Diffraction & Interference of light Test - 1

A light wave that has vibrations in more than one plane is known as unpolarized light.

• Polarized waves are light waves in which the vibrations occur in a single plane.
• A Nicol prism is made up of two prisms of calcite (CaCO3).
• Nicol prism is designed in such a way that it eliminates the ordinary rays by total internal reflection and only allow the extraordinary rays to transmit through it.

​Notes:

• Nicol prism is used for producing a polarised beam of light from an un-polarised beam. Hence, option 2 is correct.
• It is based on the principle of action which involves refraction as it passes into the lower half of the prism.
• It leaves the prism as polarised light after undergoing another refraction as it exits the far right side of the prism. Thus its action is based on double refraction.

Concept:

Brewster angle

The angle of incidence at which a beam of unpolarized light falling on a transparent surface is reflected as a beam of completely plane polarised light is called polarising or Brewster angle. It is denoted by ip.

The British Physicist David Brewster found the relationship between Brewster angle (ip) and refractive index (μ) –

μ = tan ip

This relation is known as Brewster law.

Calculation:

Given:

Critical angle,

\({i_c} = {\sin ^{ - 1}}\left( {\frac{3}{5}} \right)\;\;\;\;\therefore \sin {i_c} = \frac{3}{5}\)

As \(\mu = \frac{1}{{\sin {i_c}}} = \frac{5}{3}\) 

According to Brewster’s law

tan i_p = μ

Where i_p is the polarising angle

∴ \(\tan {i_p} = \frac{5}{3}\;{i_p} = {\tan ^{ - 1}}\left( {\frac{5}{3}} \right)\)

CONCEPT:

• Polarization: A property of transverse waves that gives the geometrical orientation of the wave oscillations is called Polarization.
  • In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave.
• Polaroids can be used to control the intensity, in sunglasses windowpane, etc. Polaroids are used in photographic cameras and 3D movie cameras.

EXPLANATION:

• When we wear sunglasses without polarization, the lenses reduce the amount of light only that is transmitted through the lens both horizontally and vertically.
• If we wear polarized lenses, the glasses absorb horizontal light waves by chemical eliminate the pattern and still allows vertical waves to pass through. Hence it reduces intensity.

• Basically, light only travels in one direction through polarized lenses which then eliminates glass.
• So, we can say that light intensity can be reduced to half by polarization.
• So the correct answer will be option 1.

CONCEPT:

• Diffraction of light at single slit: In the case of diffraction at single slit, we get central bright band with alternate bright and dark bands.
  • Width of the central maxima β₀ = 2λ D/d
  • Angular width of central maxima b sin θ = λ
  • All the secondary fringes are of same width but the central fringe having width double.
  • Width of secondary fringes = λ D/d

CALCULATION:

Given, θ = π/6 = 30° Width of the slit (d) = 1.2 μm = 1.2 × 10^-6 m

We know, b sin θ = n λ

For central maxima n = 1

⇒ b sin θ = λ

\(\lambda = 1.2 \times {10^{ - 6}} \times \sin 30^\circ = 1.2 \times {10^{ - 6}} \times \frac{1}{2} = 6 \times {10^{ - 7}}m = 6000 \times {10^{ - 10}}\;m \) = 6000 Å

Concept:

Intensity ∝ (Amplitude)²

\(\frac{{{I_1}}}{{{I_2}}}\; = \;{\left( {\frac{{{A_1}}}{{{A_2}}}} \right)^2}\)

The maximum intensity of interference:

\({I_{max}}\; = \;{\left( {{A_1} + {A_2}} \right)^2}\; = \;{\left( {\sqrt {{I_1}} + \sqrt {{I_2}} } \right)^2}\)

Minimum intensity of the interference

\({I_{min}}\; = \;{\left( {{A_1} - {A_2}} \right)^2}\; = \;{\left( {\sqrt {{I_1}} - \sqrt {{I_2}} } \right)^2}\)

Calculation:

\(\frac{{{I_1}}}{{{I_2}}}\; = \;{\left( {\frac{{{A_1}}}{{{A_2}}}} \right)^2}\; = \;\frac{2}{1} \)

\(\Rightarrow \frac{{{A_1}}}{{{A_2}}}\; = \;\sqrt {\frac{{{I_1}}}{{{I_2}}}} \; = \;\frac{{\sqrt 9 }}{1}\; = \;\frac{3}{1}\)

\( \Rightarrow {A_1}\; = \;3{A_2}\)

\(\frac{{{I_{max}}}}{{{I_{min}}}}\; = \;\frac{{{{\left( {{A_1} + {A_2}} \right)}^2}}}{{{{\left( {{A_1} - {A_2}} \right)}^2}}}\; = \;\frac{{{{\left( {\frac{{{A_1}}}{{{A_2}}} + 1} \right)}^2}}}{{{{\left( {\frac{{{A_1}}}{{{A_2}}} - 1} \right)}^2}}}\;\)

\( = \;\frac{{{{\left( {3 + 1} \right)}^2}}}{{{{\left( {3 - 1} \right)}^2}}}\; = \;\frac{{16}}{4}\; = \;\frac{4}{1}\)

CONCEPT:

• Wavefront: The locus of all particles in a medium, vibrating in the same phase is called waveFront.
• The direction of propagation of light (ray of light) is perpendicular to the waveFront.
• Every point on the given wavefront acts as a source of a new disturbance called secondary wavelets which travel in all directions with the velocity of light in the medium.
• A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new wavefront at that instant. This is called a secondary wavefront.

