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Page No 379:

Question 1:

Answer:

White light is a composition of seven colours. These are violet, indigo, blue, green, yellow, orange and red, collectively known as VIBGYOR. Spectrum of white light consists of  sever colour bands. Each band consists of some range of wavelengths or frequencies. 
For orange colour : (590 nm to 620 nm)
For red colour: (620 nm to 780 nm)

So, the colour of 620 nm and 780 nm lights may be different. But the colour of 620 nm light and 621 nm light is same.

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Question 2:

White light is a composition of seven colours. These are violet, indigo, blue, green, yellow, orange and red, collectively known as VIBGYOR. Spectrum of white light consists of  sever colour bands. Each band consists of some range of wavelengths or frequencies. 
For orange colour : (590 nm to 620 nm)
For red colour: (620 nm to 780 nm)

So, the colour of 620 nm and 780 nm lights may be different. But the colour of 620 nm light and 621 nm light is same.

Answer:

Colour of light will depend only on the frequency of light and not on the wavelength of the light. So, light will appear red to an observer under water. 

Page No 379:

Question 3:

Colour of light will depend only on the frequency of light and not on the wavelength of the light. So, light will appear red to an observer under water. 

Answer:

The width of the central band is inversely proportional to the slit width. So, as the width of the slit is increased, the central band will become less wider and further bands will start merging in them. Hence, diffraction effects will be visible less clearly.

Page No 379:

Question 4:

The width of the central band is inversely proportional to the slit width. So, as the width of the slit is increased, the central band will become less wider and further bands will start merging in them. Hence, diffraction effects will be visible less clearly.

Answer:

Light waves have the property of travelling in a straight line, unlike sound waves. When we put a cardboard between the light source and our eyes, the light waves are obstructed by the cardboard and cannot reach our eyes, which doesn't happen when the cardboard is inserted between sound source and our ear.

Page No 379:

Question 5:

Light waves have the property of travelling in a straight line, unlike sound waves. When we put a cardboard between the light source and our eyes, the light waves are obstructed by the cardboard and cannot reach our eyes, which doesn't happen when the cardboard is inserted between sound source and our ear.

Answer:

To receive TV signals transmitted from Delhi in Patna directly, one has to use antennas of great height, which will cost much. On the other hand, transmission of signals with the help of satellites requires only high frequency waves and can be done easily.

Page No 379:

Question 6:

To receive TV signals transmitted from Delhi in Patna directly, one has to use antennas of great height, which will cost much. On the other hand, transmission of signals with the help of satellites requires only high frequency waves and can be done easily.

Answer:

Young's double slit experiment can be performed with sound waves, as the sound waves also show interference pattern. To get a reasonable "fringe pattern", the separation of the slits should be of the order of the wavelength of the sound waves used.
In this experiment, the bright and dark fringes can be detected by measuring the intensity of sound using a microphone or any other detector.

Page No 379:

Question 7:

Young's double slit experiment can be performed with sound waves, as the sound waves also show interference pattern. To get a reasonable "fringe pattern", the separation of the slits should be of the order of the wavelength of the sound waves used.
In this experiment, the bright and dark fringes can be detected by measuring the intensity of sound using a microphone or any other detector.

Answer:

Interference pattern can be studied with waves of unequal intensity.
 Ι=Ι1+Ι2+2Ι1×Ι2cosϕ ,
where ϕ=phase difference.
In case of waves of unequal intensities, the contrast will not be clear, as the minima will not be completely dark.

Page No 379:

Question 8:

Interference pattern can be studied with waves of unequal intensity.
 Ι=Ι1+Ι2+2Ι1×Ι2cosϕ ,
where ϕ=phase difference.
In case of waves of unequal intensities, the contrast will not be clear, as the minima will not be completely dark.

Answer:

The interference pattern can be produced by any two coherent waves moving in the same direction. It cannot be concluded from the interference phenomenon that light is a transverse wave, as sound waves that are longitudinal in nature also interfere.

Page No 379:

Question 9:

The interference pattern can be produced by any two coherent waves moving in the same direction. It cannot be concluded from the interference phenomenon that light is a transverse wave, as sound waves that are longitudinal in nature also interfere.

Answer:

In order to get interference, the sources should be coherent, i.e. they should emit wave of the same frequency and a stable phase difference. Two candles that are placed close to each other are distinct and cannot be considered as coherent sources. Two independent sources cannot be coherent. So, two different laser sources will also not serve the purpose. 

Page No 379:

Question 10:

In order to get interference, the sources should be coherent, i.e. they should emit wave of the same frequency and a stable phase difference. Two candles that are placed close to each other are distinct and cannot be considered as coherent sources. Two independent sources cannot be coherent. So, two different laser sources will also not serve the purpose. 

Answer:

The fringe width in Young's double slit experiment depends on the separation of the slits.
χ=λDd,
where
           λ=wavelengthχ=fringe widthD=distance between slits and screend=separation between slits
On increasing d, fringe width decreases. If the separation is increased too much, the fringes will merge with each other and the fringe pattern won't be detectable.

Page No 379:

Question 11:

The fringe width in Young's double slit experiment depends on the separation of the slits.
χ=λDd,
where
           λ=wavelengthχ=fringe widthD=distance between slits and screend=separation between slits
On increasing d, fringe width decreases. If the separation is increased too much, the fringes will merge with each other and the fringe pattern won't be detectable.

Answer:

The violet filter will allow only violet light to pass through it. Now, if the double slit experiment is performed with the white light and violet light, the fringe pattern will not be the same as obtained by just using white light as the source. To have interference pattern, the light waves entering from the slits should be monochromatic. So, in this case, the violet light will superimpose with only violet light (of wavelength 400 nm) in such a way that the bright bands will be of violet colour and the minima will be completely dark.

Page No 379:

Question 1:

The violet filter will allow only violet light to pass through it. Now, if the double slit experiment is performed with the white light and violet light, the fringe pattern will not be the same as obtained by just using white light as the source. To have interference pattern, the light waves entering from the slits should be monochromatic. So, in this case, the violet light will superimpose with only violet light (of wavelength 400 nm) in such a way that the bright bands will be of violet colour and the minima will be completely dark.

Answer:

(c) both a particle and a wave phenomenon
       
Light shows photoelectric effect and Compton effect, which depicts its particle nature. It also shows interference and diffraction, which depicts the wave nature of light. 

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Question 2:

(c) both a particle and a wave phenomenon
       
Light shows photoelectric effect and Compton effect, which depicts its particle nature. It also shows interference and diffraction, which depicts the wave nature of light. 

Answer:

(d) neither on elasticity nor on inertia

The speed of light in any medium depends on the refractive index of that medium, which is an intensive property. Hence, speed of light is not affected by the elasticity and inertia of the medium.

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Question 3:

(d) neither on elasticity nor on inertia

The speed of light in any medium depends on the refractive index of that medium, which is an intensive property. Hence, speed of light is not affected by the elasticity and inertia of the medium.

Answer:

(d) electric field

Light consists of mutually perpendicular electric and magnetic fields. So, the equation of a light wave is represented by its field vector.

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Question 4:

(d) electric field

Light consists of mutually perpendicular electric and magnetic fields. So, the equation of a light wave is represented by its field vector.

Answer:

(d) Polarization 

Reflection, interference and diffraction are the phenomena shown by both transverse waves and longitudinal waves. Polarization is the phenomenon shown only by transverse waves.

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Question 5:

(d) Polarization 

Reflection, interference and diffraction are the phenomena shown by both transverse waves and longitudinal waves. Polarization is the phenomenon shown only by transverse waves.

Answer:

(c) its wavelength decreases but frequency remains unchanged

Frequency of a light wave, as it travels from one medium to another, always remains unchanged, while wavelength decreases.

Decrease in the wavelength of light entering a medium of refractive index μ is given by
 λΜ=λμ,where  λΜ=wavelength in medium             λ=wavelength in vacuum             μ=refractive index

Page No 379:

Question 6:

(c) its wavelength decreases but frequency remains unchanged

Frequency of a light wave, as it travels from one medium to another, always remains unchanged, while wavelength decreases.

Decrease in the wavelength of light entering a medium of refractive index μ is given by
 λΜ=λμ,where  λΜ=wavelength in medium             λ=wavelength in vacuum             μ=refractive index

Answer:

(b) Frequency

Frequency of a light wave doesn't change on changing the medium of propagation of light. 

Page No 379:

Question 7:

(b) Frequency

Frequency of a light wave doesn't change on changing the medium of propagation of light. 

Answer:

(a) λa>λf

An electromagnetic wave bends round the corners of an obstacle if the size of the obstacle is comparable to the wavelength of the wave. An AM wave has less frequency than an FM wave. So, an AM wave has a higher wavelength than an FM wave and it bends round the corners of a 1 m × 1m board.
λ

Page No 379:

Question 8:

(a) λa>λf

An electromagnetic wave bends round the corners of an obstacle if the size of the obstacle is comparable to the wavelength of the wave. An AM wave has less frequency than an FM wave. So, an AM wave has a higher wavelength than an FM wave and it bends round the corners of a 1 m × 1m board.
λ

Answer:

(d) A laser

Among the given sources, laser is the best coherent source providing monochromatic light with constant phase difference.

