Young's Double Slit Experiment (YDSE)

This experiment predicted the wave-like nature of light and other waves. Young's double slit experiment is pivotal in affirming the wave-like behaviour of light, validating the wave theory of light propagation. It’s findings were instrumental in developing wave optics and contributed significantly to understanding quantum mechanics, particularly wave-particle duality. The observed interference pattern directly supports the wave nature of light, which is integral to understanding phenomena like diffraction and interference. This experiment cannot be fully explained by a particle (photonic) model of light alone, highlighting the necessity of wave-based models to comprehend light's behaviour.

1.0Explanation of Young's Double Slit Experiment

• Light Source: A coherent light source, typically a laser, emits light waves of a single wavelength (monochromatic light).
• Barrier with Two Slits: A thin barrier with two very narrow parallel slits is placed in the path of the light beam. The slits are very close together, usually on the order of micrometres to millimetres.
•  Screen:  A screen is placed some distance away from the double slit where the light waves can be observed after passing through the slits.

Observations and Explanation: When the coherent light passes through the double slits, it diffracts (spreads out) as it passes through each slit. After passing through the slits, the diffracted waves from each slit interfere with each other. This interference produces an interference pattern on the screen. The interference pattern consists of alternating bright and dark fringes (bands of light and darkness) where the light waves either reinforce (constructive interference) or cancel out (destructive interference) each other.

Constructive interference-It occurs when the crests of one wave align with the crests of another wave, or when the troughs of one wave align with the troughs of another wave. This alignment results in the combined amplitudes and maximum intensity of the resulting wave.

 (ஃ Intensity $\propto$ Amplitude²)

Destructive Interference

When the crest of one wave coincides with the trough of another wave, or when the trough of one wave coincides with the crest of another, their amplitudes subtract, resulting in minimal intensity.

Crest and Trough Of a Wave

Young's Double Slit Experiment Diagram

2.0Interference  of Light Observing From YDSE

Interference

Light interference is the phenomenon where light energy is redistributed due to the superposition of light waves from two coherent sources.

3.0Superposition Principle

The superposition principle is a foundational concept in physics, particularly in studying wave phenomena like interference. It asserts that when multiple waves overlap within a given space, the resultant wave function at any point is determined by the arithmetic combination of the individual wave functions of these overlapping waves. This principle elucidates how waves interact, allowing us to predict their combined effects based on their individual properties and phases. There are two types of superposition: constructive superposition and destructive superposition, represented by a diagram.

Coherent and Incoherent Sources:

1. Coherent sources: Coherent sources are defined as two light sources emitting continuous waves of the same frequency or wavelength, maintaining either zero or a constant phase difference between them. They obtained from a single source.
2. Incoherent sources: Two sources of light which do not emit light waves with a constant phase difference are called incoherent sources.

4.0Conditions for Constructive And Destructive Interference

$\text{ ( }\therefore \text{ Phase difference }(\varphi )=\frac{2\pi }{\text{ wavelength }(x)}\times \text{ Path Difference }(\Delta x)$

Condition for Constructive Interference:

• Resultant Intensity at a point is given by $l_R=l_1+l_2+2\sqrt{I_1{I}_2}\mathrm{Cos}\varphi$
• $2\sqrt{I_1{I}_2}\mathrm{Cos}\varphi =\text{ Interference Factor }$
• For Constructive interference the resultant intensity at a point P will be maximum $\mathrm{Cos}\varphi =1\text{ or }\varphi =0,2\pi ,4\pi \dots \dots$
• $\varphi =0,2\pi ,4\pi \dots$
• $\Delta x=0,\lambda ,2\lambda ,3\lambda ,\dots ..=n\lambda$
• The resultant intensity at a point P is maximum when the phase difference between two superposing wave is an even multiple of $\pi$ or path difference is an integral multiple of wavelength $\lambda .$

Conditions for Destructive Interference:

• The resultant Intensity at point P will be minimum 
• $\cos \varphi =-1\text{ or }\varphi =\pi ,3\pi ,5\pi \dots ....$
• $\varphi =\pi ,3\pi ,5\pi \dots \dots ..=(2n-1)\pi$
• $\Delta x=\frac{\lambda }{2},\frac{3\lambda }{2},\frac{5\lambda }{2}\dots =(2n-1)\frac{\lambda }{2}$
• The resultant intensity at point is minimum when the phase difference between the two superposing waves is an odd multiple of $\pi$ or the path difference is a multiple of an odd number $\frac{\lambda }{2}$.

5.0Fringe Width Formula

Fringe width: It refers to the distance between two consecutive bright or dark fringes.

