The distance of a lightbulb from a large plane mirror is...

The distance of a lightbulb from a large plane mirror is twice the distance of a person from the plane mirror. Light from the bulb reaches the person by two paths. It travels to the mirror at an angle of incidence $\theta$ , and reflects from the mirror to the person. It also travels directly to the person without reflecting off the mirror. The total distance traveled by the light in the first case is twice the distance traveled by the light in the second case. Find the value of the angle $\theta .$

Key Concepts

Geometric Optics Path Lengths

This concept involves computing the distances that light travels along different paths, whether it takes a direct route or a route including reflection. In the context of this problem, the relationship between the direct path and the reflected path is given, and it is used to set up geometrical equations—often by using the positions of the source, the mirror, and the observer—to determine unknown parameters like the angle of incidence.

Trigonometric Relationships

Trigonometry is used to relate the angles and sides in the right?triangles formed by the distances between the light source, the mirror (or its image), and the observer. By expressing quantities such as the horizontal and vertical separations in terms of trigonometric functions (like the tangent of the angle of incidence), one can derive the value of the angle needed to meet the given path length condition.

Law of Reflection

This principle states that when light reflects off a surface, the angle of incidence (the angle between the incoming ray and the normal to the surface) is equal to the angle of reflection (the angle between the reflected ray and the same normal). It is a fundamental concept in geometrical optics and is used to determine the path of a light ray after it bounces off a mirror.

Mirror Image Method

In problems involving plane mirrors, the mirror image method simplifies analysis by replacing the reflection process with the notion of an image. The reflected ray’s path from a source to an observer via the mirror is equivalent to the straight?line path from the image of the source (reflected across the mirror) to the observer. This method makes it easier to analyze distances and angles in the reflection problem.

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