Fringe Width Formula

In Young’s double slit experiment, the position of the nth order bright fringe is given by \( y_n = \frac{n \lambda D}{d} \), where \( \lambda \) is the wavelength, D is the slit-to-screen distance, and d is the slit separation.

The fringe width \( \beta = \frac{\lambda D}{d} \) represents the separation between consecutive fringes of the same wavelength.

For different wavelengths, the 2nd order positions differ, so the separation is \( |y_{2,\lambda_2} – y_{2,\lambda_1}| = 2D \left| \frac{\lambda_2 – \lambda_1}{d} \right| \).

Step-by-Step Calculation

Convert units: d = 0.05 mm = 5 × 10^-5 m, D = 2 m, λ₁ = 450 nm = 4.5 × 10^-7 m, λ₂ = 520 nm = 5.2 × 10^-7 m.

2nd order position for λ₁: \( y_{2,1} = \frac{2 \times 4.5 \times 10^{-7} \times 2}{5 \times 10^{-5}} = 0.036 \) m.

2nd order position for λ₂: \( y_{2,2} = \frac{2 \times 5.2 \times 10^{-7} \times 2}{5 \times 10^{-5}} = 0.0416 \) m.

Separation: |0.0416 – 0.036| = 0.0056 m = 0.56 cm.

Python Verification Code

d = 0.05e-3  # m
D = 2  # m
lambda1 = 450e-9  # m
lambda2 = 520e-9  # m

y1_2nd = 2 * lambda1 * D / d
y2_2nd = 2 * lambda2 * D / d

separation = abs(y2_2nd - y1_2nd)
separation_cm = separation * 100

print(f"Separation: {separation_cm} cm")

Output: Separation: 0.56 cm

SEO-Friendly Article

In Young’s double slit experiment fringe separation calculations, understanding interference patterns from multiple wavelengths like 450 nm and 520 nm is crucial for competitive exams such as CSIR NET and JEE. This setup with 0.05 mm slit separation and 2 m screen distance demonstrates wave optics principles through precise 2nd order bright fringes positioning.

Core Principles

• Light waves from two slits interfere constructively at bright fringes where path difference δ = nλ, yielding \( y_n = \frac{n \lambda D}{d} \).
• Fringe width β remains constant for one wavelength but shifts positions for λ₁ = 450 nm (blue) and λ₂ = 520 nm (green), causing distinct patterns.
• Factors affecting separation: slit distance d inversely proportional, screen distance D directly proportional, and Δλ = 70 nm driving the difference.

Detailed Solution Walkthrough

Exam Tips

• Practice variations: If d halves, separation doubles; if D increases, fringes spread.
• For CSIR NET life sciences students exploring biophysics, this links to diffraction in microscopy.
• Common pitfalls include forgetting the ‘2’ for 2nd order or unit mismatches (nm to m, mm to m).