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Light from He-Ne laser (𝜆 = 633 nm) is used to illuminate two narrow slits. The interference pattern is observed on a screen located 3.0 m behind the slits. Eleven bright fringes are seen, spanning 5.2 cm. a. Can a small angle approximation be used? b. What is the separation between the slits?

Question

Light from He-Ne laser (𝜆 = 633 nm) is used to illuminate two narrow slits. The interference pattern is observed on a screen located 3.0 m behind the slits. Eleven bright fringes are seen, spanning 5.2 cm. a. Can a small angle approximation be used? b. What is the separation between the slits?

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Solution

a. The small angle approximation can be used when the angle is less than about 10 degrees. In this case, the angle can be calculated using the formula θ = y/L, where y is the distance spanned by the 11 fringes (5.2 cm or 0.052 m) and L is the distance from the slits to the screen (3.0 m).

θ = 0.052 m / 3.0 m = 0.0173 radians

Converting this to degrees gives θ = 0.0173 * (180/π) = 0.99 degrees, which is less than 10 degrees. So, yes, the small angle approximation can be used.

b. The separation between the slits, d, can be calculated using the formula for double-slit interference:

d = mλ / sinθ

where m is the order of the fringe (for the 11th fringe, m = 11), λ is the wavelength of the light (633 nm or 633 x 10^-9 m), and θ is the angle calculated above.

First, we need to calculate sinθ. Using the small angle approximation, sinθ ≈ θ, so sinθ ≈ 0.0173.

Then, we can substitute these values into the formula:

d = 11 * (633 x 10^-9 m) / 0.0173 = 4.05 x 10^-6 m or 4.05 μm

So, the separation between the slits is approximately 4.05 μm.

This problem has been solved

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