Question

What is the size of the smallest vertical plane mirror in which a woman of height $h$ can see her full-length image?

Short answer

The mirror should be half her height, $\frac{h}{2}$.

Step-by-step solution

Understanding the Problem

To find the size of the smallest mirror necessary for a woman to see her full-length image, we need to consider the path of light. The mirror must reflect light from the woman's feet to her eyes. We'll use principles of reflection and symmetry.

Analyzing the Light Path

When light reflects off a surface, the angle of incidence is equal to the angle of reflection. For the woman to see her feet in the mirror, light must travel from her feet, reflect off the mirror, and enter her eyes.

Applying Geometry and Symmetry

Since light behaves symmetrically, the woman only needs a mirror that surrounds the midpoint between her eyes and feet for the full-length view. This means the top edge of the mirror should be at the level of her eyes, while the bottom edge should be midway between her eyes and feet.

Calculating Mirror Height

The height of the mirror is half the woman's height. Therefore, if the woman is of height $h$, the mirror needs to be $\frac{h}{2}$ tall.

Conclusion and Verification

This height allows the woman to see herself entirely from top to bottom, as the top edge of the mirror aligns with her eyes and the bottom edge aligns exactly midway between her eyes and her feet.

Key concepts

Reflection

Reflection is a fundamental concept in optics and involves the bouncing back of light when it hits a surface. In this exercise, reflection is key to understanding how a woman sees her reflection in a mirror. By understanding the behavior of light, we can determine how reflections are created.

When light encounters a surface, such as a mirror, the angle at which it strikes the surface is known as the angle of incidence. This angle is precisely matched by the angle of reflection, according to the law of reflection. This principle is what allows a person to see their entire body in a mirror, as light reflects from different parts of the body onto the mirror and then to the eyes. Therefore, recognizing how light reflects helps us to determine the appropriate mirror size necessary to view a whole image.

Key points about reflection include:

• The angle of incidence equals the angle of reflection.
• Reflection allows us to determine the path of light from the object to the eyes.
• Understanding reflection is crucial for optimizing mirror size and placement.

Plane Mirror

Plane mirrors are flat, reflective surfaces used in various optical applications. In this scenario, a plane mirror is required to help a woman see her full image. Unlike curved mirrors, a plane mirror allows light to reflect consistently, preserving the size and shape of the object.

The properties of a plane mirror are helpful to consider when determining the size of the smallest mirror needed for a full-length reflection. For a person to observe their complete reflection, as with the woman in our problem, the mirror must extend over a specific vertical range. This range enables light from every point on her body to reflect along a path visible to her eyes.

Characteristics of a plane mirror:

• Makes it possible to see the full height of a person with a mirror of only half their height.
• Reflects light in a way that allows a person to maintain the same orientation in the reflection.
• Ideal for creating clear, proportionate images without distortion.

Light Path Analysis

Understanding light paths helps determine the correct height and position for a mirror. For the woman in the exercise, analyzing these paths is essential to see why a smaller mirror can still provide a full-size image.

Light travels from her feet to the mirror and then to her eyes. The crucial insight here is that light from her feet reaches the mirror and reflects upwards without losing information about her position. By analyzing this path, we can determine how the light reflects and returns to her eyes, maintaining the integrity of the reflection.

Key elements of light path analysis include:

• Tracing the journey of light from the object to the eyes through reflection.
• Utilizing symmetrical paths to simplify the reflection paths.
• Ensuring light is correctly aligned to form the complete image in the observer's view.

Geometry in Optics

Geometry plays a vital role in solving optical problems, such as determining the size of the required mirror. By using geometric principles, such as symmetry and proportionality, we can simplify complex optical scenarios.

In this problem, symmetry is used to determine the mirror's height. By understanding that the mirror only needs to cover half the vertical height of the person, we comprehend its geometric efficiency. Essentially, the top of the mirror matches the observer's eyes, while the bottom extends to the midpoint between their eyes and feet.

Useful geometric principles in optics include:

• Utilizing symmetry to simplify optical configurations.
• Applying proportional lines to represent light paths and mirror boundaries.
• Ensuring that the calculated size and position fulfill the optical requirement for full reflection.