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Physics · 2024 · 2 marks
CBSE 2024 · Region 2 · Set 1 · Q18
(a) Two waves, each of amplitude 'a' and frequency ' ω ' emanating from two coherent sources of light superpose at a point. If the phase difference…
(a)
Two waves, each of amplitude 'a' and frequency ' $\displaystyle \omega$ ' emanating from two coherent sources of light superpose at a point. If the phase difference between the two waves is $\displaystyle \phi$, obtain an expression for the resultant intensity at that point.
OR
(b)
What is the effect on the interference pattern in Young's double-slit experiment when (i) the source slit is moved closer to the plane of the slits, and (ii) the separation between the two slits is increased? Justify your answers.
Marking-scheme solution
$\displaystyle x_{1}=a \cos \omega t$
$\displaystyle x_{2}=a \cos (\omega t+\phi)$
$\displaystyle x=x_{1}+x_{2}$
$\displaystyle =a(\cos \omega t+\cos (\omega t+\phi))$
$\displaystyle =a\left(2 \cos \left(\omega t+\frac{\phi}{2}\right) \cos \frac{\phi}{2}\right)$
$\displaystyle =2 a \cos \frac{\phi}{2} \cos \left(\omega t+\frac{\phi}{2}\right)$
Intensity:
$\displaystyle I=K(\text { amplitude })^{2}$ where K is a constant.
$\displaystyle =K\left(2 a \cos \frac{\phi}{2}\right)^{2}$
$\displaystyle =4 I_{0} \cos ^{2} \frac{\phi}{2}$
$\displaystyle I_{0}=K a^{2}=$ intensity of each incident wave.
(b)
Sharpness of the interference pattern decreases.
$\displaystyle \frac{s}{S}<\frac{\lambda}{d}$
As S decreases, interference patterns produced by different parts of the source overlap and finally fringes disappear.
As the source slit is brought closer to the plane of the slits, the screen gets illuminated uniformly and fringes disappear.
(ii)
$\displaystyle \beta=\frac{\lambda D}{d}$
As d increases, $\displaystyle \beta$ decreases and fringes disappear.
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