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Chapter 30: Problem 10

Under what conditions will the polarizing angle be smaller than \(45^{\circ} ?\)

Short Answer

Expert verified
The polarizing angle will be smaller than \(45^{\circ}\) when the refractive index of the medium is less than 1.

Step by step solution

01

Understand Brewster's Law

Brewster's Law defines the polarizing angle, which is given by the formula: \(胃_p = atan(n)\), where 'atan' represents the inverse tangent function, and \(n\) is the refractive index of the transparent medium. Brewster's angle is always measured with respect to the normal at the point of incidence.
02

Find Expression for Condition

The question asks under what conditions the polarizing angle will be smaller than \(45^{\circ}\). Therefore, the conditions will satisfy the inequality \(胃_p < 45^{\circ}\). Substituting \(胃_p = atan(n)\), the inequality becomes: \(atan(n) < 45^{\circ}\).
03

Solve the Inequality

To solve the inequality \(atan(n) < 45^{\circ}\), you need to take the tangent on both sides (please note we can do this only because the tangent function is increasing in the interval from 0 to \(45^{\circ}\)): \(n < tan(45^{\circ})\). Since the tangent of \(45^{\circ}\) is equal to 1, the inequality simplifies to \(n < 1\).

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