In a double slit experiment, the two slits are 1 mm apart and the screen is place 1 m away. A monochromatic light of wavelength 500 nm is used. What will be the width of each slit for obtaining ten maxima of double slit within the central maxima of single slit pattern?

\\(d = 10^{-3}\:m\\)\\(D=1\:m\\)\\( \lambda = 500 \times 10^{-9}\:m\\)The width of central maxima in a single slit diffraction: \\(\dfrac{2 \lambda D}{a}\\)Fringe width in double slit pattern: \\(\beta = \dfrac{\lambda D}{d}\\)Given,\\(10 \beta = \dfrac{2 \lambda D}{a}\\)\\( \Rightarrow 10 \dfrac{\lambda D}{d} = \dfrac{2 \lambda D}{a}\\)\\( \Rightarrow a =\dfrac{d}{5}\:mm= 0.2\:mm\\)

A beam of light consisting of two wavelengths, \\(650 nm\\) and \\(520 nm\\), is used to obtain interference fringes in a Young's double-slit experiment.(a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength \\(650 nm\\).(b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide? The distance between the two slits is \\(0.28mm\\) and the screen is at a distance of \\(1.4m\\) from the slits.

GivenThe wavelength of the light is \\( \lambda_1 = 650 nm\\)The wavelength of seond light,\\( \lambda_2 = 520 nm\\)Distance between the slit and the screen is \\(1.4m\\).Distance between the slits is \\(0.28mm\\).\\((a)\\)The relation between the \\(n^{th}\\) bright fringe and the width of fringe is:\\(x = n\lambda_1\dfrac Dd\\)For third bright fringe, \\(n = 3\\)\\( x = 3 \times 650\dfrac {1.4}{0.28\times10^{-3}}= 1950 \times5\times10^3 nm\\)\\(x=9.75\times10^{-3} m\\)\\(=9.75mm\\)\\((b)\\)We can consider that \\(n^{th}\\) bright frind of \\(\lambda_2\\) and the \\((n-1)^{th}\\) bright fringe of wavelength \\(\lambda_1\\) coincide with each other.\\(n\lambda_2 = (n - 1) \lambda_1\\)\\(520n = 650n - 650\\)\\(650 = 130n\\)\\( n = 5\\)therefore, the least distance from the central maximum can be obtained as:\\(x' = n\lambda_2 \dfrac Dd \\)\\(x'= 5 \times 520 \dfrac Dd = 2600 \dfrac {1.4}{0.28\times10^{-3}} nm\\)\\(x'=1.30\times10^{-2}m=1.3cm\\)

The maximum intensity in Young's double slit experiment is \\({I}_{0}\\). Distance between the slits is \\(d=5\lambda\\), where \\(\lambda\\) is the wavelength of monochromatic light used in the experiment. What will be the intensity of light in front of one of the slits on a screen at a distance \\(D=10d\\)?

Path difference \\(=\Delta x=\dfrac{xd}{D}\\)-------(i)In front of one of the slits,\\(x=\dfrac{d}{2}\\) but \\(d=5\lambda\\)\\(x=\dfrac{5\lambda}{2}\\) and \\(D=10d=50\lambda\\)Now, from equation (i), we have:\\(\Delta x=\dfrac{5\lambda}{2}\times\dfrac{5\lambda}{50\lambda}=\dfrac{\lambda}{4}\\)So, corresponding phas defference:\\(\phi=\dfrac{2\pi}{\lambda}.(\Delta x)=\dfrac{2\pi}{\lambda}\times\dfrac{\lambda}{4}=\dfrac{\pi}{2}\\)As, \\(I=I_0\cos^2(\dfrac{\phi}{2})\\)So, \\(I=I_0\cos^2(\dfrac{\pi}{4})=\dfrac{I_0}{2}\\)

Two coherent monochromatic light beams of intensities \\(I\\) and \\(4I\\) are superposed. The maximum and minimum possible resulting intensities are :

Maximum intensity, \\(I_{max} = (\sqrt {I_{1}} + \sqrt {I_{2}})^{2}\\)\\(= (\sqrt {I} + \sqrt {4I})^{2}\\)\\(= 9I\\)Minimum intensity \\(I_{min} = (\sqrt {I_{1}} - \sqrt {I_{2}})^{2}\\)\\((\sqrt {I} - \sqrt {4I})^{2}\\)\\(= I\\)

In a Youngs double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.