Page No 379:

Question 9:

(d) A laser

Among the given sources, laser is the best coherent source providing monochromatic light with constant phase difference.

Answer:

(d) cos-11/3
On writing the given equation in the plane equation form lx + my + nz = p,
where l2 + m2 + n2 = 1 and p>0, we get:

13x+13y+13z=c3
If θ is the angle between the normal and +X axis, then 
cosθ=13θ=cos-113

Page No 379:

Question 10:

(d) cos-11/3
On writing the given equation in the plane equation form lx + my + nz = p,
where l2 + m2 + n2 = 1 and p>0, we get:

13x+13y+13z=c3
If θ is the angle between the normal and +X axis, then 
cosθ=13θ=cos-113

Answer:

(a) plane

Wave travelling from a distant source always has plane wavefront.

Page No 379:

Question 11:

(a) plane

Wave travelling from a distant source always has plane wavefront.

Answer:

(a) point source

Intensity of a point source obeys the inverse square law.
Intensity of light at distance r from the point source is given by
I=S/4πr2 ,
where S is the source strength.

Page No 379:

Question 12:

(a) point source

Intensity of a point source obeys the inverse square law.
Intensity of light at distance r from the point source is given by
I=S/4πr2 ,
where S is the source strength.

Answer:

(d) having a constant phase difference
For light waves emitted by two sources of light to remain coherent, the initial phase difference between waves should remain constant in time. If the phase difference changes continuously or randomly with time, then the sources are incoherent.  

Page No 379:

Question 13:

(d) having a constant phase difference
For light waves emitted by two sources of light to remain coherent, the initial phase difference between waves should remain constant in time. If the phase difference changes continuously or randomly with time, then the sources are incoherent.  

Answer:

(d) interference of light 

Interference effect is produced by a thin film ( coating of a thin layer of a translucent material on a medium of different refractive index which allows light to pass through it))In the present case, oil floating on water forms a thin film on the surface of water, leading to the display of beautiful colours in daylight because of the interference of sunlight.



Page No 380:

Question 1:

(d) interference of light 

Interference effect is produced by a thin film ( coating of a thin layer of a translucent material on a medium of different refractive index which allows light to pass through it))In the present case, oil floating on water forms a thin film on the surface of water, leading to the display of beautiful colours in daylight because of the interference of sunlight.

Answer:

Given:
Range of wave length is
400 nm<λ<700 nm
We know that frequency is given by f=cλ,
where c=speed of light=3×108 m/s
          f is the frequency
          λ is the wavelength
We can write wavelength as:
1700 nm<1λ<1400 nm17×10-7 m<1λ<14×10-7 m3×1087×10-7 Hz<cλ<3×1084×10-7 Hz4.3×1014 Hz<cλ<7.5×1014 Hz4.3×1014 Hz<f<7.5×1014 Hz

Hence, frequency of the range of light that is visible to an average human being is 4.3×1014 Hz to7.5×1014 Hz.

Page No 380:

Question 2:

Given:
Range of wave length is
400 nm<λ<700 nm
We know that frequency is given by f=cλ,
where c=speed of light=3×108 m/s
          f is the frequency
          λ is the wavelength
We can write wavelength as:
1700 nm<1λ<1400 nm17×10-7 m<1λ<14×10-7 m3×1087×10-7 Hz<cλ<3×1084×10-7 Hz4.3×1014 Hz<cλ<7.5×1014 Hz4.3×1014 Hz<f<7.5×1014 Hz

Hence, frequency of the range of light that is visible to an average human being is 4.3×1014 Hz to7.5×1014 Hz.

Answer:

Given:
Wavelength of sodium light in air, λa=589 nm=589×10-9 m
Refractive index of water, μw= 1⋅33
We know that f=cλ,
where c=speed of light=3×108 m/s
          f  = frequency
          λ = wavelength
(a) Frequency in air, fair=cλa
fair=3×108589×10-9      =5.09×1014 Hz

(b)
Let wavelength of sodium light in water be λw.
We know that
μaμw=λωλa,
where μa is the refractive index of air which is equal to 1 and
           λw is the wavelength of sodium light in water.
11.33=λω589×10-9      λω=443 nm

(c) Frequency of light does not change when light travels from one medium to another.
      fω=fa               =5.09×1014 Hz

(d) Let the speed of sodium light in water be νω
     and speed in air, va = c.
     Using μaμω=νωνa, we get:
          νω=μacμω                    =3×1081.33=2.25×108 m/s

Page No 380:

Question 3:

Given:
Wavelength of sodium light in air, λa=589 nm=589×10-9 m
Refractive index of water, μw= 1⋅33
We know that f=cλ,
where c=speed of light=3×108 m/s
          f  = frequency
          λ = wavelength
(a) Frequency in air, fair=cλa
fair=3×108589×10-9      =5.09×1014 Hz

(b)
Let wavelength of sodium light in water be λw.
We know that
μaμw=λωλa,
where μa is the refractive index of air which is equal to 1 and
           λw is the wavelength of sodium light in water.
11.33=λω589×10-9      λω=443 nm

(c) Frequency of light does not change when light travels from one medium to another.
      fω=fa               =5.09×1014 Hz

(d) Let the speed of sodium light in water be νω
     and speed in air, va = c.
     Using μaμω=νωνa, we get:
          νω=μacμω                    =3×1081.33=2.25×108 m/s

Answer:

Given:
Refractive index of fused quartz for light of wavelength 400 nm is 1.472.
And refractive index of fused quartz for light of wavelength 760 nm is 1.452.
We known that refractive index of a material is given by
μ = Speed of light in vacuumSpeed of light in the material=cv
Let speed of light for wavelength 400 nm in quartz be v400.
So,
  1.472=3×108v400v400=2.04×108 m/s           
Let speed of light of wavelength 760 nm in quartz be v760.
Again, 1.4521=3×108ν760
v760=2.07×108 m/s

Page No 380:

Question 4:

Given:
Refractive index of fused quartz for light of wavelength 400 nm is 1.472.
And refractive index of fused quartz for light of wavelength 760 nm is 1.452.
We known that refractive index of a material is given by
μ = Speed of light in vacuumSpeed of light in the material=cv
Let speed of light for wavelength 400 nm in quartz be v400.
So,
  1.472=3×108v400v400=2.04×108 m/s           
Let speed of light of wavelength 760 nm in quartz be v760.
Again, 1.4521=3×108ν760
v760=2.07×108 m/s

Answer:

Given:
Speed of yellow light in liquid (vL)= 2⋅4 × 108 m s−1
And  speed of yellow light in air speed = va 
Let μL be the refractive index of the liquid
Using, μL=speed of light in vaccumvelocity of light in the given medium=cvL
μL=3×1082.4×108=1.25

Hence, the required refractive index is 1.25.

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Question 5:

Given:
Speed of yellow light in liquid (vL)= 2⋅4 × 108 m s−1
And  speed of yellow light in air speed = va 
Let μL be the refractive index of the liquid
Using, μL=speed of light in vaccumvelocity of light in the given medium=cvL
μL=3×1082.4×108=1.25

Hence, the required refractive index is 1.25.

Answer:

Given:
Separation between two narrow slits, d = 1 cm = 10−2 m
Wavelength of the light, λ=5×10-7 m
Distance of the screen, D=1 m
(a)
    We know that separation between two consecutive maxima =  fringe width (β).
That is, β=λDd    ...(i)
         =5×10-7×110-2 m =5×10-5 m=0.05 mm

(b)
Separation between two consecutive maxima = fringe width
β=1 mm=10-3 m
Let the separation between the sources be 'd'
Using equation (i), we get:
d'=5×10-7×110-3 d'=5×10-4 m=0.50 mm.

Page No 380:

Question 14:

Given:
Separation between two narrow slits, d = 1 cm = 10−2 m
Wavelength of the light, λ=5×10-7 m
Distance of the screen, D=1 m
(a)
    We know that separation between two consecutive maxima =  fringe width (β).
That is, β=λDd    ...(i)
         =5×10-7×110-2 m =5×10-5 m=0.05 mm

(b)
Separation between two consecutive maxima = fringe width
β=1 mm=10-3 m
Let the separation between the sources be 'd'
Using equation (i), we get:
d'=5×10-7×110-3 d'=5×10-4 m=0.50 mm.