EXPLANATION:

• Huygens suggested that light may be a wave phenomenon produced by mechanical vibrations of an all-pervading homogeneous medium called ether just like those in solids and liquids. So each particle at a wavefront behaves as a new light source. So option 3 is correct.

Concept:

• Bright fringes or maxima are formed when there is constructive interference. It happens when the path difference between waves (difference in distance traveled) is 

∆x = n λ, n = 0, 1, 2....

λ is the wavelength of light, n is an integer representing the number of the particular interference.

• Dark fringes or minima are formed when there is destructive interference. It happens when the path difference between waves is 

\(\Delta x = (n + \frac{1}{2}) \lambda , \; n = 0, 1, 2...\)

Calculation:

We have to find find the path difference  ∆x when the second minima (dark fringe) is formed.

putting n = 1 in the formula of the dark fringe

\(\Delta x = (1 + \frac{1}{2}) \lambda \)

\(\Delta x = (\frac{3}{2}) \lambda\)

So, 3 λ/2  is the correct option.

Important Points

Young's double-slit Experiment: The Young's double-slit Experiment was done by Thomas Young in 1851.

• It is a famous experiment to understand the phenomena of interference of light.
• It exhibits the wave nature of light.
• Two light rays from a coherent source are allowed to pass through from two slits kept at a small distance to form interference patterns at a screen kept a distance from the slit. 

​

CONCEPT:

Young's double-slit experiment

• Young’s double-slit experiment helped in understanding the wave nature of light.
• The original Young’s double-slit experiment used diffracted light from a single monochromatic source of light.
• The light that comes from the monochromatic source is passed into two slits to be used as two coherent sources.
• At any point on the screen at a distance ‘y’ from the center, the waves travel distances l1 and l2 to create a path difference of Δl at that point.
• If there is a constructive interference on the point then the bright fringe occurs.
• If there is a destructive interference on the point then the dark fringe occurs.

​

Fringe width (β):

• The separation between any two consecutive bright or dark fringe is called fringe width.
• In Young’s double-slit experiment all fringes are of equal width.
• In Young’s double-slit experiment the fringe width is given as,

\(⇒ β=\frac{λ D}{d}\)

Where d = distance between slits, D = distance between slits and screen, and λ = wavelength

CALCULATION:

Given λ = 580 nm = 580 × 10^-9 m, d = 0.30 mm = 0.30 × 10-3 m, and D = 2m

• In Young’s double-slit experiment the fringe width is given as,

\(⇒ β=\frac{λ D}{d}\)     -----(1)

So the width of the central fringe is given as,

⇒ W = 2β

\(⇒ W=\frac{2λ D}{d}\)

\(⇒ W=\frac{2\times580\times10^{-9}\times2}{0.3\times10^{-3}}\)

⇒ W = 7.7 × 10–3 m

• Hence, option 4 is correct.

CONCEPT:

• Interference: When two light sources from different coherent sources meet together, then there is a distribution of energy is disturbed by each other this superposition of two light waves is called Interference of light waves.
  • The interference of light gives the idea of superposition of waves, it shows its wave nature.
  • Coherent source: The source of light have a constant phase difference is called a coherent source.
• Young's Double Slit Experiment (YDSE): Monochromatic light (single wavelength) falls on two narrow slits S1 and S2 which are very close together act as two coherent sources when waves coming from two coherent sources, (S1, S2) superimposes on each other, an interference pattern is obtained on the screen.

EXPLANATION:

• Since the interference pattern can be observed by the wave only.
• That's why we can say that the young's double-slit experiments manifest the wave nature of light. So option 2 is correct.

EXTRA POINTS:

The following are the conditions for the interference to happen:

• The source should emit light waves continuously.
• Light waves emitted should have a single wavelength.
• Waves should either have a constant phase difference or be in phase.
• The light sources should be close to each other and narrow.

• Photoelectric effect: The phenomenon in which the light energy forces a metal surface to release electrons is called the photoelectric effect.
  • When the light hits, it shows the particle theory of light, and light is defined as a stream of photons or energy packets.
  • The other phenomenon such as interference, diffraction, and polarization can only be explained when the light is treated as a wave wherein the photoelectric effect, line spectra, and the production and scattering of x rays demonstrate the particle nature of light.

CONCEPT:

• Interference: When two light sources from different coherent sources meet together, then there is a distribution of energy is disturbed by each other this superposition of two light waves is called Interference of light waves.
• Diffraction: The Phenomenon of bending of light the sharp edges of the size of its wavelength is called diffraction.

EXPLANATION:

• Since the interference and diffraction pattern can be observed by the wave only.
• That's why we can say that the young's double-slit experiments and diffraction by single slit manifest the wave nature of light. So option 2 is correct.

Additional Information

The following are the conditions for the interference to happen:

• The source should emit light waves continuously.
• Light waves emitted should have a single wavelength.
• Waves should either have a constant phase difference or be in phase.
• The light sources should be close to each other and narrow.

• Photoelectric effect: The phenomenon in which the light energy forces a metal surface to release electrons is called the photoelectric effect.
  • When the light hits, it shows the particle theory of light, and light is defined as a stream of photons or energy packets.
  • The other phenomenon such as interference, diffraction, and polarization can only be explained when the light is treated as a wave wherein the photoelectric effect, line spectra, and the production and scattering of x rays demonstrate the particle nature of light.