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Interference Of Light - Condition And Types - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Interference of light waves- 1, Interference of light waves- 2 is considered one of the most asked concept.

  • 33 Questions around this concept.

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QuestionMEDIUM

In a Young's double slit experiment using red and blue lights of wavelengths \mathrm{600 \mathrm{~nm}} and \mathrm{480 \mathrm{~nm}} respectively, the value of \mathrm{n} for which the \mathrm{\mathrm{n}^{\text {th }}} red fringe coincides with (\mathrm{n}+1)^{\mathrm{th}} blue fringe is:

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In an interference experiment using waves of same amplitude, path difference between the waves at a point on the screen is \mathrm{\lambda / 4.} The ratio of intensity at this point to that at the central bright fringe is:

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The path difference between two interfering waves at a point on the screen is \mathrm{\lambda / 8}. The ratio of the intensity at this point and that at the central fringe will be:

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Out of the following, which can be used as coherent sources required to produce interference pattern of light? 

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A plane monochromatic light falls normally on a diaphragm with two narrow slits separated by a distance \mathrm{d=2.5 \mathrm{~mm}.} A fringe pattern is formed on the screen placed at \mathrm{D=100 \mathrm{~cm}} behind the diaphragm. If one of the slits is covered by a glass plate of thickness \mathrm{10 \mu \mathrm{m},} then distance by which these fringes will be shifted will be

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In the below diagram, CP represents a wavefront and AO and BP, the corresponding two rays. Find the condition of \mathrm{\theta} for constructive interference at P between the rays BP and reflected ray AOP:

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It is found that when waves of same intensity from two coherent sources superpose at a certain point, then the resultant intensity is equal to the intensity of one wave only. This means that the phase difference between the two waves at the point is: 

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Two coherent light sources A and B are at a distance \mathrm{3 \lambda} from each other \mathrm{(\lambda= wavelength).} The distances from \mathrm{A} on the X-axis at which the interference is constructive are:

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Two coherent sources separated by distance d are radiating in phase having wavelength \lambda. A detector moves in a big circle around the two sources in the plane of the two sources. The angular position of \mathrm{n=4 } interference maxima is given as:

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Concepts Covered - 2

Interference of light waves- 1

In order to observe interference in light waves, the following conditions must be met:

  •  The sources must be coherent.
  • The source should be monochromatic (that is, of a single wavelength).

Coherent sources-

Two sources are said to be coherent if they produce waves of the same frequency with a constant phase difference.  

  • The relation between  Phase difference (\Delta \phi )  and Path difference  (\Delta x )

Phase difference (\Delta \phi ):

The difference between the phases of two waves at a point is called phase difference.

\text { i.e. if } y_{1}=a_{1} \sin \omega t \text { and } y_{2}=a_{2} \sin (\omega t+\phi) \text { so phase difference }=\phi

Path difference (\Delta x ): 

The difference in path lengths of two waves meeting at a point is called path difference between the waves at that point.

And The relation between  Phase difference (\Delta \phi )  and Path difference  (\Delta x ) is given as

   \Delta \phi =\frac{2\pi }{\lambda }\Delta x=k\Delta x

where \lambda =wavelength \ of \ waves

 

 

Interference of light waves- 2

Principle of Super Position-

According to the principle of Super Position of waves, when two or more waves meet at a point, then the resultant wave has a displacement (y) which is the algebraic sum of the displacements ( y_1 \ \ and \ \ y_2) of each wave.

i.e y=y_1+y_2

consider two waves with the equations as

 \begin{array}{l}{y_{1}=A_{1} \sin (k x-w t)} \\ {y_{2}=A_{2} \sin (k x-w t+\phi)}\end{array}

where \phi is the phase difference between waves y_1 \ \ and \ \ y_2.

And According to the principle of Super Position of waves

\begin{aligned} y=& y_{1}+y_{2}=A_{1} \sin (k x-w t)+A_{2} \sin (k x-w t+\phi) \\ &=A_{1} \sin (k x-w t)+A_{2}[\sin (k x-w t) \cos \phi+\sin \phi \cos (k x-\omega t)] \\ \Rightarrow y &=\sin (k x-w t)\left[A_{1}+A_{2} \cos \phi\right]+A_{2} \sin \phi \cos (k x-w t) \dots (1) \end{aligned}

Now let

 Acos\theta =A_{1}+A_{2} \cos \phi \\ and \ \ Asin\theta=A_{2} sin \phi

Putting this in equation (1) we get 

y=A \sin (k x-\omega t) \cos \theta+A \sin \theta \cos (k x-\omega t)

thus we get the equation of the resultant wave as

y=A \sin (k x-\omega t+\theta)

where A=Resultant amplitude of two waves

and A= \sqrt{{A_{1}}^{2}+{A_{2}}^{2}+2A_{1}A_{2}\cos \phi}

and \theta=\tan^{-1} \left (\frac{A_{2} \sin \phi }{A_{1}+A_{2} \cos \phi} \right )

where 

A_{1}= the amplitude of wave 1

A_{2}=  the amplitude of wave 2

  • \begin{array}{c} \ \ A_{\max }={A_{1}+A_{2}} \ \ and \ \ A_{\min }={A_{1}-A_{2}}\end{array}