Answer:

(c) 9 : 4

Ratio of maximum intensity and minimum intensity is given by
ImaxImin=I1 +I22I1-I22=251I1=3 and I2=2I1=9 and I2=4

Then,
I1I2=94

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Question 15:

(c) 9 : 4

Ratio of maximum intensity and minimum intensity is given by
ImaxImin=I1 +I22I1-I22=251I1=3 and I2=2I1=9 and I2=4

Then,
I1I2=94

Answer:

(b) I0/​4

Total intensity coming from the source is Iwhich is present at the central maxima. In case of two slits, the intensity is getting distributed between the two slits and for a single slit, the amplitude of light coming from the slit is reduced to half  which leads to 1/4th of intensity. 

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Question 16:

(b) I0/​4

Total intensity coming from the source is Iwhich is present at the central maxima. In case of two slits, the intensity is getting distributed between the two slits and for a single slit, the amplitude of light coming from the slit is reduced to half  which leads to 1/4th of intensity. 

Answer:

(c) remain same

On the introduction of a transparent sheet in front of one of the slits, the fringe pattern will shift slightly but the width will remain the same.

Page No 380:

Question 17:

(c) remain same

On the introduction of a transparent sheet in front of one of the slits, the fringe pattern will shift slightly but the width will remain the same.

Answer:

(a) the fringe width will decrease

As fringe width is proportional to the wavelength and wavelength of light is inversely proportional to the refractive index of the medium,
Here,   λΜ=λ/ηλΜ=wavelength in mediumλ=wavelength in vacuumη=refractive index of medium
Hence, fringe width decreases when Young's double slit experiment is performed under water.

Page No 380:

Question 1:

(a) the fringe width will decrease

As fringe width is proportional to the wavelength and wavelength of light is inversely proportional to the refractive index of the medium,
Here,   λΜ=λ/ηλΜ=wavelength in mediumλ=wavelength in vacuumη=refractive index of medium
Hence, fringe width decreases when Young's double slit experiment is performed under water.

Answer:

(a) in vacuum
(c) in a material medium

Light is an electromagnetic wave that can travel through vacuum or any optical medium.

Page No 380:

Question 2:

(a) in vacuum
(c) in a material medium

Light is an electromagnetic wave that can travel through vacuum or any optical medium.

Answer:

(b) Speed of light in water is smaller than its speed in vacuum.
(c) Light shows interference. 

Snell's Law, which states that the speed of light reduces on moving from a rarer to a denser medium, can be concluded from Huygens' wave theory and interference of light waves is based on the wave properties of light.

Page No 380:

Question 3:

(b) Speed of light in water is smaller than its speed in vacuum.
(c) Light shows interference. 

Snell's Law, which states that the speed of light reduces on moving from a rarer to a denser medium, can be concluded from Huygens' wave theory and interference of light waves is based on the wave properties of light.

Answer:

(b) have zero average value
(c) are perpendicular to the direction of propagation of light
(d) are mutually perpendicular

Light is an electromagnetic wave that propagates through its electric and magnetic field vectors, which are mutually perpendicular to each other, as well as to the direction of propagation of light. The average value of both the fields is zero.

Page No 380:

Question 4:

(b) have zero average value
(c) are perpendicular to the direction of propagation of light
(d) are mutually perpendicular

Light is an electromagnetic wave that propagates through its electric and magnetic field vectors, which are mutually perpendicular to each other, as well as to the direction of propagation of light. The average value of both the fields is zero.

Answer:

(c) find the new position of a wavefront
(d) explain Snell's Law

Huygen's wave theory explains the origin of points for the new wavefront proceeding successively. It also explains the variation in speed of light on moving from one medium to another, i.e. it proves Snell's Law.

Page No 380:

Question 5:

(c) find the new position of a wavefront
(d) explain Snell's Law

Huygen's wave theory explains the origin of points for the new wavefront proceeding successively. It also explains the variation in speed of light on moving from one medium to another, i.e. it proves Snell's Law.

Answer:

(c) vAvB = vC  
(d) νB=12vA+vC

 
Since the speed of light is a universal constant, vAvB =vC= 3×108 m/s.
νB=12vA+vC. This expression also implies that vAvB =vC.
1
νs=12

Page No 380:

Question 6:

(c) vAvB = vC  
(d) νB=12vA+vC

 
Since the speed of light is a universal constant, vAvB =vC= 3×108 m/s.
νB=12vA+vC. This expression also implies that vAvB =vC.
1
νs=12

Answer:

 (a) vv> vC
(d) vB= (vA + vC )/2 

In any other medium, the speed of light is given by v=c/η, where η is the refractive index of the medium and according to Doppler effect,  for an observer moving towards the source ,speed of light appears to be  more than the other two cases. On the other hand, it will be least when the observer is moving away from the source. 

Page No 380:

Question 7:

 (a) vv> vC
(d) vB= (vA + vC )/2 

In any other medium, the speed of light is given by v=c/η, where η is the refractive index of the medium and according to Doppler effect,  for an observer moving towards the source ,speed of light appears to be  more than the other two cases. On the other hand, it will be least when the observer is moving away from the source. 

Answer:

(a) x = c

​The wave is travelling along the X-axis. So, it'll have planar wavefront perpendicular to the X-axis.

Page No 380:

Question 8:

(a) x = c

​The wave is travelling along the X-axis. So, it'll have planar wavefront perpendicular to the X-axis.

Answer:

(b) consecutive fringes will come closer

Fringe width,  β=λD/d.
Wavelength of red light is greater than wavelength of violet light; so, the fringe width will reduce.

Page No 380:

Question 9:

(b) consecutive fringes will come closer

Fringe width,  β=λD/d.
Wavelength of red light is greater than wavelength of violet light; so, the fringe width will reduce.

Answer:

(a) The central fringe will be white.
(b) There will not be a completely dark fringe.
(d) The fringe next to the central will be violet.

The superposition of all the colours at the central maxima gives the central band a white colour. As we go from the centre to corner, the fringe colour goes from violet to red. There will not be a completely dark fringe, as complete destructive interference does not take place.

Page No 380:

Question 10:

(a) The central fringe will be white.
(b) There will not be a completely dark fringe.
(d) The fringe next to the central will be violet.

The superposition of all the colours at the central maxima gives the central band a white colour. As we go from the centre to corner, the fringe colour goes from violet to red. There will not be a completely dark fringe, as complete destructive interference does not take place.

Answer:

(a) (i) and (ii)
(d) (iii) and (iv).

The waves are travelling with the same frequencies and varying by constant phase difference. 



Page No 381:

Question 6:

(a) (i) and (ii)
(d) (iii) and (iv).

The waves are travelling with the same frequencies and varying by constant phase difference. 

Answer:

Given:
Separation between consecutive dark fringes = fringe width (β) = 1 mm = 10−3 m
Distance between screen and slit (D) = 2.5 m
The separation between slits (d) = 1 mm = 10−3 m
Let the wavelength of the light used in experiment be λ.
We know that
β=λDd
 10-3 m=2.5×λ10-3λ=12.5 10-6 m      =4×10-7 m=400 nm

Hence, the wavelength of light used for the experiment is 400 nm.

Page No 381:

Question 7:

Given:
Separation between consecutive dark fringes = fringe width (β) = 1 mm = 10−3 m
Distance between screen and slit (D) = 2.5 m
The separation between slits (d) = 1 mm = 10−3 m
Let the wavelength of the light used in experiment be λ.
We know that
β=λDd
 10-3 m=2.5×λ10-3λ=12.5 10-6 m      =4×10-7 m=400 nm

Hence, the wavelength of light used for the experiment is 400 nm.

Answer:

Given:
Separation between the two slits, d=1 mm=10-3 m
Wavelength of the light used, λ=5.0×10-7 m
Distance between screen and slit, D=1 m
(a) The distance of the centre of the first minimum from the centre of the central maximum, x = width of central maxima2
That is, x=β2=λD2d    ...(i)
         =5×10-7×12×10-3 =2.5×10-4 m=0.25 mm
(b) From equation (i),
fringe width, β=2×x=0.50 mm
So, number of bright fringes formed in one centimetre (10 mm) = 100.50=20.

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Question 8:

Given:
Separation between the two slits, d=1 mm=10-3 m
Wavelength of the light used, λ=5.0×10-7 m
Distance between screen and slit, D=1 m
(a) The distance of the centre of the first minimum from the centre of the central maximum, x = width of central maxima2
That is, x=β2=λD2d    ...(i)
         =5×10-7×12×10-3 =2.5×10-4 m=0.25 mm
(b) From equation (i),
fringe width, β=2×x=0.50 mm
So, number of bright fringes formed in one centimetre (10 mm) = 100.50=20.

Answer:

Given:
Separation between two narrow slits, d=0.8 mm=0.8×10-3 m
Wavelength of the yellow light, λ=589 nm=589×10-9 m
Distance between screen and slit, D=2.0 m
Separation between the adjacent bright bands = width of one dark fringe
That is, β=λDd    ...(i)
β=589×10-9×20.8×10-3      =1.47×10-3 m      =1.47 mm.

