Problem Statement
A diffraction grating with $d = 2.5\;\mu$m is used with light of $\lambda = 550$ nm. What is the maximum order of diffraction that can be observed?
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: Maximum order when $\sin\theta = 1$: $m_{max} = d/\lambda = 2500/550 = 4.5$.
Step 2 — Apply the relevant physical law or equation: $$\boxed{m_{max} = 4 \text{ (since order must be integer)}}$$
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\boxed{m_{max} = 4 \text{ (since order must be integer)}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\boxed{m_{max} = 4 \text{ (since order must be integer)}}}$$
Maximum order when $\sin\theta = 1$: $m_{max} = d/\lambda = 2500/550 = 4.5$.
$$\boxed{m_{max} = 4 \text{ (since order must be integer)}}$$
Answer
$$\boxed{m_{max} = 4 \text{ (since order must be integer)}}$$
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.
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