Learn the Young’s Double-Slit Experiment Derivation

Light has the properties of both the wave as well as the particle. The wave nature of light allows the light to travel in the form of waves and has different properties such as wavelength, amplitude, and interference. Interference is the property of light in which two or more waves of light meet each other and depending upon the properties they either cancel out or amplify each other. This property of light is visualized by an experiment known as Young’s Double Slit Experiment. This topic of Physics will be dealing with Young’s Double Slit Experiment derivation in detail.

But first, let’s study the experiment.

Young’s Double-Slit Experiment

Young’s double slit experiment is an experiment to demonstrate the different interference patterns of light.

In this experiment, light from a monochromatic source is made to pass through two slits in an opaque object. The slits are at a specific distance apart. The result obtained is an interference pattern upon the screen kept at a distance from the two slits.

The above image is a representation of Young’s Double Slit Experiment.

Young’s Double Slit Experiment Derivation

In Young’s Double Slit Experiment the slits are separated from each other by very less distance. To understand the interference pattern of the light, the phase of the light source coming out of the slit is not taken into consideration since both slits have light from the same source. Thus, for the interference pattern, only the phase difference of the light reaching the screen is important because the light from the two slits reaches any point on the screen after covering a different amount of distance.

Thus, if one requires to find the path difference for any interference pattern at point P, it can be derived as follows:

The angle between the centre line and the point ‘P’ is .

According to the rule of similar triangles, the angle between the slit distance ‘d’ and the perpendicular from the slit S1 is also .

The distance from the slit S2 to the foot of the perpendicular gives the path difference. This distance can be found by considering the smaller triangle made by the slit distance ‘d’, the perpendicular from S1 and the distance between the slit S2 and the foot of the perpendicular.

In this triangle

Since for constructive interference the path difference is equal to , where ‘m’ is an integer and the is the wavelength.

For constructive interference, the path difference between the two waves should be

Since for destructive interference, the path difference is equal to the odd integral which is equal to .

For destructive interference, the path difference between the two waves should be

Formula for Constructive and Destructive Interference

The formula for path difference for constructive interference is

The formula for path difference for destructive interference is

Where,

m = Integer

= Wavelength of the light 

d = Distance between the center of the slits

= Angle of the point from the center line

If the distance between the slits is larger than the wavelength of the light, then

Since

We get,

for Constructive interference

for Destructive Interference

Fringe’s Role in Young’s Double Slit Experiment

The bright and the light bands are known as the fringes of the light. The brightest fringe is at the centre line where ‘m’ is equal to 0.

To know the fringe width, which is the distance between two adjacent bright fringes the formula is as follows :

The intensity of the fringes can be known from the formula which is

The position of any bright fringe (for example nth bright fringe) from the centre is given by the formula

The position of any dark fringe (for example nth dark fringe) from the centre is given by the formula

Conditions To Get a Stable Interference Pattern

A stable interference pattern is achievable by keeping these points in mind which are:

• The two sources of light which is the slit must be close to each other
• The two sources of light must have the same phase, equal amplitudes and same intensity.
• The light sources must have the same frequency.
• The two sources of light must be monochromatic.
• The two sources of light should emit in the same direction.

Solved Examples

Example 1. Young's double slit experiment is first performed in air and then in a medium other than air. It is found that 8th bright fringe in the medium lies where 5th dark fringe lies in the air. What is the refractive index of the medium?

Solution:

In medium

For Air, 5th dark,

 

m=5, 

For bright in medium m

 

∴ 

y′=y

∴

μ=1.78

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