Hence, the adjacent bright bands in the interference pattern are 1.47 mm apart.  

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Question 9:

Given:
Separation between two narrow slits, d=0.8 mm=0.8×10-3 m
Wavelength of the yellow light, λ=589 nm=589×10-9 m
Distance between screen and slit, D=2.0 m
Separation between the adjacent bright bands = width of one dark fringe
That is, β=λDd    ...(i)
β=589×10-9×20.8×10-3      =1.47×10-3 m      =1.47 mm.

Hence, the adjacent bright bands in the interference pattern are 1.47 mm apart.  

Answer:

Given:
Wavelength of the blue-green light, λ=500×10-9 m
Separation between two slits, d=2×10-3 m,
Let angular separation between the consecutive bright fringes be θ.
Using θ= βD=λDdD=λd, we get:     θ=500×10-92×10-3       =250×10-6       =25×10-5 radian or 0.014°
Hence, the angular separation between the consecutive bright fringes is 0.014 degree. 

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Question 10:

Given:
Wavelength of the blue-green light, λ=500×10-9 m
Separation between two slits, d=2×10-3 m,
Let angular separation between the consecutive bright fringes be θ.
Using θ= βD=λDdD=λd, we get:     θ=500×10-92×10-3       =250×10-6       =25×10-5 radian or 0.014°
Hence, the angular separation between the consecutive bright fringes is 0.014 degree. 

Answer:

Given:
Wavelengths of the source of light,
λ1=480×10-9 m and  λ2=600×10-9 m
Separation between the slits, d=0.25 mm=0.25×10-3 m
Distance between screen and slit, D=150 cm=1.5 m
We know that the position of the first maximum is given by
 y=λDd
So, the linear separation between the first maximum (next to the central maximum) corresponding to the two wavelengths = y2 y1
y2-y1=Dy2-y1d

y2-y1=1.50.25×10-3600×10-9-480×10-9     y2-y1=72×10-5 m =0.72 mm      

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Question 11:

Given:
Wavelengths of the source of light,
λ1=480×10-9 m and  λ2=600×10-9 m
Separation between the slits, d=0.25 mm=0.25×10-3 m
Distance between screen and slit, D=150 cm=1.5 m
We know that the position of the first maximum is given by
 y=λDd
So, the linear separation between the first maximum (next to the central maximum) corresponding to the two wavelengths = y2 y1
y2-y1=Dy2-y1d

y2-y1=1.50.25×10-3600×10-9-480×10-9     y2-y1=72×10-5 m =0.72 mm      

Answer:

Let the separation between the slits be d and distance between screen from the slits be D.
Suppose, the mth bright fringe of violet light overlaps with the nth bright fringe of red light.
Now, the position of the mth bright fringe of violet light, yv = mλvDd
Position of the nth bright fringe of red light, yr = nλrDd
For overlapping, yv =yr .
So, as per the question,
m×400×Dd=n×700×Ddmn=74
Therefore, the 7th bright fringe of violet light overlaps with the 4th bright fringe of red light.
It can also be seen that the 14th violet fringe will overlap with the 8th red fringe.
Because, mn=74=148

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Question 12:

Let the separation between the slits be d and distance between screen from the slits be D.
Suppose, the mth bright fringe of violet light overlaps with the nth bright fringe of red light.
Now, the position of the mth bright fringe of violet light, yv = mλvDd
Position of the nth bright fringe of red light, yr = nλrDd
For overlapping, yv =yr .
So, as per the question,
m×400×Dd=n×700×Ddmn=74
Therefore, the 7th bright fringe of violet light overlaps with the 4th bright fringe of red light.
It can also be seen that the 14th violet fringe will overlap with the 8th red fringe.
Because, mn=74=148

Answer:

Given:
The refractive index of the plate is μ.
Let the thickness of the plate be 't' to produce a change in the optical path difference of λ2.
We know that optical path difference is given by μ-1t.
 μ-1t=λ2t=λ2μ-1
Hence, the thickness of a plate is λ2μ-1.

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Question 13:

Given:
The refractive index of the plate is μ.
Let the thickness of the plate be 't' to produce a change in the optical path difference of λ2.
We know that optical path difference is given by μ-1t.
 μ-1t=λ2t=λ2μ-1
Hence, the thickness of a plate is λ2μ-1.

Answer:

Given:
Refractive index of the plate is μ.
The thickness of the plate is t.
Wavelength of the light is λ.
(a)
When the plate is placed in front of the slit, then the optical path difference is given by μ-1t.

(b) For zero intensity at the centre of the fringe pattern, there should be distractive interference at the centre.
So, the optical path difference should be = λ2
i.e. μ-1 t=λ2t=λ2 μ-1

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Question 14:

Given:
Refractive index of the plate is μ.
The thickness of the plate is t.
Wavelength of the light is λ.
(a)
When the plate is placed in front of the slit, then the optical path difference is given by μ-1t.

(b) For zero intensity at the centre of the fringe pattern, there should be distractive interference at the centre.
So, the optical path difference should be = λ2
i.e. μ-1 t=λ2t=λ2 μ-1

Answer:

Given:
Refractive index of the paper, μ = 1.45
The thickness of the plate, t=0.02 mm=0.02×10-3 m
Wavelength of the light, λ=620 nm=620×10-9 m
We know that when we paste a transparent paper in front of one of the slits, then the optical path changes by μ-1t.
And optical path should be changed by λ for the shift of one fringe.
∴ Number of fringes crossing through the centre is
n=μ-1tλ  =1.45-1×0.02×10-3620×10-9  =14.5

Hence, 14.5 fringes will cross through the centre if the paper is removed.

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Question 15:

Given:
Refractive index of the paper, μ = 1.45
The thickness of the plate, t=0.02 mm=0.02×10-3 m
Wavelength of the light, λ=620 nm=620×10-9 m
We know that when we paste a transparent paper in front of one of the slits, then the optical path changes by μ-1t.
And optical path should be changed by λ for the shift of one fringe.
∴ Number of fringes crossing through the centre is
n=μ-1tλ  =1.45-1×0.02×10-3620×10-9  =14.5

Hence, 14.5 fringes will cross through the centre if the paper is removed.

Answer:

Given:
Refractive index of the mica sheet,μ = 1.6
Thickness of the plate, t=1.964 micron=1.964×10-6 m
Let the wavelength of the light used = λ.
Number of fringes shifted is given by
n=μ-1tλ
So, the corresponding shift in the fringe width equals the number of fringes multiplied by the width of one fringe.
                                         Shift =n × β=μ-1tλ×λDd=μ-1t×Dd     ...(i)
As per the question, when the distance between the screen and the slits is doubled,
i.e. D'=2D,
fringe width, β=λD'd=λ2Dd
According to the question, fringe shift in first case = fringe width in second case.
So, μ-1t×Dd=λ2Ddλ=μ-1 t2      =1.6-1×1.964×10-62      =589.2×10-9=589.2 nm

Hence, the required wavelength of the monochromatic light is 589.2 nm.

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Question 16:

Given:
Refractive index of the mica sheet,μ = 1.6
Thickness of the plate, t=1.964 micron=1.964×10-6 m
Let the wavelength of the light used = λ.
Number of fringes shifted is given by
n=μ-1tλ
So, the corresponding shift in the fringe width equals the number of fringes multiplied by the width of one fringe.
                                         Shift =n × β=μ-1tλ×λDd=μ-1t×Dd     ...(i)
As per the question, when the distance between the screen and the slits is doubled,
i.e. D'=2D,
fringe width, β=λD'd=λ2Dd
According to the question, fringe shift in first case = fringe width in second case.
So, μ-1t×Dd=λ2Ddλ=μ-1 t2      =1.6-1×1.964×10-62      =589.2×10-9=589.2 nm

Hence, the required wavelength of the monochromatic light is 589.2 nm.

