Problem 114 - Fundamentals of Physics

Question

Light travels from point \(A\) to point \(B\) via reflection at point \(O\) on the surface of a mirror. Without using calculus, show that length \(A O B\) is a minimum when the angle of incidence \(\theta\) is equal to the angle of reflection \(\phi .\) (Hint: Consider the image of \(A\) in the mirror.

Step-by-Step Solution

Answer

Length is minimized when angle of incidence equals angle of reflection.

Step 1: Reflection Symmetry of Light

To solve this problem, imagine point A's image across the mirror, labeled as A'. The junction of pathways point O is notable because it's on the mirror where light reflects to reach B.

Step 2: Visualizing the Path

On a straight-line path, let the path length be the combination of AO and OB. If light follows the shortest pathway, both spans (AOA' and BOA') are aligned in a straight line.

Step 3: Intersection of Path with Mirror

Consider A', the reflection of A over the mirror line. The shortest path from A' to B which crosses the mirror occurs precisely when light strikes the mirror orthogonally (with angle of incidence equal to the angle of reflection).

Step 4: Reflecting Principle

Connecting points A', O, and B forms a straight line when the reflection principle (angle of incidence equals the angle of reflection) is fulfilled. Any deviations longer.

Step 5: Final Simplification

Therefore, if you stand at A' looking into the mirror, O appears directly in the line connecting A' to B, confirming angle equality and leading to the shortest pathway.

Key Concepts

Angle of Incidence

When light strikes a surface, such as a mirror, it hits at a specific angle known as the angle of incidence. This is the angle formed between the incoming light ray and an imaginary line called the "normal," which is perpendicular to the surface at the point of contact.
Understanding the angle of incidence is crucial in defining how light interacts with surfaces. It determines how much light is reflected or refracted when it reaches different materials. In optical systems, controlling the angle of incidence allows engineers to design lenses that focus light effectively.
To visualize the angle of incidence, imagine standing at point A looking into a mirror on a wall. The light leaving from you to the mirror forms an angle with the normal at the point of reflection. This is your angle of incidence. The specifics of this angle also play a significant role in phenomena like rainbows and optical illusions, as they depend on light bending through media at varying angles.

Angle of Reflection

Once light hits a mirror, it doesn't just stop there; instead, it bounces off. The angle at which it bounces off is named the angle of reflection. Much like the angle of incidence, the angle of reflection is measured from the reflected light ray to the normal line, which is perpendicular to the mirror at the point of reflection.
A fundamental principle of reflection is that the angle of incidence is always equal to the angle of reflection. This means if a light ray hits a mirror at a 30-degree angle, it reflects off at that same angle on the other side of the normal. This principle holds for all smooth reflective surfaces and is the reason why mirrors can create clear images.
Visualize shining a flashlight on a mirror; the light will reflect off at the same angle it struck, ensuring that images in mirrors remain precise and undistorted. This is the essence of the activity described in the exercise: showing that the path of light is minimized when angles are equal, confirming the "law of reflection."

Mirror Reflection

Mirror reflection is the process by which light bounces off a reflective surface, such as a mirror. It is a fascinating phenomenon that allows us to see our reflections or project beams of light onto other surfaces.
When light from point A reflects off a mirror, it creates an image that appears to be behind the mirror. This virtual image, labeled as A' in the exercise, supports the law of reflection and helps us understand light travel. By picturing this image, we can see that the shortest path connecting points A to B through the mirror must consider the angle aspects we discussed.
Mirror reflections are part of daily life, from the mirrors we use while dressing to telescopes used in astronomy to see distant stars. In experiments, mirrors help in demonstrating light behavior and are integral to devices like periscopes and cameras. Knowing about mirror reflection not only enlightens our understanding of light but also enhances various technological applications.