Problem 7 - College Physics

Question

(a) When you stand between two plane mirrors on opposite walls in a dance studio, you observe (1) one, (2) two, or (3) multiple images. Explain. (b) If you stand \(3.0 \mathrm{~m}\) from the mirror on the north wall and \(5.0 \mathrm{~m}\) from the mirror on the south wall, what are the image distances for the first two images in both mirrors?

Step-by-Step Solution

Answer

(a) Multiple images are observed. (b) First images: 3.0 m (north) and 5.0 m (south); second images: 13.0 m (north) and 11.0 m (south).

Step 1: Understand Multiple Image Formation

When two plane mirrors face each other and you stand between them, they create an infinite series of reflections. Each mirror reflects both the original object and the reflections from the other mirror, producing multiple images. Thus, you observe multiple images.

Step 2: Calculate Image Distances in the North Wall Mirror (First Image)

For the first image formed in the north wall mirror, the image distance equals the object distance. Since you are standing 3.0 m from the north wall mirror, the image distance in this mirror is 3.0 m.

Step 3: Calculate Image Distances in the South Wall Mirror (First Image)

Similarly, the first image in the south wall mirror is formed at the same distance as the object distance. As you stand 5.0 m from the south wall mirror, the image distance here is 5.0 m.

Step 4: Calculate Image Distances for the Second Image in the North Wall Mirror

The second image in the north wall mirror is a reflection of the first image in the south wall mirror. As the first image in the south wall is at 5.0 m from the south wall, its reflection in the north wall is at an additional distance of 8.0 m (distance between mirrors) plus 5.0 m. Therefore, the second image in the north wall mirror is at 13.0 m.

Step 5: Calculate Image Distances for the Second Image in the South Wall Mirror

The second image in the south wall mirror is the reflection of the first image in the north wall mirror. Since the first image in the north mirror is 3.0 m from it, its reflection in the south mirror is also at 8.0 m plus 3.0 m. Thus, the second image in the south wall mirror is at 11.0 m.

Key Concepts

Plane Mirrors

Plane mirrors are flat, reflective surfaces that are commonly encountered in everyday situations, like in bathrooms or dance studios. A plane mirror works by reflecting light according to two fundamental laws:
1. The angle of incidence (the angle at which light hits the mirror) equals the angle of reflection (the angle at which light reflects off).
2. The incident ray, the reflected ray, and the normal (an imaginary line perpendicular to the surface at the point of incidence) all lie in the same plane.
Plane mirrors create virtual images of objects placed in front of them. Virtual images appear as if they are "inside" or "behind" the mirror.
Technically, no light passes through the mirror from the back side; instead, it is reflected back to the observer's eyes, creating the perception of an image.

Reflection

Reflection is the process where light bounces off a surface. In the case of plane mirrors, the surface is flat and smooth, leading to a clear and coherent image.
When light hits a plane mirror, it undergoes specular reflection (regular reflection), as opposed to diffuse reflection that occurs with rough surfaces. Specular reflection is responsible for enabling clear images in mirrors.
Light rays reflecting off the mirror follow the predictable path described by the laws of reflection. This predictability makes plane mirrors ideal for demonstrating optical principles and learning basic physics concepts. They allow us to see these processes in action.

Multiple Images

The phenomenon of multiple images occurs when two plane mirrors are placed opposite each other. This setup leads to an infinite series of reflections, creating multiple images of an object positioned between the mirrors.
For each mirror, it reflects both the original object and the images produced by the reflections from the opposite mirror. As a result, you can see a series of reflections that appear to repeat endlessly into the mirrors. The closer the mirrors are to each other, the denser the placement of the images.
Multiple images are explained by the mirrors' continuous reflection of each other’s images, where each image serves as an object for the other mirror. In theory, this creates an infinite loop of image reflections.

Image Distance Calculation

Calculating the distance of images formed by plane mirrors involves understanding how light travels and reflects between the object, mirrors, and images.
The basic principle is that the distance of the image from the mirror equals the distance of the object to the mirror. For instance, if you stand 3 meters from a mirror, your image will be 3 meters "inside" the mirror. This is because the primary reflection occurs at this distance.
When multiple images are involved, the calculation becomes more complex. In the scenario with mirrors facing each other, the first image's distance from a mirror equals the object's distance, while consecutive image distances involve adding the distance between the mirrors. For example, if mirrors are 8 meters apart, reflections result in distances calculated by summing up initial image distances with the separation distance between mirrors, creating a pattern that adds a layer of depth to solving such physics problems.