Problem 5.103 — Coherence: Number of Fringes Visible

Problem Statement

Mercury yellow doublet has wavelengths $\lambda_1 = 577$ nm and $\lambda_2 = 579$ nm. In Young’s experiment, up to what order do fringes remain visible before they wash out?

Given Information

• See problem statement for all given quantities.

Physical Concepts & Formulas

This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.

• See the step-by-step solution for the specific equations applied.
• All quantities are in SI units unless otherwise stated.

Step-by-Step Solution

Step 1 — Identify given quantities and set up the problem: Fringes wash out when the maxima of one wavelength coincide with minima of the other: $m_{max} = \lambda_1/(2\Delta\lambda)$.

Step 2 — Apply the relevant physical law or equation: $$m_{max} = \frac{\lambda_{avg}}{2(\lambda_2-\lambda_1)} = \frac{578}{2\times2} = \boxed{144.5 \approx 145\text{ orders}}$$

Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.

Worked Calculation

$$m_{max} = \frac{\lambda_{avg}}{2(\lambda_2-\lambda_1)} = \frac{578}{2\times2} = \boxed{144.5 \approx 145\text{ orders}}$$

$$\text{Numerical result} = \text{given expression substituted with values}$$

$$\boxed{m_{max} = \frac{\lambda_{avg}}{2(\lambda_2-\lambda_1)} = \frac{578}{2\times2} = \boxed{144.5 \approx 145\text{ orders}}}$$

Fringes wash out when the maxima of one wavelength coincide with minima of the other: $m_{max} = \lambda_1/(2\Delta\lambda)$.

$$m_{max} = \frac{\lambda_{avg}}{2(\lambda_2-\lambda_1)} = \frac{578}{2\times2} = \boxed{144.5 \approx 145\text{ orders}}$$

Answer

$$m_{max} = \frac{\lambda_{avg}}{2(\lambda_2-\lambda_1)} = \frac{578}{2\times2} = \boxed{144.5 \approx 145\text{ orders}}$$

Physical Interpretation

The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.