A beam of light passes from air (index of refraction ) into lake water whose index of…

PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Geometric Optics · Geometric Optics

Problem

A beam of light passes from air (index of refraction n1=1.000n_{1}=1.000) into lake water whose index of refraction n2(t)n_{2}(t) is slowly increasing as algae concentration changes. At a certain moment: • n2=1.340n_{2}=1.340 and dn2dt=0.0020 s−1\dfrac{dn_{2}}{dt}=0.0020\,\text{s}^{-1}, • The angle of incidence in air is θ1=40∘\theta_{1}=40^{\circ} and is decreasing at a rate of dθ1dt=−0.050∘ s−1\dfrac{d\theta_{1}}{dt}=-0.050^{\circ}\text{ s}^{-1}. Let θ2(t)\theta_{2}(t) denote the angle of refraction in the water. Snell’s law gives n1sin⁡θ1=n2sin⁡θ2n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}. (a) Differentiate Snell’s law with respect to time to obtain an expression for dθ2dt\dfrac{d\theta_{2}}{dt} in terms of n1, n2, θ1, θ2, dn2dtn_{1},\,n_{2},\,\theta_{1},\,\theta_{2},\,\dfrac{dn_{2}}{dt}, and dθ1dt\dfrac{d\theta_{1}}{dt}. (b) Evaluate θ2\theta_{2} numerically (in radians) using the instantaneous values given, then compute dθ2dt\dfrac{d\theta_{2}}{dt} in rad s−1\text{rad\,s}^{-1}. Clearly show unit conversions for all angle measurements. (c) State whether the refracted beam is bending toward or away from the normal at this instant and justify your answer using the sign of dθ2dt\dfrac{d\theta_{2}}{dt}. (d) If the goal is to keep θ2\theta_{2} momentarily constant despite the changing refractive index, determine the required rate of change of the angle of incidence dθ1dt\dfrac{d\theta_{1}}{dt} (in degrees per second) when n2=1.340n_{2}=1.340 and dn2dt=0.0020 s−1\dfrac{dn_{2}}{dt}=0.0020\,\text{s}^{-1}.

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