Question

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

When a plane mirror rotates by \(\theta\), the light ray deflects by \(2\theta\).

Step-by-step solution

Understand the problem setup

The problem involves a light ray reflecting off a plane mirror. Initially, the ray hits the mirror at an angle, and upon reflection, it follows the law of reflection, where the angle of incidence is equal to the angle of reflection. When the mirror is rotated by an angle \(\theta\), the change in the light ray’s path needs to be examined.

Define initial angles

Consider an incident light ray approaching the mirror at an angle \(i\) with respect to the normal, and reflecting off at the same angle \(i\) as per the law of reflection. Initially, both angle of incidence and reflection are \(i\).

Rotate the mirror

Now, rotate the mirror by \(\theta\) around an axis through its center. This rotation changes the orientation of the normal to the surface. As a result, the reflected angle also changes by \(\theta\).

Evaluate the new reflection

After rotating the mirror, the angle of incidence remains as \(i\), but the incident ray now strikes the mirror with respect to the new normal. The reflection will be \(i + \theta\) because the normal to the mirror rotated by \(\theta\).

Calculate the angle of deflection

The total deflection \(\phi\) of the reflected ray is the change in the total angle between the incident and new reflected ray paths. Originally, the path deviation from the incident ray is \(2i\). With a new path deviation of \(2(i+\theta)\), the deflected angle \(\phi = 2(i+\theta) - 2i = 2\theta\).

Visualize with a diagram

To help understand this, draw a diagram. Begin with a horizontal line to represent the mirror, a line perpendicular to this as the initial normal, and two more lines showing the incident ray making an angle \(i\) with the normal, and a reflected ray at \(i\). Then, show the normal rotated by \(\theta\) and illustrate the new reflected ray, creating an angle \(\theta\) with the initial position. The difference now forms our deflected angle, \(2\theta\).

Key concepts

Law of Reflection

The Law of Reflection is a fundamental principle in physics and optics, which declares that when a light ray encounters a reflective surface, such as a mirror, the angle at which the ray strikes the surface, known as the angle of incidence, will equal the angle at which the ray bounces away, known as the angle 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.

This straightforward principle forms the cornerstone for understanding more complex optical phenomena.

Angle of Incidence

The Angle of Incidence is the angle formed between the incoming light ray and the imaginary line known as the normal. This angle is fundamental in every reflection process and is always measured from the normal to the ray, not from the surface itself.
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.

By understanding the angle of incidence, one can predict the behavior of the reflected ray.

Plane Mirror

A Plane Mirror is any flat, smooth reflective surface that forms images by reflecting light. Think of the plane mirror as a glass window, but instead of seeing the outside world, you see a perfect replica of whatever is in front of it.
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.

Plane mirrors are helpful in simplifying the study of reflection since the properties of the images remain consistent.

Light Ray Deflection

Light Ray Deflection refers to the change in direction of a light ray upon reflection or refraction. In the context of reflection on a plane mirror, understanding deflection is all about keeping track of the angles involved.
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.

Visualizing this with diagrams can help grasp how changes in mirror alignment affect light direction, resulting in twice the rotational angle deflection.