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Chapter 33: Problem 14

Prove that a ray of light reflected from a plane mirror rotates through an angle of 2\(\theta\) when the mirror rotates through an angle \(\theta\) about an axis perpendicular to the plane of incidence.

Short Answer

Expert verified
The reflected ray rotates by 2\( \theta \) when the mirror rotates by \( \theta \).

Step by step solution

01

Understanding the Setup

Consider a ray of light hitting a plane mirror at some angle \( \theta_i \) (the angle of incidence). The angle of reflection \( \theta_r \) is equal to the angle of incidence due to the law of reflection. The ray is reflected at the same angle \( \theta_i \) with respect to the normal at the point of incidence.
02

Analyzing Rotation of Mirror

When the mirror rotates by an angle \( \theta \), the normal to the mirror rotates by the same angle \( \theta \). This changes both the angle of incidence and reflection by \( \theta \).
03

Calculating New Angles

Since the mirror has rotated by \( \theta \), the new angle of incidence becomes \( \theta_i + \theta \) and accordingly, the new angle of reflection is also \( \theta_i + \theta \) from the new normal.
04

Finding the Total Rotation

The initial angle between the incident and reflected ray was \( 2\theta_i \). After rotation of the mirror, the angle between the new reflected ray and the original reflected ray is \( (\theta_i + \theta) + (\theta_i + \theta) = 2\theta_i + 2\theta \). Thus, the difference in angle between the original and new reflected ray is \( 2\theta \).
05

Conclusion

Hence, when the mirror rotates by \( \theta \), the reflected ray rotates through an angle of \( 2\theta \). This is due to the change in the angle of both incidence and reflection by \( \theta \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plane Mirror
A plane mirror is a flat reflective surface that creates images through reflection of light. When light rays hit this smooth surface, the plane mirror reflects them back at the same angle they arrived. Plane mirrors are unique because they create virtual images that appear to originate from behind the mirror. This type of mirror is commonly found in our daily lives, such as bathroom mirrors and dressing room mirrors.
  • Flat surface reflects light uniformly.
  • Creates virtual, upright images.
  • Image distance equals object distance from the mirror.
Understanding how light behaves with plane mirrors is crucial for learning about how images are formed and reflected.
Angle of Incidence
The angle of incidence is the angle formed between the incoming ray of light and an imaginary line perpendicular to the surface of the plane mirror, called the normal. This angle is crucial because it determines how light will behave when it strikes the plane mirror. According to the law of reflection, the angle of incidence is always equal to the angle of reflection.
  • Measured from the incident ray to the normal.
  • Denoted by the symbol \(\theta_i\).
  • Key part in understanding light reflection patterns.
Grasping the concept of the angle of incidence helps one predict the path of the reflected ray.
Angle of Reflection
When a ray of light strikes a plane mirror, it bounces off the surface at a certain angle. The angle of reflection is the angle created between the reflected light ray and the normal to the mirror's surface. By the law of reflection, this angle is equal to the angle of incidence, forming a symmetrical path of the incoming and outgoing light rays.
  • Equal to the angle of incidence \(\theta_i\).
  • Measured from the reflected ray to the normal.
  • Helps in predicting the direction of the reflected ray.
Understanding the angle of reflection is essential in optics and many practical applications, such as designing optical instruments.
Rotation of Mirror
When a plane mirror is rotated around an axis perpendicular to its surface by an angle \(\theta\), it affects the direction of the reflected ray. This change causes both the new angle of incidence and angle of reflection to adjust by the same rotation amount. However, the overall effect on the reflected ray is double the rotation angle. Therefore, the rotated mirror results in a reflected ray that turns by an angle of \(2\theta\).
  • Mirror rotation affects both incidence and reflection angles equally.
  • Results in a total rotation of the reflected ray by \(2\theta\).
  • Influential in designing systems relying on precise angle change.
Knowing how the rotation of mirrors works broadens comprehension of light's behavior in dynamic systems.
Ray of Light
A ray of light is a straight line that represents the path light travels. It is usually depicted starting from a light source and traveling uninterrupted through space unless it hits an object like a plane mirror. When light hits a reflective surface, the path (or ray) changes direction according to the laws of optics.
  • Travels in straight lines until obstructed.
  • Changes direction upon reflection or refraction.
  • Models the behavior of light efficiently in diagrams and problems.
Understanding light rays is key in explaining how we see images and understand brightness and optics.

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Most popular questions from this chapter

A beam of light is traveling inside a solid glass cube having index of refraction \(1.53 .\) It strikes the surface of the cube from the inside. (a) If the cube is in air, at what minimum angle with the normal inside the glass will this light not enter the air at this surface? (b) What would be the minimum angle in part (a) if the cube were immersed in water?

Old photographic plates were made of glass with a light-sensitive emulsion on the front surface. This emulsion was some what transparent. When a bright point source is focused on the front of the plate, the developed photograph will show a halo around the image of the spot. If the glass plate is 3.10 \(\mathrm{mm}\) thick and the halos have an inner radius of \(5.34 \mathrm{mm},\) what is the index of refraction of the glass? (Hint: Light from the spot on the front surface is scattered in all directions by the emulsion. Some of it is then totally reflected at the back surface of the plate and returns to the front surface.)

A parallel beam of unpolarized light in air is incident at an angle of \(54.5^{\circ}\) (with respect to the normal) on a plane glass surface. The reflected beam is completely linearly polarized. (a) What is the refractive index of the glass? (b) What is the angle of refraction of the transmitted beam?

Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 \(\mathrm{m}\) within a plastic rod in 11.5 \(\mathrm{ns}\) . What is the refractive index of the plastic?

When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth's atmosphere, as shown in Fig. 33.53 . Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angle \(\delta\) above the sun's true position. (a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction \(n\) , and extends to a height \(h\) above the earth's surface, at which point it abruptly stops. Show that the angle 8 is given by $$ \delta=\arcsin \left(\frac{n R}{R+h}\right)-\arcsin \left(\frac{R}{R+h}\right) $$ where \(R=6378 \mathrm{km}\) is the radius of the earth. (b) Calculate \(\delta\) using \(n=1.0003\) and \(h=20 \mathrm{km}\) . How does this compare to the angular radius of the sun, which is about one quarter of a degree? (In actuality a light ray from the sun bends gradually, not abruptly, since the density and refractive index of the atmosphere change gradually with altitude.)
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