Fringe Width  Formula: Expression for Fringe in Young's Double Slit Experiment

• $\beta =\frac{D\lambda }{d}$
• $\beta$ is independent of n ( the order of fringes) all the fringes are of equal width
• $\lambda =\text{ Extremely small }$
• D is much larger than d

Position of Bright Fringes

$\left(\therefore \text{ Path difference, }(\Delta x)=S_2\mathrm{P}-S_1\mathrm{P}\right)$

• $\Delta x=\frac{xd}{D}$
• $\Delta x=\frac{xd}{D}=\mathrm{n}\lambda$
• $x_n=\frac{nD\lambda }{d}$, Position of n^th bright Fringe

Position of Dark Fringes:

• $\Delta x=\frac{xd}{D}=(2n-1)\frac{\lambda }{2}$
• $x_n^{\prime }=(2n-1)\frac{D\lambda }{2d}$, Position of nth dark Fringe

6.0Conditions for Sustained Interference

•  In an Interference pattern, the positions of the intensity maxima and minima on the observation screen that do not change with time are called sustained interference.

Several critical conditions must be met for sustained interference to be observed:

• The waves must exhibit coherence, indicating a consistent phase relationship over time. This coherence is essential for maintaining a stable and predictable interference pattern.
• Ideally, the waves should originate from a monochromatic source, meaning they have a single, well-defined wavelength λ.This simplifies the interference pattern, making distinguishing between constructive and destructive interference effects easier.
• The amplitude of the waves must remain constant. Fluctuations in amplitude can obscure or alter the interference pattern, affecting its visibility and stability.

7.0Intensity Distribution Curve for Interference

8.0Conservation of Energy in Interference

In interference phenomena, waves exhibit constructive interference when their amplitudes add together, leading to regions of higher Intensity known as bright fringes. Conversely, destructive interference occurs when their amplitudes are subtracted, resulting in regions of lower Intensity called dark fringes. Despite these variations in Intensity across the interference pattern, the total energy remains conserved. The energy that appears reduced in dark fringes due to destructive interference is precisely offset by the increased energy in bright fringes caused by constructive interference. This conservation principle ensures that energy is neither created nor lost during wave interference, maintaining the overall energy balance within the system.

9.0Solved Questions on Interference

Q-1.  No interference pattern is detected when two coherent sources are positioned infinitely close to each other.

Solution:

Fringe width, $\beta =\frac{\lambda D}{d}$

$\beta \propto \frac{1}{d}$

When $d\to 0,\beta \to \infty$

The fringe width is considerable, and an interference pattern cannot be observed.

Q-2. Find the maximum intensity resulting from the coherent interference of n identical waves, each with an intensity I₀?

Solution:

${\mathrm{I}}_{\mathrm{R},}={\mathrm{I}}_1+{\mathrm{I}}_2+2\sqrt{I_1{I}_2}\mathrm{Cos}\varphi$

For maximum intensity, $\mathrm{Cos}\varphi =1$

${\mathrm{l}}_{\mathrm{MAX}}={\mathrm{l}}_1+{\mathrm{l}}_2+2\sqrt{I_1{I}_2}={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2$

When n identical wave of each intensity I₀ interfere

${\mathrm{I}}_{\mathrm{MAX}}={\left(\sqrt{I_0}+\sqrt{I_0}+\sqrt{I_0}+\dots \dots .\mathrm{n}\text{ terms }\right)}^2={\left(\mathrm{n}\sqrt{I_0}\right)}^2$

$I_{MAX}=n^2{I}_0$

Q-3. Suppose the separation between the two slits in Young's Double Slit Experiment is decreased while maintaining the screen position unchanged. What impact will this change have on the width of the interference fringes?

Solution:

Fringe width, $\beta =\frac{\lambda D}{d}$

Separation (d)  between two slits decreases fringe width $(\beta )$ increases.

Q-4. In Young's double-slit experiment, green, red, and blue light are used individually, and the fringe width is arranged in increasing order for each colour.

Solution:

Fringe width, $\beta =\frac{\lambda D}{d}$ 

${\lambda }_{\text{red }}>{\lambda }_{\text{green }}>{\lambda }_{\text{blue }}$

${\beta }_{\text{red }}>{\beta }_{\text{gren }}>{\beta }_{\text{blue }}$

Q-5. Young's double-slit experiment employs a monochromatic source, and the appearance of the interference fringes observed on the screen is described.

Solution:

In Young's Double Slit Experiment, the interference pattern typically shows fringes that are hyperbolic in shape. However, in a narrower interference pattern, the fringes appear straight.