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Three plano-convex lenses A, B, and C are tested in pairs using Newton's ring apparatus with monochromatic light of wavelength lambda = 600 nm. The radius of the 10th dark ring is: r_AB = 4.0 mm (A and B combined), r_BC = 4.5 mm (B and C combined), r_CA = 5.0 mm (C and A combined). Find the radii of curvature of lenses A, B, and C.

1. The radius of curvature of lens A is approximately 6.28 m.
2. The radius of curvature of lens B is approximately 4.64 m.
3. The radius of curvature of lens C is approximately 6.2 m.
4. The radius of curvature of lens C is approximately 3.1 m.

Correct answer: The radius of curvature of lens A is approximately 6.28 m.

Solution

rₙ² = n*lambda*R_eff. So R_AB = r_AB²/(n*lambda) = (4.0*10⁻³)²/(10*600*10⁻⁹) = 16*10⁻⁶/6*10⁻⁶ = 2.667 m. R_BC = (4.5)²*10⁻⁶/6*10⁻⁶ = 20.25/6 = 3.375 m. R_CA = (5.0)²*10⁻⁶/6*10⁻⁶ = 25/6 = 4.167 m. 1/R_A + 1/R_B = 1/2.667; 1/R_B + 1/R_C = 1/3.375; 1/R_C + 1/R_A = 1/4.167. Let a=1/R_A, b=1/R_B, c=1/R_C: a+b=0.3750, b+c=0.2963, c+a=0.2400. Adding all: 2(a+b+c)=0.9113, a+b+c=0.4557. a=0.4557-0.2963=0.1594, R_A=1/0.1594=6.27 m ≈ 6.28 m. b=0.3750-0.1594=0.2156, R_B=1/0.2156=4.64 m. c=0.2963-0.2156=0.0807, R_C=1/0.0807=12.4 m.

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