(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A plane electromagnetic wave refracts in a nonuniform medium. The ray trajectory is known,

[tex] y = a \left[ 1 - \left( \frac{x}{2a} \right)^2 \right] [/tex]

where a=const.

a) Plot y(x)

b) From the slope of the curve find the angle [tex] \theta \left( y \right) [/tex] which thek-vector makes with respect to the surface normal (parallel to the y-axis)

c) Using Snell's law, find the refractive index as a function of height, [tex] n\left(y\right) [/tex]

2. Relevant equations

[tex] n_i sin \left( \theta_i \right) = n_t sin \left( \theta_t \right) [/tex]

3. The attempt at a solution

a) simple downward facing parabola with y-intercept at a, x-intercepts at -2a and 2a

b) It hinted at what to do by stating the slope of the curve which is the derivative of the trajectory.

[tex] \frac{dy}{dx} = \frac{d}{dx} a \left[ 1 - \left( \frac{x}{2a} \right)^2 \right] [/tex]

[tex] = a \left[ -2 \left( \frac{x}{2a} \right) \left( \frac{1}{2a} \right) \right] = \frac{-x}{2a}[/tex]

The slope is thought of as "rise over run" so if we're looking for the angle with respect to the y-axis the triangle formed from the tangential to the trajectory and the y-axis the sides of the triangle given from the slope is adjacent and opposite.

[tex] cot \theta = \frac{-x}{2a} [/tex]

[tex] \theta \left( y \right) = tan \frac{-x}{2a} = -tan \frac{x}{2a} [/tex]

c) This is where I'm stuck, Snell's law is given in the earlier section. I can foresee the need to go to infinitesimal levels as the index of refraction changes over the minute values of y through the trajectory but I cannot see how to begin this task.

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# Homework Help: Refraction through a nonuniform medium

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