0

Multiple-slit diffraction - class-XII

Description: multiple-slit diffraction
Number of Questions: 28
Created by:
Tags: superposition of waves oscillations and waves physics
Attempted 0/28 Correct 0 Score 0

In Young's double slit experiment, the phase difference between the light waves reaching third bright fringe from the central fringe will be ($\lambda =6000\mathring {A}$)

  1. $0$

  2. $2\pi$

  3. $4\pi$

  4. $6 \pi$


Correct Option: D
Explanation:

$\because n=3\ \therefore 2n\pi =2\times 3\times \pi \ \quad \quad \quad \quad \quad =6\pi $

In Y.D.S.E. two waves of equal intensity produces an intensity $I _0$ at the centre but at a point where path difference is $\frac{\lambda}{6}$ intensity is I'. Then find the ratio $\frac{I'}{I _0}$ :-

  1. 3 : 4

  2. 4 : 3

  3. 2 : 1

  4. 2 : 3


Correct Option: B

If two coherent light waves produce minima of fifth order, the path dfference between the waves is

  1. $5 \lambda$

  2. $5 \lambda / { 2 }$

  3. $7 \lambda /{ 2 }$

  4. $9 \lambda / { 2 }$


Correct Option: D

In Young's double slit experiment, the phase difference between the two waves reaching at the location of the third dark fringe is  

  1. $\pi$

  2. $\cfrac{3\pi}{2}$

  3. $5 \pi$

  4. $3 \pi$


Correct Option: C

In the above question, the intensity of the waves reaching a point P far away on the x-axis from each of the four sources is almost the same and equal to $I _0.$ Then,

  1. If $d=\lambda /4,$ the intensity at P is $4I _0.$

  2. If $d=\lambda /6,$ the intensity at P is $3I _0.$

  3. If $d=\lambda /2,$ the intensity at P is $3I _0.$

  4. None of these is true


Correct Option: B

The amount of light that falls per unit area held perpendicular to the rays in one second is called

  1. Candle power

  2. Dioptre

  3. Lumen

  4. Intensity of illumination


Correct Option: D
Explanation:

Candle power- It is the rating of light output at source. It is the the candle's luminous intensity in a particular direction.

Dioptre- It is the ability of a lens to bend the light rays. A unit of power of lens.
Lumen- The amount of energy emanating from one square meter of surface.
Intensity of illumination - The amount of light falling on per unit area, held perpendicular to the rays in one second.

The phase difference between two waves from successive half period zones or strips is :

  1. $\dfrac{\pi}{ 4}$

  2. $ \dfrac{\pi}{ 2}$

  3. $ \pi $

  4. zero


Correct Option: C
Explanation:

The half period zone is provided by an optical device known as zone plate. It is simply a plane parallel gloss plate having concentric circles of radii accurately proportional to the square roots of the consecutive natural numbers 1,2,3 ... etc. The area is given by $\pi \gamma ^{2}$. Hence the
areas are $\pi , 2\pi , 3\pi , 4\pi$,---
The phase difference is thus $\pi$ between each successive half period zones of strips.

In Young's double slit experiment, the constant phase difference between two sources is $\dfrac{\pi }{2}$. The intensity at a point equidistant from the slits in terms of maximum intensity $I _{\circ}$ is :

  1. $I _{\circ}$

  2. $I _{\circ}/2$

  3. $3I _{\circ}/4$

  4. $3I _{\circ}$


Correct Option: B
Explanation:

If there is no phase difference let
the Intensity equal to maximum Intensity be $I _{0}$
For a phase difference $\theta $ then
Intensity at a point equidistant from the two
slits is given by
$\dfrac{I _{0}}{2}(1+cos^{2}\theta )$
As $\theta =\pi /2$
we get $\dfrac{I _{0}}{2}(1+0)=\dfrac{I _{0}}{2}$
Thus Intensity $=\dfrac{I _{0}}{2}$

