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Chapter 2: Problem 49

Derive the spherical interface equation for refraction at a concave surface. (Hint: Follow the derivation in the text for the convex surface.)

Short Answer

Expert verified
The derived spherical interface equation for refraction at a concave surface is: \( \frac{n_2^2 x_2^2 - 2n_2 x_2(y_2 - R) \frac{x_2}{R} + (y_2 - R)^2}{n_1^2 x_1^2 - 2 n_1 x_1 (y_1 - R) \frac{x_1}{R} + (y_1 - R)^2} = 1 \)

Step by step solution

01

Define the geometry of the problem

Draw a concave spherical surface with its center at point C and radius R. An incident ray comes from point Q, passes through point A on the surface, and refracts towards point P. Draw a normal at point A towards the initial ray direction, which meets the center of curvature C. Denote by θ1 the angle of incidence (the angle between the incident ray and the normal), and by θ2 the angle of refraction (the angle between the refracted ray and the normal).
02

Apply Snell's Law

According to Snell's Law, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive indices of the two media: \( n_1 \sin{\theta_1} = n_2 \sin{\theta_2} \) Here, \(n_1\) and \(n_2\) are the refractive indices of medium 1 and medium 2, respectively.
03

Introduce distance variables

Let AP = x1, AQ = y1, CP = x2, and CQ = y2. By considering the triangle ACP, we can write the following relationships: For the sine of the angles: \( \sin{\theta_1} = \frac{x1}{R} \) and \( \sin{\theta_2} = \frac{x2}{R} \) For the cosine of the angles: \( \cos{\theta_1} = \frac{y1 - R}{R} \) and \( \cos{\theta_2} = \frac{y2 - R}{R} \)
04

Substitute angle relationships into Snell's Law

Replace \(sin{\theta_1}\) and \(sin{\theta_2}\) in Snell's Law using the geometry relationships: \( n_1 \frac{x1}{R} = n_2 \frac{x2}{R} \) and subsequently, we can rewrite the equation as: \( n_1 x_1 = n_2 x_2 \)
05

Apply the cosine rule

According to the cosine rule, relate the distance variables to the angles and the radius R: \( R^2 = (y1 - R)^2 + x1^2 - 2(x1)(y1 - R)\cos{\theta_1} \) \( R^2 = (y2 - R)^2 + x2^2 - 2(x2)(y2 - R)\cos{\theta_2} \)
06

Combine Snell's Law and the cosine rule equations

Divide the second cosine equation by the first: \( \frac{n_2^2 x_2^2 - 2n_2 x_2(y_2 - R) \cos{\theta_2} + (y_2 - R)^2}{n_1^2 x_1^2 - 2 n_1 x_1 (y_1 - R) \cos{\theta_1} + (y_1 - R)^2} = 1 \) Rearrange the terms to get the final equation: \( \frac{n_2^2 x_2^2 - 2n_2 x_2(y_2 - R) \frac{x_2}{R} + (y_2 - R)^2}{n_1^2 x_1^2 - 2 n_1 x_1 (y_1 - R) \frac{x_1}{R} + (y_1 - R)^2} = 1 \) This is the derived spherical interface equation for refraction at a concave surface.

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