Answer:

Given:
The thickness of the strips = t1=t2=t=0.5 mm=0.5×10-3 m
Separation between the two slits, d=0.12 cm=12×10-4 m
The refractive index of mica, μm = 1.58 and of polystyrene, μp = 1.58
Wavelength of the light, λ=590 nm=590×10-9 m,
Distance between screen and slit, D = 1 m

(a)
We know that fringe width is given by
     β=λDd 
 β =590×10-9×112×10-4        =4.9×10-4 m

(b) When both the mica and polystyrene strips are fitted before the slits, the optical path changes by
x=μm-1 t- μp-1 t      =μm-μp t      =1.58-1.55×0.5 10-3      =0.015×10-3 m
∴ Number of fringes shifted, n = xλ.
n=0.015×10-3590×10-9=25.43
∴ 25 fringes and 0.43th of a fringe.
⇒ In which 13 bright fringes and 12 dark fringes and 0.43th of a dark fringe.
So, position of first maximum on both sides is given by
On one side,
x=0.43×4.91×10-4   β=4.91×10-4 m      =0.021 cm           
On the other side,
x'=1-0.43×4.91×10-4   =0.028 cm

Page No 381:

Question 17:

Given:
The thickness of the strips = t1=t2=t=0.5 mm=0.5×10-3 m
Separation between the two slits, d=0.12 cm=12×10-4 m
The refractive index of mica, μm = 1.58 and of polystyrene, μp = 1.58
Wavelength of the light, λ=590 nm=590×10-9 m,
Distance between screen and slit, D = 1 m

(a)
We know that fringe width is given by
     β=λDd 
 β =590×10-9×112×10-4        =4.9×10-4 m

(b) When both the mica and polystyrene strips are fitted before the slits, the optical path changes by
x=μm-1 t- μp-1 t      =μm-μp t      =1.58-1.55×0.5 10-3      =0.015×10-3 m
∴ Number of fringes shifted, n = xλ.
n=0.015×10-3590×10-9=25.43
∴ 25 fringes and 0.43th of a fringe.
⇒ In which 13 bright fringes and 12 dark fringes and 0.43th of a dark fringe.
So, position of first maximum on both sides is given by
On one side,
x=0.43×4.91×10-4   β=4.91×10-4 m      =0.021 cm           
On the other side,
x'=1-0.43×4.91×10-4   =0.028 cm

Answer:

Given:
Refractive index of the two slabs are µ1 and µ2.
Thickness of both the plates is t.
When both the strips are fitted, the optical path changes by
x=μ1-1 t- μ2-1 t      =μ1-μ2 t    
For minimum at P0, the path difference should be λ2.
i.e. x=λ2
So,λ2=μ1-μ2tt=λ2μ1-μ2
Therefore, minimum at point P0 is λ2μ1-μ2.

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Question 18:

Given:
Refractive index of the two slabs are µ1 and µ2.
Thickness of both the plates is t.
When both the strips are fitted, the optical path changes by
x=μ1-1 t- μ2-1 t      =μ1-μ2 t    
For minimum at P0, the path difference should be λ2.
i.e. x=λ2
So,λ2=μ1-μ2tt=λ2μ1-μ2
Therefore, minimum at point P0 is λ2μ1-μ2.

Answer:

Given:
The thickness of the thin paper, t=0.02 mm=0.02×10-3 m
Refractive index of the paper, μ=1.45.
Wavelength of the light,  λ=600 nm=600×10-9 m
(a)
Let the intensity of the source without paper = I1
and intensity of source with paper =I2
Let a1 and a2 be corresponding amplitudes.
As per the question,
I2=49I1
We know that
I1I2=a12a22      Ia2a1a2=32   
Here, a is the amplitude.
We know that  ImaxImin=a1+a22a1-a22. ImaxImin=3+223-22             =251Imax:Imin=25 : 1

(b)
Number of fringes that will cross through the centre is given by n=μ-1tλ.
 n=1.45-1×0.02×10-3600×10-9       =0.45×0.02×1046=15

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Question 19:

Given:
The thickness of the thin paper, t=0.02 mm=0.02×10-3 m
Refractive index of the paper, μ=1.45.
Wavelength of the light,  λ=600 nm=600×10-9 m
(a)
Let the intensity of the source without paper = I1
and intensity of source with paper =I2
Let a1 and a2 be corresponding amplitudes.
As per the question,
I2=49I1
We know that
I1I2=a12a22      Ia2a1a2=32   
Here, a is the amplitude.
We know that  ImaxImin=a1+a22a1-a22. ImaxImin=3+223-22             =251Imax:Imin=25 : 1

(b)
Number of fringes that will cross through the centre is given by n=μ-1tλ.
 n=1.45-1×0.02×10-3600×10-9       =0.45×0.02×1046=15

Answer:

Given:
Separation between two slits, d=0.28 mm=0.28×10-3 m
Distance between screen and slit (D) = 48 cm = 0.48 m
Wavelength of the red light, λa=700 nm in vaccum =700 ×10-9 m
Let the wavelength of red light in water = λω
We known that refractive index of water (μw =4/3), 
μw = Speed of light in vacuumSpeed of light in the water
So, μw=vavω=λaλω43=λaλωλω= 3λa4=3×7004=525 nm
So, the fringe width of the pattern is given by
β=λωDd   =525×10-9×0.480.28×10-3   =9×10-4=0.90 mm

Hence, fringe-width of the pattern formed on the screen is 0.90 mm.

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Question 20:

Given:
Separation between two slits, d=0.28 mm=0.28×10-3 m
Distance between screen and slit (D) = 48 cm = 0.48 m
Wavelength of the red light, λa=700 nm in vaccum =700 ×10-9 m
Let the wavelength of red light in water = λω
We known that refractive index of water (μw =4/3), 
μw = Speed of light in vacuumSpeed of light in the water
So, μw=vavω=λaλω43=λaλωλω= 3λa4=3×7004=525 nm
So, the fringe width of the pattern is given by
β=λωDd   =525×10-9×0.480.28×10-3   =9×10-4=0.90 mm

Hence, fringe-width of the pattern formed on the screen is 0.90 mm.

Answer:

Let the two slits are S1 and S2 with separation d as shown in figure.

 

The wave fronts reaching P0from S1 and S2 will have a path difference of S1X = ∆x.
In ∆S1S2X, sinθ=S1XS1S2=xd
x=dsinθ
Using θ=sin-1 λ2d, we get,
x=d×λ2d=λ2
Hence, there will be dark fringe at point P0 as the path difference is an odd multiple of λ2.

Page No 381:

Question 21:

Let the two slits are S1 and S2 with separation d as shown in figure.

 

The wave fronts reaching P0from S1 and S2 will have a path difference of S1X = ∆x.
In ∆S1S2X, sinθ=S1XS1S2=xd
x=dsinθ
Using θ=sin-1 λ2d, we get,
x=d×λ2d=λ2
Hence, there will be dark fringe at point P0 as the path difference is an odd multiple of λ2.

Answer:

(a) The phase of a light wave reflecting from a surface differs by 'π' from the light directly coming from the source.

Thus, the wave fronts reaching just above the mirror directly from the source and after reflecting from the mirror have a phase difference of π, which is the condition of distractive interference. So, the intensity at a point just above the mirror is zero.

(b) Here, separation between two slits is 2d.
Wavelength of the light is λ.
Distance of the screen from the slit is D.
Consider that the bright fringe is formed at position y. Then,
path difference, x=y×2dD=nλ.
After reflection from the mirror, path difference between two waves is λ2.
y×2dD=λ2+nλFor first order, put n=0.y=λD4d



Page No 382:

Question 22:

(a) The phase of a light wave reflecting from a surface differs by 'π' from the light directly coming from the source.

Thus, the wave fronts reaching just above the mirror directly from the source and after reflecting from the mirror have a phase difference of π, which is the condition of distractive interference. So, the intensity at a point just above the mirror is zero.

(b) Here, separation between two slits is 2d.
Wavelength of the light is λ.
Distance of the screen from the slit is D.
Consider that the bright fringe is formed at position y. Then,
path difference, x=y×2dD=nλ.
After reflection from the mirror, path difference between two waves is λ2.
y×2dD=λ2+nλFor first order, put n=0.y=λD4d

Answer:

Given:
Separation between two slits, d'=2d=2 mm=2×10-3 m (as d = 1 mm)
Wavelength of the light used, λ=700 nm=700×10-3 m     
Distance between the screen and slit (D) = 1.0 m
It is a case of Lloyd's mirror experiment.
Fringe width, β=λDd'                  =700×10-9×12×10-3                  =0.35×10-3 m=0.35 mm

Hence, the width of the fringe is 0.35 mm.

Page No 382:

Question 23:

Given:
Separation between two slits, d'=2d=2 mm=2×10-3 m (as d = 1 mm)
Wavelength of the light used, λ=700 nm=700×10-3 m     
Distance between the screen and slit (D) = 1.0 m
It is a case of Lloyd's mirror experiment.
Fringe width, β=λDd'                  =700×10-9×12×10-3                  =0.35×10-3 m=0.35 mm

Hence, the width of the fringe is 0.35 mm.

Answer:

Given:
The mirror reflects 64% of the energy or intensity of light.
Let intensity of source = I1.
And intensity of light after reflection from the mirror = I2.
Let a1 and a2 be corresponding amplitudes of intensity I1 and I2.
According to the question,
I2=I1×64100I2I1=64100=1625And I2I1=a22a12a2a1=45   
We know that ImaxImin=a1+a22a1-a22                         =5+425-42Imax:Imin=81 : 1

Hence, the required ratio is 81 : 1.

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Question 24:

Given:
The mirror reflects 64% of the energy or intensity of light.
Let intensity of source = I1.
And intensity of light after reflection from the mirror = I2.
Let a1 and a2 be corresponding amplitudes of intensity I1 and I2.
According to the question,
I2=I1×64100I2I1=64100=1625And I2I1=a22a12a2a1=45   
We know that ImaxImin=a1+a22a1-a22                         =5+425-42Imax:Imin=81 : 1

Hence, the required ratio is 81 : 1.

