21. The ratio of resolving powers of an optical microscope for two wavelengths λ ₁ = 4000 Å and λ ₂ = 6000 Å is

1. 1. 9 : 4

   
   
   

   2. 3 : 2

   
   
   

   3. 16 : 81

   
   
   

   4. 8 : 27

   
   
   

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   Resolving power of a microscope
   

   =	2μ sinθ
   	λ

   
   

   ie., R ∝	1
   	λ

   
   

   or,	R1	=	λ2
   	R2		λ1

   
   Given that the two wavelengths,
   ​λ₁ = 4000 Å
   ​λ₂ = 6000 Å
   

   ∴	R1	=	6000 Å	=	3
   	R2		4000 Å		2

   Correct Option: B

   Resolving power of a microscope
   

   =	2μ sinθ
   	λ

   
   

   ie., R ∝	1
   	λ

   
   

   or,	R1	=	λ2
   	R2		λ1

   
   Given that the two wavelengths,
   ​λ₁ = 4000 Å
   ​λ₂ = 6000 Å
   

   ∴	R1	=	6000 Å	=	3
   	R2		4000 Å		2

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22. In a diffraction pattern due to a single slit of width 'a', the first minimum is observed at an angle 30° when light of wavelength 5000 Å is incident on the slit. The first secondary maximum is observed at an angle of :

1. 1. sin-1		1
      		4

   
   
   

   2. sin-1		2
      		3

   
   
   

   3. sin-1		1
      		2

   
   
   

   4. sin-1		3
      		4

   
   
   

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   For the first minima,
   

   θ =	ηλ
   	a

   
   

   sin30° =	λ	=	1
   	a		2

   
   First secondary maxima will be at
   

   sin θ =	3λ	=	3			1
   	2a		2			2

   
   

   ⇒ θ = sin-1 =		3
   		4

   Correct Option: D

   For the first minima,
   

   θ =	ηλ
   	a

   
   

   sin30° =	λ	=	1
   	a		2

   
   First secondary maxima will be at
   

   sin θ =	3λ	=	3			1
   	2a		2			2

   
   

   ⇒ θ = sin-1 =		3
   		4

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23. For a parallel beam of monochromatic light of wavelength 'λ', diffraction is produced by a single slit whose width 'a' is of the wavelength of the light. If 'D' is the distance of the screen from the slit, the width of the central maxima will be :

1. 1. 	Dλ
      	a

   
   
   

   2. 	Da
      	λ

   
   
   

   3. 	2Da
      	λ

   
   
   

   4. 	2Dλ
      	a

   
   
   

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   Linear width of central maxima y
   

   = D(2q) = 2Dq =	2Dλ
   	a

   
   

   ∴ q =	λ
   	a

   
   

   Correct Option: D

   Linear width of central maxima y
   

   = D(2q) = 2Dq =	2Dλ
   	a

   
   

   ∴ q =	λ
   	a

   
   

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24. At the first minimum adjacent to the central maximum of a single-slit diffraction pattern, the phase difference between the Huygen's wavelet from the edge of the slit and the wavelet from the midpoint of the slit is :

1. 1. 	π	radian
      	2

   
   
   

   2. π radian

   
   
   

   3. 	π	radian
      	8

   
   
   

   4. 	π	radian
      	4

   
   
   

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   For first minima at P
   AP - BP = λ
   

   AP - MP =	λ
   	2

   
   
   So phase difference,
   

   φ =	2π	×	λ	= π radian
   	λ		2

   Correct Option: B

   For first minima at P
   AP - BP = λ
   

   AP - MP =	λ
   	2

   
   
   So phase difference,
   

   φ =	2π	×	λ	= π radian
   	λ		2

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25. A beam of light of λ = 600 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 distance between first dark fringes on either side of the central bright fringe is:

1. 1. 1.2 cm

   
   
   

   2. 1.2 mm

   
   
   

   3. 2.4 cm

   
   
   

   4. 2.4 mm

   
   
   

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   Given: D = 2m; d = 1 mm = 1 × 10^{– 3} m
   ​λ = 600 nm = 600 × 10^{– 6} m ​
   Width of central bright fringe (= 2β)
   

   =	2λD	=	2 × 600 × 10-6 × 2	m
   	d		1 × 10-3

   
   = 2.4 × 10^{– 3} m = 2.4 mm

   Correct Option: D

   Given: D = 2m; d = 1 mm = 1 × 10^{– 3} m
   ​λ = 600 nm = 600 × 10^{– 6} m ​
   Width of central bright fringe (= 2β)
   

   =	2λD	=	2 × 600 × 10-6 × 2	m
   	d		1 × 10-3

   
   = 2.4 × 10^{– 3} m = 2.4 mm

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