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University Physics Volume 3

Challenge Problems

University Physics Volume 3Challenge Problems

Challenge Problems

90.

Light shows staged with lasers use moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction by 2θ2θ when the mirror is rotated by an angle θθ.

91.

Consider sunlight entering Earth’s atmosphere at sunrise and sunset—that is, at a 90.0°90.0° incident angle. Taking the boundary between nearly empty space and the atmosphere to be sudden, calculate the angle of refraction for sunlight. This lengthens the time the Sun appears to be above the horizon, both at sunrise and sunset. Now construct a problem in which you determine the angle of refraction for different models of the atmosphere, such as various layers of varying density. Your instructor may wish to guide you on the level of complexity to consider and on how the index of refraction varies with air density.

92.

A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown below. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.

93.

A light ray falls on the left face of a prism (see below) at the angle of incidence θθ for which the emerging beam has an angle of refraction θθ at the right face. Show that the index of refraction n of the glass prism is given by

n = sin 1 2 ( α + ϕ ) sin 1 2 ϕ n = sin 1 2 ( α + ϕ ) sin 1 2 ϕ

where ϕϕ is the vertex angle of the prism and αα is the angle through which the beam has been deviated. If α=37.0°α=37.0° and the base angles of the prism are each 50.0°,50.0°, what is n?

94.

If the apex angle ϕϕ in the previous problem is 20.0°20.0° and n=1.50n=1.50, what is the value of αα?

95.

The light incident on polarizing sheet P1P1 is linearly polarized at an angle of 30.0°30.0° with respect to the transmission axis of P1P1. Sheet P2P2 is placed so that its axis is parallel to the polarization axis of the incident light, that is, also at 30.0°30.0° with respect to P1P1. (a) What fraction of the incident light passes through P1P1? (b) What fraction of the incident light is passed by the combination? (c) By rotating P2P2, a maximum in transmitted intensity is obtained. What is the ratio of this maximum intensity to the intensity of transmitted light when P2P2 is at 30.0°30.0° with respect to P1P1?

96.

Prove that if I is the intensity of light transmitted by two polarizing filters with axes at an angle θθ and I′I′ is the intensity when the axes are at an angle 90.0°−θ,90.0°−θ, then I+I′=I0,I+I′=I0, the original intensity. (Hint: Use the trigonometric identities cos90.0°−θ=sinθcos90.0°−θ=sinθ and cos2θ+sin2θ=1.)cos2θ+sin2θ=1.)

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