Distance between the slits, \\(d=0.28\times10^{-3}\ m\\)Distance between the slits and the screen, \\(D = 1.4 m\\)Distance between the central fringe and the fourth \\((n = 4)\\) fringe, \\(u=1.2\times10^{-2}\ m\\)In case of a constructive interference, we have the relation for the distance between the two fringes as:\\(u=n\lambda D/d\\)\\(\Rightarrow \lambda=ud/nD=6\times10^{-7}m=600nm\\)

The interference pattern is obtained with two coherent light source of intensity ratio n. In the interference pattern, the ratio \\(\dfrac{I_{max}-I_{min}}{I_{max}+I_{min}}\\) will be

Given : \\(I_2 = n I_1\\)Maximum intensity of interference \\(I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2\\)\\(\therefore\\) \\(I_{max} = (\sqrt{I_1} + \sqrt{nI_1})^2 = (1+\sqrt{n})^2 I_1 = (1+ n +2\sqrt{n } ) I_1\\)Minimum intensity of interference \\(I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2\\)\\(\therefore\\) \\(I_{min} = (\sqrt{I_1} - \sqrt{nI_1})^2 = (1-\sqrt{n})^2 I_1 = (1+ n - 2\sqrt{n})I_1\\)\\(\therefore\\) \\(\dfrac{I_{max} -I_{min}}{I_{max} + I_{min}} = \dfrac{2\sqrt{n} - (-2\sqrt{n})}{2(1+n) } = \dfrac{2\sqrt{n}}{1+n}\\)

Derive the expression for the intensity at a point where interference of light occurs. Arrive at the conditions for the maximum and zero intensity.

Let \\(y_1\\) and \\(y_2\\) be displacement of two waves having same amplitude a and phase difference \\(\phi\\) between them. \\(y_1=a\sin\omega t\\)\\(y_2=a\sin(\omega t+\phi)\\)Resultant displacement is: \\(y=y_1+y_2\\)\\(y=a\sin\omega t+a\sin(\omega t+\phi)=a\sin\omega t(1+\cos\phi)+\cos\omega t(a\sin\phi) \\)\\(R\cos \theta=a(1+\cos\phi)\\)\\(R\sin \theta=a\sin\phi\\)\\(y=R\sin(\omega t +\theta)\\)Where, \\(R\\) is resultant amplitude at \\(P\\), \\(I\\) is intensity, squaring the equations we get,\\(I=R^2=a^2(1+\cos\phi)^2+a^2(\sin\phi)^2=2a^2(1+\cos\phi)=4a^2\cos^2\dfrac{\phi}{2}\\)Maximum intensity:\\(\cos^2\dfrac{\phi}{2}=1\\)\\(\phi=2n\pi\\) where \\(n=0,1,2,3,\\)....Therefore, \\(I_{max}=4a^2\\)Minmum intensity:\\(\cos^2\dfrac{\phi}{2}=0\\)\\(\phi=(2n+1)\pi\\) where \\(n=0,1,2,3,\\)....Therefore, \\(I_{max}=0\\)

In Young's double slit experiment, the slits are 2mm apart are illuminated by photons of two wavelengths \\({ I }_{ 1 }=12000\mathring { A } \\) and \\({ I }_{ 2 }=10000\mathring { A } \\). at what minimum distance from the common central bright fringe on the screen 2m from the slit will a bright fringe from one interference pattern coincide with a bright fringe from the other?

Now, from the question we can infer that\\(n_1\lambda_1 = n_2\lambda_2\\)so,\\(\frac{n_1}{n_2} = \frac{\lambda_2}{\lambda_1}\\)or \\(\frac{n_1}{n_2} = \frac{10000A}{12000A}\\)thus,we have\\(\frac{n_1}{n_2} = \frac{5}{6}\\)5th and 6th fringes will coincide respectively.the minimum distance is given as\\(X_{min} =\frac{ n_1\lambda_1D}{d}\\)here,\\(n_1 = 5\\)D = 2md = 2mm = 2\\(\times10^{-3}m\\)so,\\(X_{min} = \frac{[5\times12000\times10^{-10}\times2]}{2\times10^{-3}}\\)thus, we get\\(X_{min}\\) = 6mm hence B option is the correct answer