The maximum intensity produced by two coherent sources of intensity $I _{1}$ and $I _{2}$ constructively will be:

  1. $I _{I}+I _{2}$

  2. $I _{I}^{2}+I _{2}^{2}$

  3. $I _{I}+I _{2}+2\sqrt{I _{1}I _{2}}$

  4. zero


Correct Option: C
Explanation:

If the angle between two coherent sources is $\theta $ then the Intensity after Interference is given by $I _{1}+I _{2}+2\sqrt{I _{1}I _{2}} cos \theta $. For constructive Interference $\theta =2n\pi $ Thus Intensity is $I _{1}+I _{2}+2\sqrt{I _{1}I _{2}}$

The intensity ratio for the two interfering beam of light is $\beta$. What is the value of 
$\dfrac{I _{max}-I _{min}}{I _{max}+I _{min}}$ ?

  1. 2$\sqrt{\beta}$

  2. $\dfrac{2\sqrt{\beta}}{1+\beta}$

  3. $\dfrac{2}{1+\beta}$

  4. $\dfrac{1+\beta}{2\sqrt{\beta}}$


Correct Option: B
Explanation:

Given $\cfrac { { I } _{ 1 } }{ { I } _{ 2 } } =\beta $

${ I } _{ max }={ I } _{ 1 }+{ I } _{ 2 }+2\sqrt { { I } _{ 1 }{ I } _{ 2 } } $
${ I } _{ min }={ I } _{ 1 }+{ I } _{ 2 }-2\sqrt { { I } _{ 1 }{ I } _{ 2 } } $
$\therefore \cfrac { { I } _{ max }-{ I } _{ min } }{ { I } _{ max }+{ I } _{ min } } \quad =\cfrac { 4\sqrt { { I } _{ 1 }{ I } _{ 2 } }  }{ 2({ I } _{ 1 }+{ I } _{ 2 }) } \quad =\frac { 2\sqrt { { I } _{ 2 }^{ 2 }\beta  }  }{ { I } _{ 2 }(\beta +1) } \quad =\cfrac { 2\sqrt { \beta  }  }{ \beta +1 } $

Let ${a _1}$ and ${a _2}$ be the amplitudes of two light waves of same frequency and ${\alpha _1}$ and ${\alpha _2}$ be their initial phases. The resultant amplitude due to the superposition of two light waves is

  1. $R = \sqrt {a _1^2 + a _2^2 + 2{a _1}{a _2}} $

  2. $R = {a _1} - {a _2}$

  3. $R = \sqrt {a _1^2 + a _2^2 + 2{a _1}{a _2}\cos \left( {{\alpha _1} - {\alpha _2}} \right)} $

  4. $R = \sqrt {a _1^2 + a _2^2 - 2{a _1}{a _2}} $


Correct Option: C
Explanation:

The angle between two light waves $={ \alpha  } _{ 1 }-{ \alpha  } _{ 2 }$

Resultant $=\sqrt { { a } _{ 1 }^{ 2 }+{ a } _{ 2 }^{ 2 }+2{ a } _{ 1 }.{ a } _{ 2 }\cos { ({ \alpha  } _{ 1 }-{ \alpha  } _{ 2 }) }  } $

In the interference of waves from two sources of intensities $I _o$ and $4I _o$, the intensity at a point where the phase difference is $\pi$, is?

  1. $I _o$

  2. $2I _o$

  3. $3I _o$

  4. $4I _o$


Correct Option: A
Explanation:

$I=I _1+I _2+2\sqrt{I _1I _2}\cos\theta =I _o+4I _o+2\sqrt{(I _o\times 4I _o)}\cos\pi =I _o$
Hence (A) is correct.