Answer:

Given:
Separation between the two slits = d
Wavelength of the coherent light =λ
Distance between the slit and mirror is D1.
Distance between the slit and screen is D2.
Therefore,
apparent distance of the screen from the slits,
D=2D1+D2Fringe width, β=λDd=2D1+D2 λd

Hence, the required fringe width is 2D1+D2 λd.

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Question 25:

Given:
Separation between the two slits = d
Wavelength of the coherent light =λ
Distance between the slit and mirror is D1.
Distance between the slit and screen is D2.
Therefore,
apparent distance of the screen from the slits,
D=2D1+D2Fringe width, β=λDd=2D1+D2 λd

Hence, the required fringe width is 2D1+D2 λd.

Answer:

Given: Separation between two slits,
d=0.5 mm=0.5×10-3 m
Wavelength of the light, λ=400 nm to 700 nm
Distance of the screen from the slit, D=50 cm=0.5 m,
Position of hole on the screen, yn=1 mm=1×10-3 m

(a) The wavelength(s) will be absent in the light coming from the hole, which will form a dark fringe at the position of hole.
yn=2n+1λn2Dd , where n = 0, 1, 2, ...

λn=22n+1 yndD       =22n+1×10-3×0.05×10-30.5       =22n+1×10-6 m        =22n+1×103 nm

For n=1,λ1=23×1000 =667 nmFor n=2,λ2=25×1000=400 nm
Thus, the light waves of wavelength 400 nm and 667 nm will be absent from the light coming from the hole.

(b) The wavelength(s) will have a strong intensity, which will form a bright fringe at the position of the hole.
 So, yn=nλnDdλn=yndnDFor n=1, λ1=yndD     =10-3×0.5×10-30.5     =10-6 m=1000 nm.

But 1000 nm does not fall in the range 400 nm − 700 nm.

Again, for n=2,λ2=ynd2D=500 nm
So, the light of wavelength 500 nm will have strong intensity.

Page No 382:

Question 26:

Given: Separation between two slits,
d=0.5 mm=0.5×10-3 m
Wavelength of the light, λ=400 nm to 700 nm
Distance of the screen from the slit, D=50 cm=0.5 m,
Position of hole on the screen, yn=1 mm=1×10-3 m

(a) The wavelength(s) will be absent in the light coming from the hole, which will form a dark fringe at the position of hole.
yn=2n+1λn2Dd , where n = 0, 1, 2, ...

λn=22n+1 yndD       =22n+1×10-3×0.05×10-30.5       =22n+1×10-6 m        =22n+1×103 nm

For n=1,λ1=23×1000 =667 nmFor n=2,λ2=25×1000=400 nm
Thus, the light waves of wavelength 400 nm and 667 nm will be absent from the light coming from the hole.

(b) The wavelength(s) will have a strong intensity, which will form a bright fringe at the position of the hole.
 So, yn=nλnDdλn=yndnDFor n=1, λ1=yndD     =10-3×0.5×10-30.5     =10-6 m=1000 nm.

But 1000 nm does not fall in the range 400 nm − 700 nm.

Again, for n=2,λ2=ynd2D=500 nm
So, the light of wavelength 500 nm will have strong intensity.

Answer:


From the figure, AB=BO and AC=CO.
Path difference of the wave front reaching O,
x=AB+BO-AC+CO     =2 AB-AC     =2 D2+d2-D
For dark fringe to be formed at O, path difference should be an odd multiple of λ2.
So,x=2n+1 λ2 2D2+d2-D=2n+1 λ2D2+d2=D+2n+1 λ4D2+d2=D2+2n+12λ216+2n+1 λD2
Neglecting, 2n+12λ216, as it is very small, we get:
d=2n+1 λD2For minimum d, putting, n=0dmin=λD2, we get:
Thus, for dmin=λD2 there is a dark fringe at O.

Page No 382:

Question 27:


From the figure, AB=BO and AC=CO.
Path difference of the wave front reaching O,
x=AB+BO-AC+CO     =2 AB-AC     =2 D2+d2-D
For dark fringe to be formed at O, path difference should be an odd multiple of λ2.
So,x=2n+1 λ2 2D2+d2-D=2n+1 λ2D2+d2=D+2n+1 λ4D2+d2=D2+2n+12λ216+2n+1 λD2
Neglecting, 2n+12λ216, as it is very small, we get:
d=2n+1 λD2For minimum d, putting, n=0dmin=λD2, we get:
Thus, for dmin=λD2 there is a dark fringe at O.

Answer:



Let P be the point of minimum intensity.
For minimum intensity at point P,

        S1P-S2P=x=2n+1 λ2

Thus, we get:

Z2+2λ2-Z=2n+1 λ2Z2+4λ2=Z2 2n+12 λ24+2Z 2n+1 λ2Z=4λ2-2n+1 2λ2/42n+1 λ      =16λ2-2n+14 2n+1 λ                ... (i)

When n=0,        Z=15λ4            n=-1,    Z=-15λ4            n=1,       Z=7λ12            n=2 ,      Z=-9λ20

Thus, Z=7λ12 is the smallest distance for which there will be minimum intensity.

Page No 382:

Question 28:



Let P be the point of minimum intensity.
For minimum intensity at point P,

        S1P-S2P=x=2n+1 λ2

Thus, we get:

Z2+2λ2-Z=2n+1 λ2Z2+4λ2=Z2 2n+12 λ24+2Z 2n+1 λ2Z=4λ2-2n+1 2λ2/42n+1 λ      =16λ2-2n+14 2n+1 λ                ... (i)

When n=0,        Z=15λ4            n=-1,    Z=-15λ4            n=1,       Z=7λ12            n=2 ,      Z=-9λ20

Thus, Z=7λ12 is the smallest distance for which there will be minimum intensity.

Answer:

(a) Given:
Wavelength of light = λ
Path difference of wave fronts reaching from A and B is given by
xB=BP0-AP0=λ3D2+d2-D=λ3D2+d2= D2+λ29 +2λD3

We will neglect the term λ29, as it has a very small value.
d=2λD3

(b) To calculating the intensity at P0, consider the interference of light waves coming from all the three slits.
Path difference of the wave fronts reaching from A and C is given by
CP0-AP0=D2+2d2-D               =D2+8λD3-D   Using the value of d from part a               =D1+8λ3D12-DExpanding the value using binomial theorem and neglecting the higher order terms, we get:               D1+12×8λ3D+...-D               
CP0-AP0=4λ3
So, the corresponding phase difference between the wave fronts from A and C is given by
ϕc=2πxCλ=2π×4λ3λϕc=8π3 or 2π+2π3ϕc=2π3     ...(1)Again, ϕB=2π xBλϕB=2πλ3λ=2π3    ...(2)

So, it can be said that light from B and C are in the same phase, as they have the same phase difference with respect to A.

Amplitude of wave reaching P0 is given by
A=2a2+a2+2a×acos2π3       =4a2+a2+2a23lpo=K 3 r2=3 Kr2=3l
Here, I is the intensity due to the individual slits and Ipo is the total intensity at P0.
Thus, the resulting amplitude is three times the intensity due to the individual slits.

Page No 382:

Question 29:

(a) Given:
Wavelength of light = λ
Path difference of wave fronts reaching from A and B is given by
xB=BP0-AP0=λ3D2+d2-D=λ3D2+d2= D2+λ29 +2λD3

We will neglect the term λ29, as it has a very small value.
d=2λD3

(b) To calculating the intensity at P0, consider the interference of light waves coming from all the three slits.
Path difference of the wave fronts reaching from A and C is given by
CP0-AP0=D2+2d2-D               =D2+8λD3-D   Using the value of d from part a               =D1+8λ3D12-DExpanding the value using binomial theorem and neglecting the higher order terms, we get:               D1+12×8λ3D+...-D               
CP0-AP0=4λ3
So, the corresponding phase difference between the wave fronts from A and C is given by
ϕc=2πxCλ=2π×4λ3λϕc=8π3 or 2π+2π3ϕc=2π3     ...(1)Again, ϕB=2π xBλϕB=2πλ3λ=2π3    ...(2)

So, it can be said that light from B and C are in the same phase, as they have the same phase difference with respect to A.

Amplitude of wave reaching P0 is given by
A=2a2+a2+2a×acos2π3       =4a2+a2+2a23lpo=K 3 r2=3 Kr2=3l
Here, I is the intensity due to the individual slits and Ipo is the total intensity at P0.
Thus, the resulting amplitude is three times the intensity due to the individual slits.

