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The index of refraction of a glass plate is 1.48 at $θ_{1} = 30^{0} C$ and varies linearly with temperature with a coefficient of $2.5 \times 10^{- 5}^{0} C^{- 1}$ . The coefficient of linear expansion of the glass is $5 \times 10^{- 5}^{0} C^{- 1}$ at $30^{0} C$ , the length of the glass plate is 3cm. this plate is placed in front of one of the sits in Young’s double-slit experiment. If the plate is being heated so that its temperature increases at a rate of $5^{0} C^{- 1} \min$ , the light source has wavelength $λ = 589 n m$ and the glass plate initially is at $θ = 30^{0} C$ . The number of fringes that shift on the screen in each minute is nearly (use approximation)

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110

$1.1 \times 10^{3}$

answer is B.

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Detailed Solution

$μ_{0} = r e f r a c t i v e i n d e x o f g l a s s p l a t e a t 30 ° C$

$μ = f i n a l r e f r a c t i v e i n d e x o f g l a s s p l a t e a f t e r h e a t i n g$

$t_{0} = i n i t i a l t h i c k n e s s o f g l a s s s p l a t e a t 30 ° C$

$t = f i n a l t h i c k n e s s o f g l a s s p l a t e a f t e r h e a t i n g$

Path difference $= μ t - μ_{0} t_{0} = n λ_{0}$

Where n is the number of fringes that shift on the screen

$\Rightarrow \frac{μ_{0} (1 + α_{1} θ) t_{0} (1 + α_{2} θ) - μ_{0} t_{0}}{λ_{0}} = n$

$\frac{μ_{0} t_{0} (α_{1} + α_{2}) θ}{λ_{0}} = n$

Given, $μ_{0} = 1.48, t_{0} = 3 \times 10^{- 2} m$

$α_{1} = 2.5 \times 10^{- 5}^{0} C^{- 1}$

$α_{2} = 0.5 \times 10^{- 8}^{0} C^{- 1}, θ = 5^{0} \min^{- 1}$

$λ_{0} = 589 n m$

$∴ n = \frac{1.48 \times 3 \times 10^{- 2} (3 \times 10^{- 5}) \times 5}{589 \times 10^{- 9}} = 11$

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