--- title: "Coherent sources and Newton's ring law" subject: "Engineering Physics" chapter: 4 key: "semester_1/engineering_physics:newton-rings-coherent-sources-diameter-law" source: https://www.bctengineer.com/q/semester_1/engineering_physics/newton-rings-coherent-sources-diameter-law site: BCT Engineering --- # Coherent sources and Newton's ring law Engineering Physics · Chapter 4: Physical Optics - Asked in: 2072 Kartik, 5 marks - Video: [newton rings - interference complete tutorial](https://www.youtube.com/watch?v=cA8g_pjoTrg) — e Tution - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/newton-rings-coherent-sources-diameter-law - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-4 ## Question What are the coherent sources of light? How can such sources be developed in the lab? Show that the square of the diameters of the nth dark ring by the reflected light of Newton's ring is directly proportional to the natural number. ## Short answer Coherent sources emit with constant phase difference, obtained by division of wavefront (e.g., Young’s double slit) or amplitude (thin films). For Newton’s rings in reflected light, dark ring condition 2t = nλ with t ≈ r²/(2R) leads to D_n² = 4nλR, so D_n² ∝ n. ## Answer #### Coherent sources Two sources are **coherent** if they emit light waves of the same frequency with a constant phase difference between them. Two independent ordinary sources are not coherent, because their atoms emit light in random, short bursts lasting only about 10⁻⁸ s, so the phase difference keeps changing. #### How to obtain them Both coherent sources are derived from **one** source: 1. **Division of wavefront**: Young's double slit, Fresnel's biprism, Lloyd's mirror. The wavefront from a single slit is divided into two parts, which act as two (real or virtual) coherent sources. 2. **Division of amplitude**: the beam is partly reflected and partly transmitted by a thin film, wedge or beam splitter (Newton's rings, Michelson interferometer). #### D² ∝ n in Newton's rings ```text lens (radius R) _______|_______ \ r | t / ________\____|_____/_____ glass plate O film ``` 1. A plano-convex lens of radius of curvature *R* rests on a plane glass plate. At distance *r* from the contact point, the air film has thickness *t*. 2. Geometry: R² = (R − t)² + r² → r² = 2Rt − t² ≈ 2Rt, so **t = r²/(2R)**. 3. In reflected light, ray 2 (reflected at the plate) gets an extra λ/2, so the path difference is 2t + λ/2 (normal incidence, air). 4. **Dark ring**: 2t + λ/2 = (2n+1)λ/2 → **2t = nλ**. 5. Substitute t: r²/R = nλ → r² = nλR. With D = 2r: ```text D_n² = 4 n λ R ``` Since λ and R are constants, **D_n² ∝ n**: the squares of the diameters of the dark rings are in the ratio 1 : 2 : 3 ... of the natural numbers, so the rings crowd closer together as n increases. The centre (n = 0) is dark. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Two independent ordinary sources are not coherent because: A. Atoms emit randomly for about 10^-8 s, so the phase difference keeps changing B. They have exactly the same frequency C. They are too far apart D. They emit no light Answer: A. Random atomic emission means no constant phase relationship. (Key point: Coherent sources) 2. Which method is division of amplitude? A. Young's double slit B. Fresnel's biprism C. Thin film / Newton's rings / Michelson interferometer D. Lloyd's mirror Answer: C. Partial reflection and transmission splits the amplitude; the others divide the wavefront. (Key point: Obtaining coherent sources) 3. From R^2 = (R − t)^2 + r^2, with t << R, the film thickness is: A. t = r^2/R B. t = r^2/(2R) C. t = 2r^2/R D. t = r/(2R) Answer: B. r^2 = 2Rt − t^2 ≈ 2Rt, so t = r^2/(2R). (Key point: Film thickness) 4. What does D_n^2 = 4nλR imply? A. D_n is proportional to n B. D_n^2 is proportional to n, so rings crowd as n increases C. Ring spacing is constant D. D_n is independent of λ Answer: B. Squares of diameters are in ratio 1:2:3..., so successive rings get closer. 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