Question 64: Wave Optics

Question

Given below are two statements :
Statement I
: In a Young's double slit experiment, the angular separation of fringes will increase as the screen is moved away from the plane of the slits
Statement II
: In a Young's double slit experiment, the angular separation of fringes will increase when monochromatic source is replaced by another monochromatic source of higher wavelength
In the light of the above statements, choose the
correct answer
from the options given below :

Accepted Answer

In YDSE, the linear position of the $\mathrm{n}^{	ext {th }}$ bright fringe on the screen is given by :
$$ \mathrm{y}_{\mathrm{n}}=\frac{\mathrm{n} \lambda \mathrm{D}}{\mathrm{~d}} $$
Where :
$\lambda=$ wavelength of light used.
$\mathrm{D}=$ distance between the slits and the screen.
$\mathrm{d}=$ distance between the two slits.
The linear fringe width $(\beta)$, which is the distance between two consecutive bright or dark fringes, is :
$$ \beta=y_{n+1}-y_n=\frac{\lambda D}{d} $$
The angular separation (or angular fringe width) $	heta_\beta$ is the angle subtended by the fringe width at the slits. For small angles :
$$ 	heta_\beta \approx \frac{\beta}{D} $$
Substituting the value of $\beta$ :
$$ 	heta_\beta=\frac{\left(\frac{\lambda \mathrm{D}}{\mathrm{~d}}ight)}{\mathrm{D}}=\frac{\lambda}{\mathrm{d}} $$
Statement I says that angular separation $\left(	heta_\betaight)$ increases as the screen is moved away (D increases).
From the formula $	heta_\beta=\lambda / d$, we can see that $	heta_\beta$ depends only on the wavelength ( $\lambda$ ) and the slit separation (d).
$$ 	heta_\beta \propto \lambda \ldots 	ext { (i), } \quad 	heta_\beta \propto \frac{1}{d} \ldots 	ext { (ii) } $$
It is independent of the distance to the screen (D).
Therefore, Statement I is false.
Statement II says that $	heta$ increases when a source of higher wavelength is used.
From the formula $	heta_\beta \propto \lambda$, i.e., if $\lambda$ increases, $	heta$ must also increase.
Therefore, Statement II is true.
Therefore, Statement I is false and Statement II is true.
Hence, the correct option is (C).

Answer

Both
Statement I
and
Statement II
are true

Answer

Statement I
is true but
Statement II
is false

Answer

Statement I
is false but
Statement II
is true

Answer

Both
Statement I
and
Statement II
are false

Wave Optics

Choose the correct answer: