Question

Using the expression $2d\;sin\theta=\lambda$, one calculates the value of $d$ b measuring the corresponding angles $\theta$ in the range $0^{\circ}$ to $90^{\circ}$. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all values of $\theta$. As $\theta$ increases from $0^{\circ}$,
A. the absolute error in $d$ remains constant.
B. the absolute error in $d$ increases
C. the fractional error in $d$ remains constant
D. the fractional error in $d$ decreases

Answer 1

Hint: Young’s double slit experiment explains the wave nature of light using the interference of the light waves coming from two slits. The distance between the two slits is comparable to the magnitude of the wavelength.
Formula: $2d\;sin\theta=\lambda$

Complete answer:
Given that $2d\;sin\theta=\lambda$ and angles $\theta$ in the range $0^{\circ}$ to $90^{\circ}$.
Clearly from the given equation, we can say that $d\propto\dfrac{1}{sin\theta}$.
Thus, as the value increases from the range $0^{\circ}$ to $90^{\circ}$, the value of the $d$ decreases. Then we can also say that the fractional error in $d$ also decreases.

Hence, the answer is D. the fractional error in $d$ decreases.

Additional information:
In Young's double slit experiment; the distance between the two coherent sources is comparable to the wavelength. Due to the path difference between the light coming from both the slits, interference pattern is observed at the screen which is placed away from the sources.
The path difference is given as $\Delta x=\dfrac{xd}{D}$, where $x$is the position of the fringe from the origin, $d$ is the distance between the fringes and $D$ is the distance between the slits and source.
Then during constructive interference,$\Delta x=n\lambda$.i.e. causes the bright fringe and during destructive interference $\Delta x=(2n+1)\dfrac{\lambda}{2}$.i.e. cause the dark fringe. Where $\lambda$ is the wavelength of the coherent source.

Note:
Constructive interference pattern occurs when the troughs or the crests of the two coherent sources interfere. These results in addition to their amplitude, hence the fringe is bright. Similarly, Destructive interference pattern occurs when one trough and one crest of the two coherent sources interfere. These result in decreasing their amplitude, hence the fringe is dark.