: Distance of the screen from the slits, $D=1 \mathrm{~m}$ Wavelength of light used, $\lambda_{1}=600 \mathrm{~nm}$ Angular width of the fringe in air $\theta_{1}=0.2^{\circ}$ Ang...
Read More →A beam of light consisting of two wavelengths,
Question: A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes in a Youngs double-slit experiment. (a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 650 nm. (b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide? Solution: Wavelength of the light beam, $\lambda_{1}=650 \mathrm{~nm}$ Wavelength of another light beam, $\lambda_{2}=5...
Read More →In Young's double-slit experiment using the monochromatic
Question: In Young's double-slit experiment using the monochromatic light of wavelength $\lambda$, the intensity of light at a point on the screen where path difference is $\lambda$, is $\mathrm{K}$ units. What is the intensity of light at a point where path difference is $\frac{\lambda}{3} ?$ Solution: Let $I_{1}$ and $I_{2}$ be the intensity of the two light waves. Their resultant intensities can be obtained as: $I^{\prime}=I_{1}+I_{2}+2 \sqrt{I_{1} I_{2}} \cos \phi$ Where, $\phi=$ Phase diffe...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{5 x-2}{1+2 x+3 x^{2}} d x$ Solution: $I=\int \frac{5 x-2}{3 x^{2}+2 x+1} d x$ 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 $3 x^{2}+2 x+1$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(3 \mathrm{x}^{2}+2 \mathrm{x}+1\right)=6 \mathrm{x}+2$ $\therefore$ Let, $5 x-2=A(6 x+2)+B$ $\Rightarrow 5 x-2=6 A x+2...
Read More →The mean and standard deviation of 18 observations are found to be 7 and 4 respectively.
Question: The mean and standard deviation of 18 observations are found to be 7 and 4 respectively. On rechecking it was found that an observation 12 was misread as 21. Calculate the correct mean and standard deviation. Solution: Given that number of observations (n) = 18 Incorrect Mean $(\bar{x})=7$ and Incorrect Standard deviation, $(\sigma)=4$ We know that, $\overline{\mathrm{x}}=\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{x}_{\mathrm{i}}$ $\Rightarrow 7=\frac{1}{18} \sum_{i=...
Read More →In Young’s double-slit experiment,
Question: In Youngs double-slit experiment, 0.28mm separation between the slits and the screen is placed 1.4m away. 1.2cm is the distance between the central bright fringe and the fourth bright fringe. Determine the wavelength of light used in the experiment. Solution: Distance between the slits and the screen,D = 1.4 m and the distance between the slits, d = 0.28 mm = 0.28 x 10-3m Distance between the central fringe and the fourth (n = 4) fringe, u = 1.2cm= 1.2 10-2m For constructive interferen...
Read More →(i) The refractive index of glass is 1.5.
Question: (i) The refractive index of glass is 1.5. What is the speed of light in glass? Speed of light in a vacuum is ( 3.0 x 108m s-1) (ii) Is the speed of light in glass Independent of the colour of light? If not, which of the two colours red and violet travels slower in a glass prism? Solution: (i) Refractive Index of glass, $\mu=1.5$ Speed of light, c = 3 108ms-1 The relation for the speed of light in a glass is: $v=\frac{c}{\mu}$ The relation for the speed of light in a glass is: $v=\frac{...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{x+2}{2 x^{2}+6 x+5} d x$ Solution: $I=\int \frac{x+2}{2 x^{2}+6 x+5} d x$ 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 $2 x^{2}+6 x+5$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(2 \mathrm{x}^{2}+6 \mathrm{x}+5\right)=4 \mathrm{x}+6$ $\therefore$ Let, $x+2=A(4 x+6)+B$ $\Rightarrow x+2=4 A x+6 A+B$ On...
Read More →Monochromatic light having a wavelength of 589nm
Question: Monochromatic light having a wavelength of 589nm from the air is incident on a water surface. Find the frequency, wavelength and speed of (i) reflected and (ii) refracted light? [1.33 is the Refractive index of water] Solution: Monochromatic light incident having wavelength, $\lambda=589 \mathrm{~nm}=589 \times 10^{-9} \mathrm{~m}$ Speed of light in air,c = 3 x 108m s-1 Refractive index of water, $\mu=1.33$' (i)In the same medium through which incident ray passed the ray will be reflec...
Read More →