A laser beam passes through a thin slit.When the pattern is viewed on a screen 1.25 m past the slit,you observe that the fifth-order dark fringes occur at ±2.41 cm from the central bright fringe.The entire experiment is now performed within a liquid,and you observe that each of the fifth-order dark fringes is 0.790 cm closer to the central fringe than it was in air.What is the index of refraction of this liquid?
A) 1.33
B) 1.40
C) 1.49
D) 1.62
E) 3.05

Question

Quizplus · Accepted Answer

The distance between the fifth-order dark fringes and the central bright fringe is given by:
y = (nλL)/d
where n is the order of the fringe, λ is the wavelength of light, L is the distance from the slit to the screen, and d is the width of the slit.

From the given data, we have:
y5 = 2.41 cm = (5λL)/d  --- (1)

Now, when the experiment is performed in a liquid, the wavelength of light changes according to:
λ' = λ/n
where n is the refractive index of the liquid.

Also, the distance between the fringes is reduced by a factor of n:
y'5 = y5/n

From equation (1), we have:
d = (5λL)/y5

Substituting for d in terms of y'5 and λ' in terms of λ, we get:
y'5 = (5λ'L)/(n^2y5)

Substituting the given values, we get:
0.790 cm = (5λ'L)/(n^2(2.41 cm))
=> n = sqrt[(5λ'L)/(0.790 cm(2.41 cm))] = 1.49

Therefore, the index of refraction of the liquid is 1.49 (option C).

A Laser Beam Passes Through a Thin Slit