Consider Fraunhoffer diffraction pattern obtained with a single slit at normal incidence. At the angular position of first diffraction minimum, the phase difference between the wavelets from the opposite edges of the slit is :
A single slit of width 0.20 mm is illuminated with light of wavelength 500 nm. The observing screen is placed 80 cm from the slit. The width of the central bright fringe will be :
The first diffraction minimum due to a single slit diffraction is at θ = 30° for a light of wavelength 5000Å. The width of the slit is :
In a diffraction pattern due to a single slit of width \(a\), the first minimum is observed at an angle \(30^\circ\) when light of wavelength \(\lambda\) is incident on the slit. The first secondary maximum is observed at an angle
For first minimum, \(a sin(30^\circ) = \lambda \Rightarrow a = 2\lambda\). For first secondary maximum, \[a sin\theta = \frac{3}{2}\lambda \Rightarrow sin\theta = \frac{3\lambda}{2(2\lambda)} = \frac{3}{4}\].
A beam of light of wavelength \(\lambda = 500 \text{ nm}\) from a distant source falls on a single slit 1 mm wide and the resulting diffraction pattern is observed on a screen 2 m away. The width of central maxima is
The width of the central maximum in single slit diffraction is given by \(w = \frac{2\lambda D}{d}\). Substituting the values: \(w = \frac{2 \times 500 \times 10^{-9} \times 2}{10^{-3}} = 2.0 \times 10^{-3} \text{ m} = 2.0 \text{ mm}\).
A screen is placed 50 cm from a single slit, which is illuminated with 6000 Å light. If distance between the first and third minima in the diffraction patten is 3 mm, the width of the slit is
The position of \(n^{\text{th}}\) minimum in single-slit diffraction is \(y_n = \frac{n\lambda D}{a}\). The distance between the 1st and 3rd minima is \(Delta y = \frac{2\lambda D}{a} ⇒ a = \frac{2\lambda D}{\Delta y} = \frac{2 \times 6000 \times 10^{-10} \times 0.50}{3 \times 10^{-3}} = 0.2\text{ mm}\).
A screen is placed \(50\text{ cm}\) from a single slit, which is illuminated with \(6000\text{ A^0}\) light. If distance between the first and third minima in the diffraction pattern is \(3\text{ mm}\), the width of the slit is
The distance between the first and third minima on the same side of central maximum is \(\Delta y = \frac{2\lambda D}{d}\). Substituting the values: \(3 \times 10^{-3}\text{ m} = \frac{2 \times 6000 \times 10^{-10}\text{ m} \times 0.5\text{ m}}{d}\), we get \(d = 2 \times 10^{-4}\text{ m} = 0.2\text{ mm}\).
Assertion (A): In case of single slit diffraction intensity of higher order maxima decreases.
Reason (R): Higher order maxima are at larger distance.
Assertion (A) is true; the intensity of higher order maxima in single slit diffraction decreases rapidly. Reason (R) is also true; higher order maxima occur at larger distances from the central maximum. However, R is not the correct explanation for A, as the decrease in intensity is due to the smaller effective aperture contributing to these maxima, not their distance.
Assertion (A): On increasing wavelength of light used, resolving power increases.
Reason (R): On increasing wavelength, width of central maxima decreases.
Assertion (A) is false. Resolving power of optical instruments (e.g., telescope, microscope) is inversely proportional to the wavelength (\(RP \propto 1/\lambda\)). So, increasing wavelength decreases resolving power. Reason (R) is also false. In diffraction, the width of the central maximum is directly proportional to the wavelength (\(w \propto \lambda\)). Thus, increasing wavelength increases the width of the central maximum.
Assertion (A): In single slit diffraction arrangement, instead of keeping the screen far away, often a converging lens is placed after the slit and a screen is placed at its focus.
Reason (R): Lens doesn’t introduce any extra path difference for a parallel beam.
Assertion (A) is true. Using a converging lens to focus the diffraction pattern at its focal plane is standard for Fraunhofer diffraction, simulating far-field conditions. Reason (R) is false. A lens works by introducing varying optical path lengths across its aperture to achieve focusing, thus creating path differences.