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Wave Optics - Coherent and Incoherent Addition of Waves

Grade 12CBSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Principle of Superposition states that when two or more wave pulses overlap, the resultant displacement at any point is the algebraic sum of the displacements due to the individual waves, represented as y=y1+y2y = y_1 + y_2.

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Coherent sources are sources that emit light waves of the same frequency and have a constant phase difference ϕ\phi between them. Coherence is a necessary condition for observing a stable interference pattern.

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Incoherent sources are those where the phase difference between the waves changes randomly and rapidly with time. Most common light sources (like two independent bulbs) are incoherent.

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Constructive Interference occurs when two waves meet in phase (phase difference ϕ=2nπ\phi = 2n\pi, where n=0,1,2,...n = 0, 1, 2, ...). This results in maximum intensity.

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Destructive Interference occurs when two waves meet out of phase (phase difference ϕ=(2n+1)π\phi = (2n+1)\pi, where n=0,1,2,...n = 0, 1, 2, ...). This results in minimum intensity.

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For incoherent addition, the average value of the interference term 2I1I2cos⁡ϕ2\sqrt{I_1 I_2} \cos \phi over time is zero because ϕ\phi varies randomly. Thus, the resultant intensity is simply the sum of individual intensities: I=I1+I2I = I_1 + I_2.

📐Formulae

y=a1sin⁡(ωt)+a2sin⁡(ωt+ϕ)y = a_1 \sin(\omega t) + a_2 \sin(\omega t + \phi) Rose

A2=a12+a22+2a1a2cos⁡ϕA^2 = a_1^2 + a_2^2 + 2a_1 a_2 \cos \phi

I=I1+I2+2I1I2cos⁡ϕI = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi

Imax=(I1+I2)2∝(a1+a2)2I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \propto (a_1 + a_2)^2

Imin=(I1−I2)2∝(a1−a2)2I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \propto (a_1 - a_2)^2

Phase Difference (ϕ)=2πλ×Path Difference (Δx)\text{Phase Difference } (\phi) = \frac{2\pi}{\lambda} \times \text{Path Difference } (\Delta x)

💡Examples

Problem 1:

Two coherent sources of light have an intensity ratio of 81:181:1. Calculate the ratio of the maximum to minimum intensity in the interference pattern.

Solution:

Given I1I2=811\frac{I_1}{I_2} = \frac{81}{1}. Since I∝a2I \propto a^2, the ratio of amplitudes is a1a2=811=9\frac{a_1}{a_2} = \sqrt{\frac{81}{1}} = 9. Using the formula for intensity ratio: ImaxImin=(a1+a2)2(a1−a2)2\frac{I_{max}}{I_{min}} = \frac{(a_1 + a_2)^2}{(a_1 - a_2)^2} Substituting the values: ImaxImin=(9+1)2(9−1)2=10282=10064=2516\frac{I_{max}}{I_{min}} = \frac{(9 + 1)^2}{(9 - 1)^2} = \frac{10^2}{8^2} = \frac{100}{64} = \frac{25}{16}

Explanation:

First, we find the amplitude ratio from the intensity ratio by taking the square root. Then, we apply the formula for maximum and minimum intensity which depends on the sum and difference of the amplitudes respectively.

Problem 2:

Two waves of equal intensity I0I_0 from two coherent sources meet at a point with a path difference of λ3\frac{\lambda}{3}. What is the resultant intensity at that point?

Solution:

First, find the phase difference ϕ\phi: ϕ=2πλ×Δx=2πλ×λ3=2π3\phi = \frac{2\pi}{\lambda} \times \Delta x = \frac{2\pi}{\lambda} \times \frac{\lambda}{3} = \frac{2\pi}{3} The resultant intensity formula is: I=I1+I2+2I1I2cos⁡ϕI = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi Given I1=I2=I0I_1 = I_2 = I_0: I=I0+I0+2I02cos⁡(2π3)I = I_0 + I_0 + 2\sqrt{I_0^2} \cos\left(\frac{2\pi}{3}\right) I=2I0+2I0(−12)=2I0−I0=I0I = 2I_0 + 2I_0 \left(-\frac{1}{2}\right) = 2I_0 - I_0 = I_0

Explanation:

We calculate the phase difference using the given path difference. Then, we substitute the phase difference into the general intensity equation for coherent sources.