Question

Light from a helium-neon laser \((630 \mathrm{nm})\) is incident on a pair of slits. In the interference pattern on a screen \(1.5 \mathrm{m}\) from the slits, the bright fringes are separated by \(1.35 \mathrm{cm} .\) What is the slit separation? [Hint: Is the small angle approximation justified?]

Short answer

Answer: The slit separation in the given double-slit interference experiment is 70 μm.

Step-by-step solution

Recall the double-slit interference formula

The formula for double-slit interference is: mλ = dsinθ Where: m is the order of the bright fringe (m = 0, 1, 2, ...) λ is the wavelength of the light d is the slit separation θ is the angle between the central maximum and the mth-order fringe

Apply the small angle approximation

The small angle approximation states that for very small angles (less than around 5 degrees), sinθ ≈ tanθ ≈ θ (where θ is in radians). In this problem, we are given the distance from the screen and the separation between bright fringes, so we can use this approximation to find θ.

Determine the separation between the central maximum and mth-order fringe on the screen

Let's call the separation between bright fringes "y". We have y = 1.35 cm. Since consecutive bright fringes are separated by y, we can use the linear relationship: y = Ltanθ Where: L is the distance from the slits to the screen (1.5 m) In our case, with the small angle approximation, we have: y = Lθ

Calculate the angle θ

Now we can solve for θ using the given values: θ = y / L = 0.0135 m / 1.5 m = 0.009 radians Since the angle is very small, the small angle approximation is justified.

Substitute the known values in the double-slit interference formula

We are given λ = 630 nm. We'll convert this to meters (recall that 1 nm = 10^{-9} m): λ = 630 * 10^{-9} m Now, we can plug in the values for m, λ, and θ into the double-slit interference formula: 1λ = d * sinθ Since the small angle approximation applies, we can write: 1(630 * 10^{-9} m) = d * 0.009

Solve for the slit separation d

Now, we just need to solve for d: d = (630 * 10^{-9} m) / 0.009 ≈ 7 * 10^{-5} m = 70 μm So, the slit separation is 70 μm.