(b) OR: (i) Two light waves from coherent sources, each of amplitud...

Question

(b) OR: (i) Two light waves from coherent sources, each of amplitude 'a' and frequency $\omega$, have displacements $y_1=a\cos\omega t$ and $y_2=a\cos(\omega t+\phi)$, where $\phi$ is the phase difference between them. These waves superpose at a point. Obtain the expression for the resultant intensity at that point.

(ii) In Young's double-slit experiment, when waves from the two slits reach two points on a screen with path differences of (i) $\lambda/6$ and (ii) $\lambda/12$, find the ratio of intensities at these points.

Accepted Answer

Step 1 (i): Resultant $y=y_1+y_2$; resultant amplitude $R=\sqrt{a^2+a^2+2a^2\cos\phi}=2a\cos(\phi/2)$, so $I=4I_0\cos^2(\phi/2)$, where $I_0\propto a^2$.

Step 2 (ii): Phase difference $\phi=\dfrac{2\pi}{\lambda}\times(\text{path difference})$.

Step 3: For path difference $\lambda/6$: $\phi_1=\dfrac{2\pi}{\lambda}\times\dfrac\lambda6=\dfrac\pi3$; $I_1=4I_0\cos^2(\pi/6)=4I_0\times\dfrac34=3I_0$.

Step 4: For path difference $\lambda/12$: $\phi_2=\dfrac{2\pi}{\lambda}\times\dfrac\lambda{12}=\dfrac\pi6$; $I_2=4I_0\cos^2(\pi/12)$. $\cos(\pi/12)=\cos15°\approx0.966$, so $\cos^2(15°)\approx0.933$; $I_2\approx4I_0\times0.933=3.73I_0$.

Step 5: Ratio $I_1:I_2 = 3I_0 : 3.73I_0 \approx 0.80:1$ (or equivalently $I_2/I_1\approx1.24$).

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