Problem 6 - College Physics

Question

A woman fixing the hair on the back of her head holds a plane mirror \(30 \mathrm{~cm}\) in front of her face so as to look into a plane mirror on the bathroom wall behind her. She is \(90 \mathrm{~cm}\) from the wall mirror. (a) The image of the back of her head will be from (1) only the front mirror, (2) only the wall mirror, or (3) both mirrors. (b) Approximately how far does the image of the back of her head appear in front of her?

Step-by-Step Solution

Answer

(a) The image is formed by both mirrors. (b) The image appears about 210 cm in front of her.

Step 1: Understanding the Scenario

The woman uses two mirrors to see the back of her head, one held in front of her and one on the wall. She checks her reflection in this setup.

Step 2: Determine Image Sources

To see the back of her head, light from the back of her head must reflect off the wall mirror and then be reflected by the mirror she's holding. Therefore, the image forms due to interactions of light with both mirrors.

Step 3: Visualizing the Path of Light

First, light travels from the back of her head to the wall mirror, where it reflects towards the mirror held in her hand. This second mirror then reflects the light towards her eyes.

Step 4: Calculate the Image Distance

The light travels a total path consisting of the distance to the wall mirror and back to her face. This distance is twice the distance between her and the wall mirror plus the distance from her to the handheld mirror: \[2 imes 90 ext{ cm} + 30 ext{ cm} = 180 ext{ cm} + 30 ext{ cm} = 210 ext{ cm}.\]

Step 5: Determine Effective Image Position

Because the image from the wall mirror appears as if it is behind and further processed by the mirror in her hand, the effective distance from the back of her head to her face via the mirrors equates to approximately 210 cm in front of her face.

Key Concepts

Plane Mirrors

Plane mirrors are flat and reflective surfaces that reflect light to form images. They are commonly used in everyday situations, like the bathroom mirrors. The key characteristics of a plane mirror include that it only changes the direction of light without altering its energy or speed. When light hits a plane mirror, it reflects off the surface at the same angle it arrived, which is known as the law of reflection.
In the scenario we analyzed, the woman uses two plane mirrors to view the back of her head. One is a handheld mirror and the other is a wall mirror. Both mirrors play a crucial role in the formation of her image, reflecting light in a specific path allowing her to see a reflection of herself that isn't visible directly. By understanding how plane mirrors work, one can easily determine how and where images will form.

Reflection of Light

Reflection of light is the process by which light rays bounce off a surface. For plane mirrors, the angle at which the light hits the mirror (the angle of incidence) equals the angle at which it reflects (the angle of reflection). This fundamental principle is crucial in determining the path light takes when encountering reflective surfaces.
In our specific example, the woman's reflection is enabled through this predictable behavior of light. Light from the back of her head travels towards the wall mirror. When it hits this mirror, it reflects back towards the mirror that she holds. Each interaction with the mirrors involves the reflection of light, ultimately allowing her to observe the back of her head. Without this property of light, mirror-based image formation wouldn't be possible.

Image Formation

Image formation in plane mirrors is a result of reflected light rays converging and forming a visible representation of an object. **Characteristics of images in plane mirrors:**

• They are virtual, meaning they can't be cast on a screen.
• They are the same size as the object.
• They appear upright but reversed from left to right.

In the exercise, the woman's goal is to form a visible image of the back of her head using both mirrors. Considering the reflection paths, the final image she views in the handheld mirror is a virtual, upright version seen only after light has bounced between both mirrors. Her understanding of the process allows her to determine the best positioning of the mirrors and her head to achieve a clear image.

Light Path Analysis

Analyzing the path taken by light is key to understanding image formation with mirrors. The light path dictates where and how images will appear.
In this case study, light starts from the back of the woman's head, traveling towards the wall mirror first. Upon hitting this mirror, the light reflects towards the handheld mirror, which she uses to view the reflection. The path is crucial: light first covering a round-trip of twice the distance from the wall mirror to her head, then an additional path to the handheld mirror.
An effective way to calculate the distance of the image from the woman's face involves doubling the distance to the wall mirror (90 cm each way) and adding the distance to the handheld mirror (30 cm), resulting in a total apparent distance of 210 cm. This detailed analysis of the light's journey helps in predicting where exactly she will see her image, making the problem scenario more comprehensible.