In Young's double slit experiment, when two light waves form third minimum, they have 

  1. Phase difference of $3 \pi$

  2. Path difference of $3 \lambda $

  3. Phase difference of $\dfrac{5\pi}{2}$

  4. Path difference of $\dfrac{5\lambda }{2}$


Correct Option: D
Explanation:

For minima , path difference=$\dfrac{(2n-1)\lambda}{2}$........(1)

          $n=3$
 so, putting in (1),
we get, path difference=$\dfrac{5\lambda}{2}$

Phase diffrence $\dfrac{\Delta\phi}{2 \pi}=\dfrac {\Delta X}{\lambda}$
 
$\Delta \phi =2\pi \dfrac {\Delta X}{\lambda} $

 $\Delta \phi =2\pi \dfrac {5 \lambda /2}{\lambda} =5 \pi$

At two points P and Q on screen in Young's double shit experiment, waves from slits $S _1$ and $S _2$ have a path difference of O and $\frac{\lambda}{4}$ respectively, the ratio of intenstine at P and Q will be: 

  1. $3: 2$

  2. $2: 1$

  3. $\sqrt2: 1$

  4. $4: 1$


Correct Option: A


The displacement of two interfering light wave are $ y _1 = 4 sin \omega t and y _2 = 3 cos(\omega t) $
The amplitude of the resultant wave is and $ y _2 $ are:(in CGS system)

  1. 5 cm

  2. 7 cm

  3. 1 cm

  4. zero


Correct Option: A
Explanation:

Given that,

$ {{y} _{1}}=4\sin \omega t $

$ {{y} _{2}}=3\cos \omega t $

Amplitude of first and second wave is 4 cm and 3 cm. So, the amplitude of resultant wave is

$ {{y}^{'}}=\sqrt{{{(4)}^{2}}+{{(3)}^{2}}} $

$ =5\,cm $

A light wave is incident normally over slit of width $24\times 10^{-5}$ cm. The angular position of second dark fringe from the central maximum is 30$^{0}$. What is the wavelength of light ?

  1. 6000 $A^0 $

  2. 5000 $A^0 $

  3. 3000 $A^0 $

  4. 1500 $A^0 $


Correct Option: A

Two coherent sources of intensity ratio of interfere in interference parteren $\frac { \mathrm { I } _ { \max } - \mathrm { I } _ { \min } } { \mathrm { I } _ { \max } + \mathrm { I } _ { \min } }$ is equal to

  1. $\frac { 2 \alpha } { 1 + \alpha }$

  2. $\frac { 2 \sqrt { a } } { 1 + \alpha }$

  3. $\frac { 2 \alpha } { 1 \sqrt { \alpha } }$

  4. $\frac { 1 + \alpha } { 2 \alpha }$


Correct Option: B
Explanation:

$\begin{array}{l} { { { I } } _{ \max   } }={ \left( { \sqrt { { I _{ 1 } } } +\sqrt { { I _{ 2 } } }  } \right) ^{ 2 } } \ ={ I _{ 1 } }+{ I _{ 2 } }+2\sqrt { { I _{ 1 } }{ I _{ 2 } } }  \ { I _{ \min   } }={ I _{ 1 } }+{ I _{ 2 } }-2\sqrt { { I _{ 1 } }{ I _{ 2 } } }  \ \therefore \dfrac { { { { { I } } _{ \max   } }-{ { { I } } _{ \min   } } } }{ { { I _{ \max   } }+{ { { I } } _{ \min   } } } } =\dfrac { { 4\sqrt { { I _{ 1 } }{ I _{ 2 } } }  } }{ { 2\left( { { I _{ 1 } }+{ I _{ 2 } } } \right)  } }  \ =\dfrac { { 2\sqrt { { I _{ 1 } }{ I _{ 2 } } }  } }{ { \left( { { I _{ 1 } }+{ I _{ 2 } } } \right)  } }  \ { I _{ 1 } }=1 \ { I _{ 2 } }=\alpha  \ =\dfrac { { 2\sqrt { \alpha  }  } }{ { 1+\alpha  } }  \ \therefore \, \, Option\, \, \left( B \right) \, \, is\, \, correct\, . \end{array}$