Young's Double Slit Experiment | Definition, Examples, Diagrams

diagram

Intensity vs height on screen in YDSE

diagram

Plot intensity vs height on screen in YDSE

example

Write intensity as a function of height from centre of screen in YDSE for two coherent light sources of equal intensity

Example: A point source is emitting light of wavelength

6000

A o

is placed at a very small height

h

above a flat reflecting surface MN as shown in the figure. The intensity of the reflected light is

36 %

of the incident intensity. Interference fringes are observed on a screen placed parallel to the reflecting surface at a very large distance

D

from it. Find the shape of the interference fringes, on the screen.

Solution:

S^{'}

represents the another point source forms due to reflection from the mirror.

S^{'} P = (x + h)^{2} + y^{2}

and

S P = (x - h)^{2} + y^{2}

Let path difference at point P be

Δ

Using,

Δ + S P = S^{'} P

Squaring and adding,

Δ^{2} + 4 h x = - 2 Δ (x - h)^{2} + y^{2}

Squaring and adding again,

Δ^{4} + 16 h^{2} x^{2} + 8 h x Δ^{2} = 4 Δ^{2} (x^{2} + h^{2} - 2 h x + y^{2})

⟹ 4 (Δ^{2} - 4 h^{2}) x^{2} - 16 Δ^{2} h x + y^{2} + 4 Δ^{2} h^{2} - Δ^{4} = 0

which represents equation of circle as

X^{2} + Y^{2} = R^{2}

Thus the fringes will be of Circular shape centered at O.

example

Problem on maximum and minimum intensity ratio calculation during interference

Example:
A point source is emitting light of wavelength

6000 A o

is placed at a very small height

h

above a flat reflecting surface MN as shown in the figure. The intensity of the reflected light is

36 %

of the incident intensity. Inference fringes are observed on a screen placed parallel to the reflecting surface at a very large distance

D

from it. Find the ratio of maximum to minimum intensities at P.

Solution:
Before reflection, intensity of light is

I_{o}

(say) and after reflection it becomes

0.36 I_{o}

I_{m a x} = I_{o} + 0.36 I_{o} + 2 I_{o} \times 0.36 I_{o} = 2.56 I_{o}

I_{m i n} = I_{o} + 0.36 I_{o} - 2 I_{o} \times 0.36 I_{o} = 0.16 I_{o}

So,

\frac{I _{m a x}}{I _{m i n}} = \frac{2 5 6}{1 6} = 16 : 1

example

Problems on points on screen with minimum and maximum intensity in YDSE

Example: A screen is placed

2 m

away from a single narrow slit. Find the slit width if the first minimum lies

5 m m

on either side of the central maximum. (wave length

= 5000 A^{\circ}

)

Solution:

x = \frac{λ \times D}{ω}

\Rightarrow

ω = \frac{λ \times D}{x}

= \frac{5 0 0 0 \times 1 0 ^{- 10} \times 2}{5 \times 1 0 ^{- 3}}

= 2 \times 1 0^{- 4} m

= 0.02 c m

diagram

Interference Pattern for white light

definition

Describe the importance of YDSE

Historically, Young's Double slit experiment, when it was first carried out by Young in 1801, demonstrated that light was a wave, which seemed to settle for once and all the debate whether light was corpuscular (particle-like) or wave. Up until that time, Newton's corpuscular was the prevailing view of light, in spite of alternate explanations such as Huygens' wave front model (which had some serious shortcomings). After Young's demonstration of the wave properties of light, followed by work by others such as Fresnel studying the interference and diffraction properties of it, and especially after Maxwell's brilliant equations of electromagnetism, physicists in the 19th century rejected Newton's corpuscular theory and believed that light was a wave.

Young's Double Slit Experiment

diagram

Intensity vs height on screen in YDSE

diagram

Plot intensity vs height on screen in YDSE

example

Write intensity as a function of height from centre of screen in YDSE for two coherent light sources of equal intensity

example

Problem on maximum and minimum intensity ratio calculation during interference

example

Problems on points on screen with minimum and maximum intensity in YDSE

diagram

Interference Pattern for white light

definition

Describe the importance of YDSE

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Important Quantities Derivation from YDSE

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