Light travels inside a glass prism of refractive index towards a boundary with air

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Where a function vanishes, and what lies between the points

Problem

Light travels inside a glass prism of refractive index n=1.5n = 1.5 towards a boundary with air. The critical angle θc\theta_c for total internal reflection satisfies sin⁡θc=1/n\sin\theta_c = 1/n, so with f(θ)=sin⁡θ−1/nf(\theta) = \sin\theta - 1/n the root of ff on [0,π/2][0, \pi/2] is the angle wanted. Someone argues: *"Take f(θ)=sin⁡θ−0.6667f(\theta) = \sin\theta - 0.6667 and start Newton's method at θ0=π/2=1.5708 rad\theta_0 = \pi/2 = 1.5708\,\mathrm{rad}, since that is the natural upper limit of the angle. Then f′(θ0)=cos⁡(π/2)=0f'(\theta_0) = \cos(\pi/2) = 0, so the Newton step θ1=θ0−f(θ0)/f′(θ0)\theta_1 = \theta_0 - f(\theta_0)/f'(\theta_0) divides by zero and the iteration blows up. Since Newton's method cannot produce a finite answer here, the equation sin⁡θ=1/n\sin\theta = 1/n has no solution for n=1.5n=1.5, and a critical angle does not exist for this glass."* ((a)) Identify which of the three named failures of Newton's method has occurred. ((b)) Explain what the failure does and does not show about the existence of a root. ((c)) Give a bracket for the true root from the physics, and state the critical angle θc\theta_c to four decimal places (in radians), obtained by a safeguarded or better-started Newton iteration.

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