Resultant Intensity of two waves (I)-

Using I \ \ \alpha \ \ A^2

we get  I= I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos \phi

where 

I_{1}= The intensity of wave 1

I_{2}= The intensity of wave 2

  • \begin{aligned} \ \ I_{\max } &=I_{1}+I_{2}+2 \sqrt{I_{1} I_{2}} \Rightarrow I_{\max } &=(\sqrt{I_{1}}+\sqrt{I_{2}})^{2} \end{aligned}
  • \begin{array}{c}{I_{\min }=I_{1}+I_{2}-2 \sqrt{I_{1} I_{2}}} \Rightarrow {I_{\min }=(\sqrt{I_{1}}-\sqrt{I_{2}})^{2}} \end{array}
  • For identical sources-

          I_{1}=I_{2}=I_{0} \Rightarrow I=I_{0}+I_{0}+2 \sqrt{I_{0} I_{0}} \cos \phi=4 I_{0} \cos ^{2} \frac{\phi}{2}

  • \text { Average intensity : } I_{a v}=\frac{I_{\max }+I_{\min }}{2}=I_{1}+I_{2}
  • The ratio of maximum and minimum intensities

\frac{I_{\max }}{I_{\min }}=\left(\frac{\sqrt{I_{1}}+\sqrt{I_{2}}}{\sqrt{I_{1}}-\sqrt{I_{2}}}\right)^{2}=\left(\frac{\sqrt{I_{1} / I_{2}}+1}{\sqrt{I_{1} / I_{2}}-1}\right)^2=\left(\frac{a_{1}+a_{2}}{a_{1}-a_{2}}\right)^{2}=\left(\frac{a_{1} / a_{2}+1}{a_{1} / a_{2}-1}\right)^{2}

or

 \sqrt{\frac{I_{1}}{I_{2}}}=\frac{a_{1}}{a_{2}}=\left(\frac{\sqrt{\frac{I_{\max }}{I_{\min }}}+1}{\sqrt{\frac{I_{\max }}{I_{\min }}-1}}\right)

Interference of Light-

It is of the following two types.

1. Constructive interference-

  • When the waves meet a point with the same phase, constructive interference is obtained at that point.

            i.e we will see bright fringe/spot.

  • The phase difference between the waves at the point of observation is  \phi=0^{\circ} \text { or } 2 n \pi

  • Path difference between the waves at the point of observation is \Delta x=n \lambda(i . e . \text { even multiple of } \lambda / 2)

  • The resultant amplitude at the point of observation will be maximum

      \begin{array}{c} i.e \ \ A_{\max }={a_{1}+a_{2}} \\ {\text { If } a_{1}=a_{2}=a_{0} \Rightarrow A_{\max }=2 a_{0}}\end{array}

  • Resultant intensity at the point of observation will be maximum

   \begin{aligned} \ \ i.e \ \ I_{\max } &=I_{1}+I_{2}+2 \sqrt{I_{1} I_{2}} \\ I_{\max } &=(\sqrt{I_{1}}+\sqrt{I_{2}})^{2} \\ \text { If } \quad I_{1} &=I_{2}=I_{0} \Rightarrow I_{\max }=4 I_{0} \end{aligned}

 

2. Destructive interference-

  • When the waves meet a point with the opposite phase, Destructive interference is obtained at that point.

            i.e we will see dark fringe/spot.

  • The phase difference between the waves at the point of observation is 

        \begin{array}{l}{\phi=180^{\circ} \text { or }(2 n-1) \pi ; n=1,2, \ldots} \\ {\text { or }(2 n+1) \pi ; n=0,1,2 \ldots .}\end{array}  

  • Path difference between the waves at the point of observation is \Delta x=(2 n-1) \frac{\lambda}{2}(\text { i.e. odd multiple of } \lambda / 2)

  • The resultant amplitude at the point of observation will be minimum

      \begin{array}{c} i.e \ \ A_{\min }={A_{1}-A_{2}} \\ {\text { If } A_{1}=A_{2} \Rightarrow A_{\min }=0}\end{array}

  • Resultant intensity at the point of observation will be minimum

    \begin{array}{c}{I_{\min }=I_{1}+I_{2}-2 \sqrt{I_{1} I_{2}}} \\ {I_{\min }=(\sqrt{I_{1}}-\sqrt{I_{2}})^{2}} \\ {\text { If } I_{1}=I_{2}=I_{0} \Rightarrow I_{\min }=0}\end{array}

Study it with Videos

Interference of light waves- 1
Interference of light waves- 2

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