Difficulty: Medium
Correct Answer: 230 km
Explanation:
Introduction / Context:
Coherence length characterizes the path-length over which a wave maintains a predictable phase relationship. In radar/optics, it is inversely related to spectral width. When bandwidth is given in hertz, a useful first-order estimate is l_coh ≈ c / Δf, where c is the speed of light and Δf is the effective spectral width.
Given Data / Assumptions:
Concept / Approach:
For a finite spectral width, coherence time τ_coh ≈ 1 / Δf. Coherence length is l_coh ≈ c * τ_coh ≈ c / Δf. Substitute Δf = 1300 Hz to compute l_coh and convert to kilometres.
Step-by-Step Solution:
τ_coh = 1 / Δf = 1 / 1300 s ≈ 7.6923 * 10^-4 sl_coh = c * τ_coh ≈ (3.0 * 10^8 m/s) * (7.6923 * 10^-4 s)l_coh ≈ 2.3077 * 10^5 mConvert to km: 2.3077 * 10^5 m ≈ 230.8 kmRounded to the nearest option ≈ 230 km
Verification / Alternative check:
Using the compact form l_coh = c / Δf directly: 3 * 10^8 / 1300 ≈ 2.31 * 10^5 m, confirming ≈ 230 km.
Why Other Options Are Wrong:
Common Pitfalls:
Confusing range bandwidth (MHz) with Doppler bandwidth (Hz) or inserting an unnecessary factor of 2 (which would correspond to certain coherence definitions) will change the numerical result.
Final Answer:
230 km
Discussion & Comments