Suppose you are 96 cm from a plane mirror. What area of the mirror is used to reflect the rays entering one eye from a point on the tip of your nose if your pupil diameter is 4.5 mm?

검증된 단계별 안내

Step 1: Understand the geometry of the situation. The plane mirror reflects light from the tip of your nose to your eye. The distance between you and the mirror is 96 cm, and the pupil diameter is 4.5 mm. The light rays form a cone with the pupil as the base and the tip of your nose as the vertex.

Step 2: Calculate the total distance the light travels. Since the mirror reflects the light, the total distance is twice the distance between you and the mirror. This means the total path length is 2 × 96 cm = 192 cm.

Step 3: Use the concept of similar triangles. The cone formed by the light rays has two similar triangles: one with the pupil diameter as the base and the other with the area of the mirror used as the base. The ratio of the pupil diameter to the total distance is the same as the ratio of the mirror area width to the distance from the mirror to the tip of your nose.

Step 4: Express the relationship mathematically. Let the width of the mirror area used be \( w \). The ratio can be written as \( \frac{4.5 \text{ mm}}{192 \text{ cm}} = \frac{w}{96 \text{ cm}} \). Solve for \( w \) to find the width of the mirror area used.

Step 5: Calculate the area of the mirror used. The area is approximately rectangular, with the width \( w \) and height equal to the pupil diameter (4.5 mm). Use the formula for the area of a rectangle: \( \text{Area} = w \times \text{pupil diameter} \). Substitute the values to find the area.

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주요 개념

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Plane Mirror Reflection

A plane mirror reflects light according to the law of reflection, which states that the angle of incidence equals the angle of reflection. This means that light rays coming from an object, such as the tip of your nose, will reflect off the mirror and travel to your eye, allowing you to see the object. Understanding this principle is crucial for determining how light interacts with the mirror.

Field of View

The field of view refers to the extent of the observable area that can be seen at any given moment. In this context, it is important to consider how the size of the pupil and the distance from the mirror affect the amount of light entering the eye. The diameter of the pupil (4.5 mm) will determine how much of the reflected light can be captured, influencing the area of the mirror that is effectively used for reflection.

Geometry of Reflection

The geometry of reflection involves understanding the spatial relationships between the object, the mirror, and the observer's eye. By applying basic geometric principles, one can calculate the area of the mirror that reflects light from the tip of the nose to the eye. This includes considering the distances involved and the angles at which light rays travel, which are essential for determining the effective area of the mirror used in the reflection.