In Young's double slit experiment if the maximum intensity of light is $I _{max}$, then the intensity at path difference $\dfrac{\lambda}{2}$ will be

  1. $I _{max}$

  2. $\displaystyle\frac{I _{max}}{2}$

  3. $\displaystyle\frac{I _{max}}{4}$

  4. zero


Correct Option: D
Explanation:

Destructive interference occurs when the difference is an odd multiple of $\pi$ ,

for path difference of $\lambda/2 $ , $\phi =\pi$

Two coherent waves are represented by $y _1=a _1\cos\omega t$ and $y _2=a _2\cos\omega t$. The maximum intensity due to interference will be proportional to

  1. $(a _1+a _2)$

  2. $(a _1-a _2)$

  3. $(a^2 _1+a^2 _2)$

  4. $(a^2 _1-a^2 _2)$


Correct Option: C
Explanation:

 $intensity \ \alpha \ (amplitude)^{2}$
so maximum intensity is proportional to $a^{2} _{1}+a^{2} _{2}$
option $C$ is correct 

The max. intensity produced by two coherent sources of intensity  $I _2$ and $I _2$ will be 

  1. I$ _1 + I _2$

  2. $ I _1^2 + I _2^2$

  3. $ I _1 + I _2$ + 2$\sqrt{I _1I _2}$

  4. zero


Correct Option: C
Explanation:

As R$^{2}$ = a$^{2}$ + b$^{2}$ + 2 ab cos $\phi$
$\therefore$ I$ _{max}$ = I$ _1$ + I$ _2$ + 2$\sqrt{I _1I _2}$ cos 0$^{o}$ 
= I$ _1$ + I$ _2$ + 2$\sqrt{I _1I _2}$

Young's double slit experiment is conducted with light of wavelength $\lambda $. The intensity of the bright fringe is $I _{o}$ . The intensity at a point, where path difference is $\lambda $ /4 is given by :

  1. $zero$

  2. $I _{o}/8 $

  3. $I _{o}/4 $

  4. $I _{o}/2 $


Correct Option: D
Explanation:

In YDSE, intensity from both slits are same ,
So, $I _{0}=I _{max}=I+I+2\sqrt{II}=4I$
Now at path difference $\dfrac{\lambda }{4}$,
$\Delta \phi =\dfrac{2\pi }{4}=\dfrac{\pi }{2}$
So, $I=I+I+2\sqrt{\pm I}cos(\dfrac{\pi }{2})$
$=2I+2\sqrt{II}(0)    (\because cos\pi /2=0)$
$=2I$
So, $I=\dfrac{I _{0}}{2}$

In Young's double slit experiment, the intensity of light at a point on the screen where the path difference '$\lambda $' is 'K' units. The intensity of light at a point where the path difference is $\dfrac{\lambda}{3} $ is ($\lambda $ being the wavelength of light used)

  1. K/2

  2. K/4

  3. K

  4. K/3


Correct Option: B
Explanation:

phase difference corresponding to path difference of $\lambda /3$ is
$\phi =\dfrac{2\pi }{3}$
So, $I=k cos^{2}(\dfrac{2\pi /3}{2})     (\because I=I _{0} cos^{2}(\phi /2))$
         $=k(+1/2)^{2}     (\because cos \pi /3=1/2)$
         $=\dfrac{k}{4}$

In an interference experiment, phase difference for points where the intensity is minimum is (n = 1, 2, 3 ...)

