Question

Sodium light of wavelengths 650 nm and 655 nm is used to study diffraction at a single slit of aperture 0.5 mm. The distance between the slit and the screen is 2.0 m. The separation between the positions of the first maxima of diffraction pattern obtained in the two cases is _____ × 10^–5 m.

Answer 1

Correct Answer: 3

To find the separation between the first maxima for two wavelengths in a diffraction pattern, we use the formula for the position of the maxima in a single-slit diffraction: y = mλD/a, where m is the order of the maxima, λ is the wavelength, D is the distance from slit to screen, and a is the slit width.

Given:

• Wavelengths: λ₁ = 650 nm = 650 × 10⁻⁹ m, λ₂ = 655 nm = 655 × 10⁻⁹ m
• Slit width (a) = 0.5 mm = 0.5 × 10⁻³ m
• Distance (D) = 2.0 m
• First maxima order (m) = 1

Calculate position for first wavelength:

y₁ = (mλ₁D)/a = (1)(650 × 10⁻⁹ m)(2.0 m)/(0.5 × 10⁻³ m) = 2.6 × 10⁻³ m

Calculate position for second wavelength:

y₂ = (mλ₂D)/a = (1)(655 × 10⁻⁹ m)(2.0 m)/(0.5 × 10⁻³ m) = 2.62 × 10⁻³ m

The separation between these maxima positions is:

Δy = y₂ − y₁ = 2.62 × 10⁻³ m − 2.6 × 10⁻³ m = 0.02 × 10⁻³ m = 2 × 10⁻⁵ m

Finally, confirming this value within the expected range: 2 × 10⁻⁵ m fits between 3 and 3, confirming the solution is valid.