The equation 1/ p 1/ i = 2/ r for spherical mirrors is an approximation that is valid if the image is formed by rays that make only small angles with the central axis. In reality, many of the angles are large, which smears the image a little. You can determine how much. Refer to Fig. 34-22 and consider a ray that leaves a point source (the object) on the central axis and that makes an angle α with that axis. First, find the point of intersection of the ray with the mirror. If the coordinates of this intersection point are x and y and the origin is placed at the center of curvature, then y = ( x + p – r ) tan α and x 2 + y 2 + r 2 , where p is the object distance and r is the mirror’s radius of curvature. Next, use tan β = y / x to find the angle β at the point of intersection, and then use α + γ = 2 β to find the value of γ . Finally, use the relation tan γ = y /( x + i – r ) to find the distance i of the image. (a) Suppose r = 12 cm and p = 20 cm. For each of the following values of α , find the position of the image — that is, the position of the point where the reflected ray crosses the central axis: 0.500, 0.100, 0.0100 rad. Compare the results with those obtained with the equation 1/ p + 1/ i = 2/ r . (b) Repeat the calculations for p = 4.00 cm.
The equation 1/ p 1/ i = 2/ r for spherical mirrors is an approximation that is valid if the image is formed by rays that make only small angles with the central axis. In reality, many of the angles are large, which smears the image a little. You can determine how much. Refer to Fig. 34-22 and consider a ray that leaves a point source (the object) on the central axis and that makes an angle α with that axis. First, find the point of intersection of the ray with the mirror. If the coordinates of this intersection point are x and y and the origin is placed at the center of curvature, then y = ( x + p – r ) tan α and x 2 + y 2 + r 2 , where p is the object distance and r is the mirror’s radius of curvature. Next, use tan β = y / x to find the angle β at the point of intersection, and then use α + γ = 2 β to find the value of γ . Finally, use the relation tan γ = y /( x + i – r ) to find the distance i of the image. (a) Suppose r = 12 cm and p = 20 cm. For each of the following values of α , find the position of the image — that is, the position of the point where the reflected ray crosses the central axis: 0.500, 0.100, 0.0100 rad. Compare the results with those obtained with the equation 1/ p + 1/ i = 2/ r . (b) Repeat the calculations for p = 4.00 cm.
The equation 1/p 1/i = 2/r for spherical mirrors is an approximation that is valid if the image is formed by rays that make only small angles with the central axis. In reality, many of the angles are large, which smears the image a little. You can determine how much. Refer to Fig. 34-22 and consider a ray that leaves a point source (the object) on the central axis and that makes an angle α with that axis.
First, find the point of intersection of the ray with the mirror. If the coordinates of this intersection point are x and y and the origin is placed at the center of curvature, then y = (x + p – r) tan α and x2 + y2 + r2, where p is the object distance and r is the mirror’s radius of curvature. Next, use tan β = y/x to find the angle β at the point of intersection, and then use α + γ = 2 β to find the value of γ. Finally, use the relation tan γ = y/(x + i – r) to find the distance i of the image.
(a) Suppose r = 12 cm and p = 20 cm. For each of the following values of α, find the position of the image — that is, the position of the point where the reflected ray crosses the central axis: 0.500, 0.100, 0.0100 rad. Compare the results with those obtained with the equation 1/p + 1/i = 2/r. (b) Repeat the calculations for p = 4.00 cm.
Two upright plane mirrors touch along one edge, where their planes make an angle of αα. A beam of light is directed onto one of the mirrors at an angle of incidence of ββ (ββ<αα) and is reflected onto the other mirror.
a) Will the angle of reflection of the beam from the second mirror be
b) If αα = 60∘∘ and ββ = 40∘∘, what will be the angle of refection of the beam from the second mirror?
An aquarium is filled with water to a height of h
A ray of light passes through the air above (n1
down into the water (n2
01 = 24°. It reflects off the bottom surface of the glass and
travels upward back toward the water-air boundary. When
7.5 cm.
1.00) and
1.33) with an angle of incidence
A
this ray of light emerges from the water and back into the air,
how far will it be from the point where it first entered the
water?
A light beam reflects off two parallel reflecting surfaces that are = 134 cm long and separated by a distance of s = 65.0 cm. If the beam of light enters this
region as shown, with the angle = 23.7°, determine the following,
0
(a) the number of times the light beam reflects off the left surface
times
(b) the number of times the light beam reflects off the right surface
times
K
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