A Short Note on Derivation of Young’s Double Slit Experiment

Interference and monochromatic light are two of the most important concepts in Young’s Double Slit Experiment. Interference is the phenomenon that occurs when two waves meet, and their amplitudes add together. Monochromatic light is light that has a single wavelength. In this blog post, we will derive Young’s Double Slit Experiment using interference and monochromatic light. Stay tuned!

What is  Young’s Double Slit Experiment?

The double-slit experiment is a classic physics demonstration that shows the wave-like nature of light. A monochromatic light source (usually a laser) is directed at a screen with two parallel slits cut into it. The interference pattern created by the light passing through the two slits is projected onto a second screen placed equidistant from the slits.

Derivation of Young’s Double Slit Experiment

Here is the derivation of Young’s Double Slit Experiment:

We consider two slits of width a, separation d and illuminated by monochromatic light of wavelength λ. The light from each slit travels equal distances L to reach point P on the screen as shown in the figure. Let us denote the path difference between the two waves by Δp = |OP₁ – OP₂|. We have,

OP₁ = L + y₁ sinθ₁ and OP₂ = L + y₂ sinθ₂

We also know that the slits are equidistant from the screen, i.e., y₁ = -y₂ = a/sinθ

Substituting this in the above expression, we get

OP₁ = L – a cotθ₁ and OP₂ = L + a cotθ₁

Therefore, the path difference is given by,

Δp = |OP₁ – OP₂| = |L – a cotθ₁ + L + a cotθ₁| = |- a cotθ₁ + a cotθ₁| = 0

This implies that the path difference is always zero and hence there is no interference.

However, if we consider the interference pattern formed on the screen, we see that it consists of bright and dark fringes. This can be explained as follows:

The intensity of light at any point on the screen is given by,

I = I₀ [sin²(π/λΔp) + cos²(π/λΔp)]

where I₀ is the intensity of light incident on the screen.

Now, for path difference Δp = 0, we have

I = I₀ [sin²(0) + cos²(0)] = I₀ (cos² 0 + sin² 0) = I₀

Therefore, the intensity is maximum at points where Δp = 0. These are called interference maxima or bright fringes.

Similarly, for path difference Δp = λ/n (where n is an integer), we have

I = I₀ [sin²(nπ/λ) + cos²(nπ/λ)] = I₀ [sin²(nπ) + cos²(nπ)] = I₀ (0 + (-(-))^n)

= I₀ (0 + (-•)^n) = I₀ (0 – 0) = 0

Therefore, the intensity is zero at points where Δp = λ/n. These are called interference minima or dark fringes.

From the above discussion, we can see that interference can only occur when the path difference between the two waves is equal to an integer multiple of the wavelength λ. This condition is known as the interference condition.

Now, we will derive the interference condition for Young’s double-slit experiment.

Suppose we have two slits of width a and separation d as shown in the figure. The light from each slit travels equal distances L to reach point P on the screen. Let us denote the path difference between the two waves by Δp = |OP₁ – OP₂|.

Now, we know that the slits are equidistant from the screen, i.e., y₁ = -y₂ = a/sinθ

Substituting this in the expression for path difference, we get

Δp = |OP₁ – OP₂| = |L + a/sinθ – (L – a/sinθ)| = |a(cotθ₁ + cotθ₂)|

Now, for interference to occur, the path difference Δp must be equal to an integer multiple of the wavelength λ. Therefore, we have

Δp = nλ ⇒ a(cotθ₁ + cotθ₂) = nλ

This is the interference condition for Young’s double-slit experiment.

We can see from the interference condition that if the path difference Δp is equal to an integer multiple of the wavelength λ, then interference will occur. This condition is known as the interference condition.

Conclusion

In conclusion, the interference pattern in Young’s double-slit experiment can be derived by considering the wave nature of light. When monochromatic light shines through two equidistant slits, the waves from each slit interfere with each other to create a characteristic interference pattern on a screen. This interference is due to constructive and destructive interference of the waves and is a result of the wave nature of light. This interference pattern is a clear demonstration of the wave nature of light and is one of the most famous experiments in physics. It has been used to study a variety of phenomena, including diffraction and interference. The experiment is also a great example of how two seemingly simple objects, like two slits, can produce such a complex interference pattern. Thank you for reading!