Question

In Newton's experiment the radii of the \[{m^{th}}\] and \[{\left( {m + 4} \right)^{th}}\] dark rings are respectively \[\sqrt 5 \;{\rm{mm}}\] and \[\sqrt 7 \;{\rm{mm}}\]. What is the value of m?
A. 2
B. 4
C. 8
D. 10

Answer 1

Hint: The above problem can be resolved by using the fundamentals of the newton's rings experiment to determine the pattern of the interference created by the reflection of light between the two specified surfaces. The mathematical relation for the radius of the dark ring is used, which significantly relates to the aperture radius, and the light's wavelength is reflected. The two conditions can be applied. The data regarding the radius of the dark ring is given. The given data can be used for the substitution in the formula, and the desired result obtained by taking the ratio of the equations.

Complete step by step answer:
We know that the expression for the radius of the \[{m^{th}}\] dark ring is given as,
\[{r_m} = \sqrt {mR\lambda } \]
Here, R is the radius of the aperture and \[\lambda \] is the wavelength of light.
From the given condition, the radius of \[{m^{th}}\] fringe is,
\[\sqrt 5 \;{\rm{mm}} = \sqrt {mR\lambda } ...............................................\left( 1 \right)\]
 Similarly, the radius of \[{\left( {m + 4} \right)^{th}}\] ring is,
\[\sqrt 7 \;{\rm{mm}} = \sqrt {\left( {m + 4} \right)R\lambda } ...............................................\left( 2 \right)\]
Dividing the equation 2 and 1 as,
\[\begin{array}{l}
\sqrt {\dfrac{{m + 4}}{m}} = \sqrt {\dfrac{7}{5}} \\
\dfrac{{m + 4}}{m} = \dfrac{7}{5}\\
5m + 20 = 7m\\
m = 10
\end{array}\]

So, the correct answer is “Option D”.

Note:
Try to understand the fundamentals and Newton's ring experiment's applications to carry out the conclusion to determine the interference pattern. The interference is that phenomena in which the light rays of different or a similar wavelength and the frequency coincide to provide a resultant ray. Moreover, the interference patterns are determined by taking the rings' radius during the experiment.