Answer:

Given:
Separation between the slits, d=2 mm=2×10-3 m     
Wavelength of the light, λ=600 nm=6×10-7 m
Distance of the screen from the slits, D = 2⋅0 m
Imax=0.20 W/m2For the point at a position y=0.5 cm=0.5×10-2 m,path difference,x=ydD.x=0.5×10-2×2×10-32         =5×10-6 m

So, the corresponding phase difference is given by
ϕ=2πxλ=2π×5×10-66×10-7    =50π3=16π+2π3or ϕ=2π3
So, the amplitude of the resulting wave at point y = 0.5 cm is given by
A=a2+a2+2a2 cos 2π3  =a2+a2-a2 =a 
Similarly, the amplitude of the resulting wave at the centre is 2a.
Let the intensity of the resulting wave at point y = 0.5 cm be I.
Since IImax=A22a2, we have:I0.2=A24a2=a24a2I=0.24=0.05 W/m2
Thus, the intensity at a point 0.5 cm away from the centre along the width of the fringes is 0.05 W/m2.

Page No 382:

Question 30:

Given:
Separation between the slits, d=2 mm=2×10-3 m     
Wavelength of the light, λ=600 nm=6×10-7 m
Distance of the screen from the slits, D = 2⋅0 m
Imax=0.20 W/m2For the point at a position y=0.5 cm=0.5×10-2 m,path difference,x=ydD.x=0.5×10-2×2×10-32         =5×10-6 m

So, the corresponding phase difference is given by
ϕ=2πxλ=2π×5×10-66×10-7    =50π3=16π+2π3or ϕ=2π3
So, the amplitude of the resulting wave at point y = 0.5 cm is given by
A=a2+a2+2a2 cos 2π3  =a2+a2-a2 =a 
Similarly, the amplitude of the resulting wave at the centre is 2a.
Let the intensity of the resulting wave at point y = 0.5 cm be I.
Since IImax=A22a2, we have:I0.2=A24a2=a24a2I=0.24=0.05 W/m2
Thus, the intensity at a point 0.5 cm away from the centre along the width of the fringes is 0.05 W/m2.

Answer:

Given:
Separation between the two slits = d
Wavelength of the light = λ
Distance of the screen = D
(a) When the intensity is half the maximum:
Let Imax be the maximum intensity and I be the intensity at the required point at a distance y from the central point.
So, I=a2+a2+2a2cosϕ
Here, ϕ is the phase difference in the waves coming from the two slits.
So, I=4a2cos2ϕ2
IImax=124a2cos2ϕ24a2=12cos2ϕ2=12cosϕ2=12ϕ2=π4ϕ=π2Corrosponding path difference, x=λ4y=xDd=λD4d

(b) When the intensity is one-fourth of the maximum:

IImax=144a2cos2ϕ2=14cos2 ϕ2=14cosϕ2=12ϕ2=π3So, corrosponding path difference, x=λ3and position, y=xDd=λD3d .

Page No 382:

Question 31:

Given:
Separation between the two slits = d
Wavelength of the light = λ
Distance of the screen = D
(a) When the intensity is half the maximum:
Let Imax be the maximum intensity and I be the intensity at the required point at a distance y from the central point.
So, I=a2+a2+2a2cosϕ
Here, ϕ is the phase difference in the waves coming from the two slits.
So, I=4a2cos2ϕ2
IImax=124a2cos2ϕ24a2=12cos2ϕ2=12cosϕ2=12ϕ2=π4ϕ=π2Corrosponding path difference, x=λ4y=xDd=λD4d

(b) When the intensity is one-fourth of the maximum:

IImax=144a2cos2ϕ2=14cos2 ϕ2=14cosϕ2=12ϕ2=π3So, corrosponding path difference, x=λ3and position, y=xDd=λD3d .

Answer:

Given:
Separation between the two slits, d=1 mm=10-3 m
Wavelength of the light, λ=500 nm=5×10-7 m
Distance of the screen, D=1 m
Let Imax be the maximum intensity and I be the intensity at the required point at a distance y from the central point.
So, I=a2+a2+2a2cosϕ
Here, ϕ is the phase difference in the waves coming from the two slits.
So, I=4a2cos2ϕ2
IImax=124a2cos2ϕ24a2=12cos2ϕ2=12cosϕ2=12ϕ2=π4ϕ=π2Corrosponding path difference, x=14y=xDd=λD4d
y=5×10-7×14×10-3      =1.25×10-4 m

∴ The required minimum distance from the central maximum is 1.25×10-4 m.

Page No 382:

Question 32:

Given:
Separation between the two slits, d=1 mm=10-3 m
Wavelength of the light, λ=500 nm=5×10-7 m
Distance of the screen, D=1 m
Let Imax be the maximum intensity and I be the intensity at the required point at a distance y from the central point.
So, I=a2+a2+2a2cosϕ
Here, ϕ is the phase difference in the waves coming from the two slits.
So, I=4a2cos2ϕ2
IImax=124a2cos2ϕ24a2=12cos2ϕ2=12cosϕ2=12ϕ2=π4ϕ=π2Corrosponding path difference, x=14y=xDd=λD4d
y=5×10-7×14×10-3      =1.25×10-4 m

∴ The required minimum distance from the central maximum is 1.25×10-4 m.

Answer:

Given:
Separation between two slits = d
Wavelength of the light = λ
Distance of the screen = D
Let Imax be the maximum intensity and I be half the maximum intensity at a point at a distance y from the central point.
So, I=a2+a2+2a2cosϕ
Here, ϕ is the phase difference in the waves coming from the two slits.
So, I=4a2cos2ϕ2
IImax=124a2cos2ϕ24a2=12cos2ϕ2=12cosϕ2=12ϕ2=π4ϕ=π2Corrosponding path difference, x=14y=xDd=λD4d
The line-width of a bright fringe is defined as the separation between the points on the two sides of the central line where the intensity falls to half the maximum.
So, line-width = 2y
=2Dλ4d=Dλ2d

Thus, the required line width of the bright fringe is Dλ2d.



Page No 383:

Question 33:

Given:
Separation between two slits = d
Wavelength of the light = λ
Distance of the screen = D
Let Imax be the maximum intensity and I be half the maximum intensity at a point at a distance y from the central point.
So, I=a2+a2+2a2cosϕ
Here, ϕ is the phase difference in the waves coming from the two slits.
So, I=4a2cos2ϕ2
IImax=124a2cos2ϕ24a2=12cos2ϕ2=12cosϕ2=12ϕ2=π4ϕ=π2Corrosponding path difference, x=14y=xDd=λD4d
The line-width of a bright fringe is defined as the separation between the points on the two sides of the central line where the intensity falls to half the maximum.
So, line-width = 2y
=2Dλ4d=Dλ2d

Thus, the required line width of the bright fringe is Dλ2d.

Answer:

Given:
Separation between the two slits = d
Wavelength of the light =λ
Distance of the screen = D
The fringe width (β) is given by  β=λDd.
At S3, the path difference is zero. So, the maximum intensity occurs at amplitude = 2a.
(a) When z=Dλ2d:
The first minima occurs at S4, as shown in figure (a).
With amplitude = 0 on screen ∑2, we get:
lmaxlmin=2a+022a-02=1


(b) When z=Dλd:
The first maxima occurs at S4, as shown in the figure.



With amplitude = 2a on screen ∑2, we get:
lmaxlmin=2a+2a22a-2a2=


(c) When z=Dλ4d:



The slit S4 falls at the mid-point of the central maxima and the first minima, as shown in the figure.
Intensity=lmax2 Amplitude=2a
lmaxlmin=2a+2a22a-2a2=34

Page No 383:

Question 34:

Given:
Separation between the two slits = d
Wavelength of the light =λ
Distance of the screen = D
The fringe width (β) is given by  β=λDd.
At S3, the path difference is zero. So, the maximum intensity occurs at amplitude = 2a.
(a) When z=Dλ2d:
The first minima occurs at S4, as shown in figure (a).
With amplitude = 0 on screen ∑2, we get:
lmaxlmin=2a+022a-02=1


(b) When z=Dλd:
The first maxima occurs at S4, as shown in the figure.



With amplitude = 2a on screen ∑2, we get:
lmaxlmin=2a+2a22a-2a2=


(c) When z=Dλ4d:



The slit S4 falls at the mid-point of the central maxima and the first minima, as shown in the figure.
Intensity=lmax2 Amplitude=2a
lmaxlmin=2a+2a22a-2a2=34

Answer:

Given:
Fours slits S1, S2, S3 and S4.
The separation between slits S3 and S4 can be changed.
Point P is the common perpendicular bisector of S1S2 and S3S4.



(a) For z=λDd:
The position of the slits from the central point of the first screen is given by
y=OS3=OS4=z2=λD2d
The corresponding path difference in wave fronts reaching S3 is given by
x=ydD=λD2d×dD=λ2
Similarly at S4, path difference, x=ydD=λD2d×dD=λ2
i.e. dark fringes are formed at S3 and S4.
So, the intensity of light at S3 and S4 is zero. Hence, the intensity at P is also zero.

