Question

In a Fresnel’s biprism experiment, the refracting angles of the prism were ${{1.5}^{{}^\circ }}$ and the refractive index of the glass was 1.5. With the single slit 5 cm from the biprism, and using light of wavelength 580 nm, fringes were formed on a screen 1 m from the single slit. The fringe width is
(a). 0.44 cm
(b). 4.4 cm
(c). 0.044 cm
(d). 0.0044 cm

Answer 1

- Hint: Equation used for finding the fringe width is, $\beta =\dfrac{D\lambda }{d}$. Distance between the coherent sources has to be found out from the following equation, $d=2a(n-1)A$. Deviations produced by each half of the prism is $(n-1)A$.

Complete step-by-step solution -

The diagram below shows Fresnel's biprism experiment.

Let’s write the given data from the question.
Refractive index of the glass, $n=1.5$
Distance between the single slit and the bi-prism, $a=5\times {{10}^{-2}}$
Refracting angle of the prism, $A={{1.5}^{{}^\circ }}$,
For the calculation purposes, A should be in the radian form, so $A=1.5\times \dfrac{\pi }{180}$
That is, $A=0.0259$
Wavelength of the light, $\lambda =580\times {{10}^{-9}}m$
Distance between the screen and single slit, $D=1m$
Fringe width, $\beta =\dfrac{D\lambda }{d}$……. (equation 1), where $d$ is the distance between the sources.
$d=2a(n-1)A$……… (equation 2)
Assigning the values into the equation 2, $d=2\times 5\times {{10}^{-2}}\times (1.5-1)\times 0.0259$
$d=0.1295\times {{10}^{-2}}$
Assigning this value into the equation 1,
Then, $\beta =\dfrac{1\times 580\times {{10}^{-9}}}{0.1295\times {{10}^{-2}}}$
$\beta =0.0447\times {{10}^{-2}}m=0.0447cm$
So, the final answer is option (c) $\beta =0.044cm$

Additional information:
Fresnel’s bi-prism is a method for measuring the wavelength of light. In this method monochromatic light falls on the bi-prism and creates two virtual images. These two virtual images will behave like the sources. Their overlap will produce an interference pattern on the screen. These fringes appeared brighter than the Young’s slits, since the large amount of light passage through the prism.

Note: Angle should be in the radian form, otherwise the answer will be wrong. Candidates have to identify correctly the d and D from the question. Both are used to specify certain distances. D represented as the distance between the screen slit, while d used for the distance between the coherent sources. Don’t misplace the values in the equation.