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Chapter 23: Problem 81

A plane mirror reflects a beam of light. Show that the rotation of the mirror by an angle \(\alpha\) causes the beam to rotate through an angle \(2 \alpha\)

Short Answer

Expert verified
Answer: The reflected beam rotates through an angle of 2α.

Step by step solution

01

Understanding the mirror reflection law

Begin by recalling the law of reflection for plane mirrors: the angle of incidence (\(\textit{i}\)) is equal to the angle of reflection (\(\textit{r}\)). Mathematically, this can be represented as: \[ i = r \]
02

Initial position of the mirror and beam

Let's consider an initial position where the mirror is set horizontally and a beam of light is incident on the mirror at an angle \(\textit{i}\). Since \(\textit{i = r}\), the reflected beam makes an angle \(\textit{r}\) with the horizontal.
03

Rotate the mirror by angle \(\alpha\)

Now, we rotate the mirror clockwise by an angle \(\alpha\). Since we only care about the final angles, and not the actual position of the mirror, we can consider the problem in an equivalent way: we keep the mirror fixed and rotate the incident beam counterclockwise by \(\alpha\). In this case, the incident angle would now be \((i + \alpha)\).
04

Apply the reflection law to the new incident angle

Following the reflection law, the new angle of reflection will be equal to the new angle of incidence. Therefore, we have: \[ r' = i + \alpha \] Where \(r'\) is the new angle of reflection of the beam.
05

Calculate the rotation of the reflected beam

We want to find the change in the angle of the reflected beam. This is simply the difference between the new reflection angle \(r'\) and the initial reflection angle \(r\): \[ \Delta r = r' - r = (i + \alpha) - r \]
06

Use the initial condition to find the rotation of the beam

Recall that initially, the angle of incidence was equal to the angle of reflection \((i = r)\). Substitute this condition into our previous equation to find the rotation in the reflected angle (\(\Delta r\)): \[ \Delta r = (i + \alpha) - i = \alpha \] But, since we effectively rotated the incident angle counterclockwise by \(\alpha\), the actual change in the reflected beam angle would be twice the angle we calculated above. Hence, the rotation in the reflected beam is: \[ \Delta r = 2 \alpha \]
07

Conclusion

We have shown that when a plane mirror is rotated by an angle \(\alpha\), the reflected beam rotates through an angle of \(2 \alpha\). This relationship holds true for any plane mirror and confirms the given statement in the exercise.

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Most popular questions from this chapter

(a) Sunlight reflected from the smooth ice surface of a frozen lake is totally polarized when the incident light is at what angle with respect to the horizontal? (b) In what direction is the reflected light polarized? (c) Is any light incident at this angle transmitted into the ice? If so, at what angle below the horizontal does the transmitted light travel?
Apply Huygens's principle to a 5 -cm-long planar wavefront approaching a reflecting wall at normal incidence. The wavelength is \(1 \mathrm{cm}\) and the wall has a wide opening (width \(=4 \mathrm{cm}\) ). The center of the incoming wavefront approaches the center of the opening. Repeat the procedure until you have wavefronts on both sides of the wall. Without worrying about the details of edge effects, what are the general shapes of the wavefronts on each side of the reflecting wall? Repeat Problem 2 for an opening of width $0.5 \mathrm{cm} .$
An object \(2.00 \mathrm{cm}\) high is placed \(12.0 \mathrm{cm}\) in front of a convex mirror with radius of curvature of \(8.00 \mathrm{cm} .\) Where is the image formed? Draw a ray diagram to illustrate.
(a) Sunlight reflected from the still surface of a lake is totally polarized when the incident light is at what angle with respect to the horizontal? (b) In what direction is the reflected light polarized? (c) Is any light incident at this angle transmitted into the water? If so, at what angle below the horizontal does the transmitted light travel?
In order to read his book, Stephen uses a pair of reading glasses. When he holds the book at a distance of \(25 \mathrm{cm}\) from his eyes, the glasses form an upright image a distance of \(52 \mathrm{cm}\) from his eyes. (a) Is this a converging or diverging lens? (b) What is the magnification of the lens? (c) What is the focal length of the lens?
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