Chapter 30: Problem 4
Prove mathematically that when a plane mirror is rotated an angle \(\theta\) about an axis through its center (Part A of the experiment), the angle of deflection \(\phi\) of a light ray is equal to \(2 \theta\). Draw a diagram and show the work involved in your proof. Attach an additional sheet, if necessary.
Short Answer
Step by step solution
Understand the problem setup
Define initial angles
Rotate the mirror
Evaluate the new reflection
Calculate the angle of deflection
Visualize with a diagram
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Law of Reflection
Imagine the reflective surface as a calm lake and the light rays as pebbles. When you throw a pebble, it skips off the water at an angle that mirrors the angle it hit the water. This same principle applies to light rays hitting a mirror.
- The incident ray is the incoming light ray that approaches the mirror.
- The normal line is an imaginary straight line that is perpendicular to the surface of the mirror at the point of incidence.
- The reflected ray is the outgoing ray that departs from the mirror after reflection.
Angle of Incidence
Consider the angle of incidence as a bowler’s approach to the pins. The path of the bowling ball (light ray) towards the pins (mirror surface) forms an angle with the lane’s centerline (normal). To ensure the ball strikes precisely, understanding this angle is crucial.
- Measure the angle of incidence from the normal, not from the mirror's surface.
- At an angle of 0 degrees, the light ray is perfectly perpendicular to the mirror, causing no lateral deviation.
- Different angles result in distinct reflective paths.
Plane Mirror
Due to their flat surfaces, plane mirrors reflect light symmetrically, which simplifies understanding how light behaves. The reflections you see are considered virtual images, where the object appears to be behind the mirror.
- Plane mirrors create virtual images that are the same size and shape as the object.
- The distance of the image from the mirror is equal to the distance of the object from the mirror.
- These mirrors do not alter the size of the images they reflect.
Light Ray Deflection
When a plane mirror is rotated by an angle \theta, it affects how light interacts with the mirror's surface. As the mirror's position changes, the normal line rotates as well, and the reflected path of the light ray adjusts accordingly.
- The initial path deviation between the incident and reflected ray is given by twice the angle of incidence, \((2i)\).
- When the mirror rotates, the new angle deflection becomes \((2(i+\theta))\).
- Thus, the change in path results in a deflected angle equivalent to \((2\theta)\), twice the angle by which the mirror rotated.