(b) For z=3λD2d
The position of the slits from the central point of the first screen is given by
y=OS3=OS4=z2=3λD4d
The corresponding path difference in wave fronts reaching S3 is given by
x=ydD=3λD4d×dD=3λ4
Similarly at S4, path difference, x=ydD=3λD4d×dD=3λ4
Hence, the intensity at P is I.

(c) Similarly, for z=2Dλd,
the intensity at P is 2I.

Page No 383:

Question 35:

Given:
Fours slits S1, S2, S3 and S4.
The separation between slits S3 and S4 can be changed.
Point P is the common perpendicular bisector of S1S2 and S3S4.



(a) For z=λDd:
The position of the slits from the central point of the first screen is given by
y=OS3=OS4=z2=λD2d
The corresponding path difference in wave fronts reaching S3 is given by
x=ydD=λD2d×dD=λ2
Similarly at S4, path difference, x=ydD=λD2d×dD=λ2
i.e. dark fringes are formed at S3 and S4.
So, the intensity of light at S3 and S4 is zero. Hence, the intensity at P is also zero.

(b) For z=3λD2d
The position of the slits from the central point of the first screen is given by
y=OS3=OS4=z2=3λD4d
The corresponding path difference in wave fronts reaching S3 is given by
x=ydD=3λD4d×dD=3λ4
Similarly at S4, path difference, x=ydD=3λD4d×dD=3λ4
Hence, the intensity at P is I.

(c) Similarly, for z=2Dλd,
the intensity at P is 2I.

Answer:

Given:
Thickness of soap film, d =0.0011 mm = 0.0011 × 10−3 m
Wavelength of light used, λ=580 nm=580×10-9 m
Let the index of refraction of the soap solution be μ.
The condition of minimum reflection of light is 2μd = ,
where n is an interger = 1 , 2 , 3 ...
μ=nλ2d=2nλ4d     =580×10-9×2n4×11×10-7     =5.82n44=0.1322n
As per the question, μ has a value between 1.2 and 1.5. So,
n=5So, μ=0.132×10=1.32

Therefore, the index of refraction of the soap solution is 1.32.

Page No 383:

Question 36:

Given:
Thickness of soap film, d =0.0011 mm = 0.0011 × 10−3 m
Wavelength of light used, λ=580 nm=580×10-9 m
Let the index of refraction of the soap solution be μ.
The condition of minimum reflection of light is 2μd = ,
where n is an interger = 1 , 2 , 3 ...
μ=nλ2d=2nλ4d     =580×10-9×2n4×11×10-7     =5.82n44=0.1322n
As per the question, μ has a value between 1.2 and 1.5. So,
n=5So, μ=0.132×10=1.32

Therefore, the index of refraction of the soap solution is 1.32.

Answer:

Given:
Wavelength of  light used, λ=560×10-9 m
Refractive index of the oil film, μ=1.4
Let the thickness of the film for strong reflection be t.
The condition for strong reflection is
2μt=2n+1λ2t=2n+1λ4μ
where n is an integer.
For minimum thickness, putting n = 0, we get:
     t=λ4μ     =560×10-94×1.4     =10-7m=100 nm
Therefore, the minimum thickness of the oil film so that it strongly reflects the light is 100 nm.

Page No 383:

Question 37:

Given:
Wavelength of  light used, λ=560×10-9 m
Refractive index of the oil film, μ=1.4
Let the thickness of the film for strong reflection be t.
The condition for strong reflection is
2μt=2n+1λ2t=2n+1λ4μ
where n is an integer.
For minimum thickness, putting n = 0, we get:
     t=λ4μ     =560×10-94×1.4     =10-7m=100 nm
Therefore, the minimum thickness of the oil film so that it strongly reflects the light is 100 nm.

Answer:

Given,
Wavelength of  light used, λ=400×10-9 m to 700×10-9 nm
Refractive index of water, μ=1.33
The thickness of film, t=10-4 cm=10-6 m
The condition for strong transmission: 2μt=nλ,
where n is an integer.
 λ=2μtn
 λ=2×1.33×10-6n       =2660×10-9n m
Putting n = 4, we get, λ1 = 665 nm.
Putting n = 5, we get, λ2 = 532 nm.
Putting n = 6, we get, λ3 = 443 nm.
Therefore, the wavelength (in visible region) which are strongly transmitted by the film are 665 nm, 532nm and 443 nm.

Page No 383:

Question 38:

Given,
Wavelength of  light used, λ=400×10-9 m to 700×10-9 nm
Refractive index of water, μ=1.33
The thickness of film, t=10-4 cm=10-6 m
The condition for strong transmission: 2μt=nλ,
where n is an integer.
 λ=2μtn
 λ=2×1.33×10-6n       =2660×10-9n m
Putting n = 4, we get, λ1 = 665 nm.
Putting n = 5, we get, λ2 = 532 nm.
Putting n = 6, we get, λ3 = 443 nm.
Therefore, the wavelength (in visible region) which are strongly transmitted by the film are 665 nm, 532nm and 443 nm.

Answer:

Given:
Wavelength of  light used, λ=400×10-9 to 750×10-9 m
Refractive index of oil, μoil, is 1.25 and that of glass, μg, is 1.50.
The thickness of the oil film, d=1×10-4 cm=10-6m,
The condition for the wavelengths which can be completely transmitted through the oil film is given by
λ=2μdn+12  =2×10-6×1.25×22n+1  =5×10-6 2n+1 m λ=50002n+1 nm
Where n is an integer.
For wavelength to be in visible region i.e (400 nm to 750 nm)
When n = 3, we get,
λ=50002×3+1 =50007=714.3 nm
When, n = 4, we get,
λ=50002×4+1  =50009=555.6 nm
When, n = 5, we get,
λ=50002×5+1  =500011=454.5 nm
Thus the wavelengths of light in the visible region (400 nm − 750 nm) which are completely transmitted by the oil film under normal incidence are 714 nm, 556 nm, 455  nm.

Page No 383:

Question 39:

Given:
Wavelength of  light used, λ=400×10-9 to 750×10-9 m
Refractive index of oil, μoil, is 1.25 and that of glass, μg, is 1.50.
The thickness of the oil film, d=1×10-4 cm=10-6m,
The condition for the wavelengths which can be completely transmitted through the oil film is given by
λ=2μdn+12  =2×10-6×1.25×22n+1  =5×10-6 2n+1 m λ=50002n+1 nm
Where n is an integer.
For wavelength to be in visible region i.e (400 nm to 750 nm)
When n = 3, we get,
λ=50002×3+1 =50007=714.3 nm
When, n = 4, we get,
λ=50002×4+1  =50009=555.6 nm
When, n = 5, we get,
λ=50002×5+1  =500011=454.5 nm
Thus the wavelengths of light in the visible region (400 nm − 750 nm) which are completely transmitted by the oil film under normal incidence are 714 nm, 556 nm, 455  nm.

Answer:

Given:
Width of the slit, b = 5.0 cm
First diffraction minimum is formed at θ = 30°.
For the diffraction minima, we have:
bsinθ =
For the first minima, we put n = 1.
 5×sin30°=1×λλ=52=2.5 cm
Therefore, the wavelength of the microwaves is 2.5 cm.

Page No 383:

Question 40:

Given:
Width of the slit, b = 5.0 cm
First diffraction minimum is formed at θ = 30°.
For the diffraction minima, we have:
bsinθ =
For the first minima, we put n = 1.
 5×sin30°=1×λλ=52=2.5 cm
Therefore, the wavelength of the microwaves is 2.5 cm.

Answer:

Given:
Wavelength of the light used, λ=560 nm=560×10-9 m
Diameter of the pinhole, d = 0.20 mm = 2 × 10−4 m
Distance of the wall, D = 2m
We know that the radius of the central bright spot is given by
R=1.22λDd   =1.22×560×10-9×22×10-4   =6.832×10-3 m or=0.683 cm

Hence, the diameter 2R of the central bright spot on the wall is 1.37 cm.

Page No 383:

Question 41:

Given:
Wavelength of the light used, λ=560 nm=560×10-9 m
Diameter of the pinhole, d = 0.20 mm = 2 × 10−4 m
Distance of the wall, D = 2m
We know that the radius of the central bright spot is given by
R=1.22λDd   =1.22×560×10-9×22×10-4   =6.832×10-3 m or=0.683 cm

Hence, the diameter 2R of the central bright spot on the wall is 1.37 cm.

Answer:

Given:
Wavelength of the light used, λ=620 nm=620×10-9m
Diameter of the convex lens, d=8 cm=8×10-2 m
Distance from the lens where light is to be focused, D=20 cm=20×10-2 m
The radius of the central bright spot is given by
R=1.22λDd  =1.22×620×10-9×20×10-28×10-2   = 1891×10-9 m1.9×10-6m
∴ Diameter of the central bright spot, 2R = 3.8 × 10−6 m.



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