  1. $n \pi$

  2. $(n\, +\, 1) \pi$

  3. $(2n\, +\, 1) \pi$

  4. zero


Correct Option: C
Explanation:

Intensity at a point due to interference of beams of intensities $I _1,I _2$ with a phase difference $\phi$ between them=$I _1+I _2+2\sqrt{I _1I _2}cos\phi$

The value of the resultant intensity is minimum for $cos\phi=-1$
$\implies \phi=(2n+1)\pi$ for positive integer values of $n$

Two identical light waves, propagating in the same direction, have a phase difference $\delta $. After they superpose the intensity of the resulting wave will be proportional to

  1. $\cos { \delta } $

  2. $\cos { \left( \dfrac { \delta }{ 2 } \right) } $

  3. $\cos ^{ 2 }{ \left( \dfrac { \delta }{ 2 } \right) } $

  4. $\cos ^{ 2 }{ \delta } $


Correct Option: C
Explanation:

Maximum intensity,
     $I=4{ I } _{ 0 }\cos ^{ 2 }{ \left( \dfrac { \delta  }{ 2 }  \right)  } $
$\Rightarrow I\propto \cos ^{ 2 }{ \left( \dfrac { \delta  }{ 2 }  \right)  } $

In the Young's double slit experiment, the resultant intensity at a point on the screen is 75% of the maximum intensity of the bright fringe. Then the phase difference between the two interfering rays at that point is 

  1. $\frac {\pi}{6}$

  2. $\frac {\pi}{4}$

  3. $\frac {\pi}{3}$

  4. $\frac {\pi}{2}$


Correct Option: C
Explanation:

The correct answer is option(C).

We know, The resultant intensity,
$I _R=4I _{max}cos^2\left( \frac \phi 2\right)$
$or, cos^2\left( \frac \phi 2\right)=\frac {I _R}{4I _max}=\frac {0.75}4=0.1875$
$\Rightarrow cos\left(\frac \phi 2\right)=\sqrt {0.1875}=0.5$
$\Rightarrow \frac \phi 2 = cos _{-1}(0.5)=\frac \pi 3$

Two beams of light having intensities I and 4 I interface to produce a fringe pattern on a screen. The phase difference between the beams is $\dfrac { \pi  }{ 2 }$ at point A and $\pi$ at point B, Then the difference between the resultant intensities at A and B is

  1. 2 I

  2. 4 I

  3. 5 I

  4. 7I


Correct Option: B
Explanation:

Resultant intensity at a point, $I _R= I _1+ I _2 +2\sqrt{I _1I _2} cos \phi$

where $I _1$ and $I _2$ be the intensities of two sources and $\phi$ be the phase differences of the sources at that point.
Here, $I _1= I$ and $I _2=4I$

At point $A$, $\phi= \dfrac{\pi}2$
So,Resulant intensity at point $A$,   $I _A= 5I+ 2\sqrt{4I^2}cos \dfrac{\pi}2= 5I$   

At point $B$, $\phi=\pi$

So,Resulant intensity at point $B$,   $I _B= 5I+ 2\sqrt{4I^2}cos\pi= 5I-4I= I$

Hence, Required difference between the intensities at $A$ and $B= I _A-I _B= 5I-I= 4I$

In Young's double slit experiment, the two slits act as coherent sources of waves of equal amplitude $A$ and wavelength $\lambda$. In another experiment with the same arrangement, the two slits are made to act as incoherent sources of waves of same amplitude and wavelength. If the intensity at the middle point of the screen in the first case is $I _1$ and in the second case is $I _2$, then the ratio $I _1/I _2$ is: 

  1. $0.5$

  2. $4$

  3. $2$

  4. $1$


Correct Option: A

Two beams of light having intensities $I$ and $4I$ interfere to produce a fringe pattern on a screen.If the phase difference between the beams is $\dfrac{\pi }{2}$ at point A and $\pi$ at point B then the difference between the resultant intensities at A and B is 

  1. 4I

  2. 2I

  3. 5I

  4. 7I


Correct Option: A
Explanation:

The resultant intensity is given by:
$I={I} _{1}+{I} _{2}+2\sqrt{{I} _{1}{I} _{2}}\cos{\phi}$
Thus difference is given by:
${I} _{A}-{I} _{B}=2\sqrt{{I} _{1}{I} _{2}}(\cos{\dfrac{\pi}{2}}-\cos{\pi})=2\sqrt{4{I}^{2}}\times1=4I$

- Hide questions