Problem 3 - University Physics with Modern Physics

Question

A pencil that is 9.0 \(\mathrm{cm}\) long is held perpendicular to the surface of a plane mirror with the tip of the pencil lead 12.0 \(\mathrm{cm}\) from the mirror surface and the end of the eraser 21.0 \(\mathrm{cm}\) from the mirror surface. What is the length of the image of the pencil that is formed by the mirror? Which end of the image is closer to the mirror surface: the tip of the lead or the end of the eraser?

Step-by-Step Solution

Answer

The image is 9.0 cm long. The tip of the lead is closer to the mirror.

Step 1: Understand the Problem

The pencil is held perpendicular to a plane mirror. We need to find out the length of its image and determine which end of the image is closer to the mirror.

Step 2: Calculate the Position of the Images

For a plane mirror, the image forms directly opposite the object at the same distance behind the mirror. Therefore, the image of the pencil tip is 12.0 cm behind the mirror and the image of the eraser end is 21.0 cm behind the mirror.

Step 3: Calculate the Length of the Image

To find the length of the pencil image in the mirror, we subtract the position of the pencil tip's image from the position of the eraser's image: \[ \text{Image Length} = 21.0 \text{ cm} - 12.0 \text{ cm} = 9.0 \text{ cm} \] Thus, the length of the image is the same as the actual pencil length, which is 9.0 cm.

Step 4: Determine Which End is Closer to the Mirror

Since the image of the pencil tip is at 12.0 cm from the mirror and the image of the eraser is at 21.0 cm from the mirror, the image of the pencil tip is closer to the mirror than the image of the eraser.

Key Concepts

Image Formation

When it comes to image formation in plane mirrors, the process is quite straightforward. Plane mirrors produce a virtual image that is upright and of the same size as the actual object. This is because the image forms at the point where the light rays appear to diverge from behind the mirror. The conversion of light rays results in an image that looks real, even though the light doesn't actually reach that spot. This is why the image is described as "virtual."
In practice, when you place an object in front of a plane mirror, every point on that object has a corresponding point on the image formed by the mirror. This means if you hold a pencil upright in front of a mirror, like in the original problem, you can predict exactly where the image will form based on these properties.
Because the image is virtual, it cannot be projected onto a screen, but it will appear to the observer looking into the mirror as if it is located at the same distance behind the mirror as the object is in front of it.

Mirror Image Distance

Understanding mirror image distance is crucial to solving problems involving reflections from plane mirrors. The key principle is that the distance from the object to the mirror is equal to the distance from the image to the mirror.

• If an object is 12 cm in front of a mirror, its image will appear 12 cm behind the mirror.
• This concept holds true for every point on the object, like the pencil tip and eraser in our original exercise.

For the pencil in the exercise, the tip is 12 cm from the mirror and its image will also be 12 cm behind the mirror. Similarly, the eraser end, which is 21 cm from the mirror, will have its image 21 cm behind the mirror. So the mirror image distance concept explains how and why these distances are mirrored exactly opposite on the other side of the plane mirror.

Optics

Optics is the study of light and its behavior. In the context of mirror reflections, optics helps us understand how light interacts with surfaces to form images. Plane mirrors are a fundamental topic in optics. They reflect light according to the law of reflection: the angle of incidence is equal to the angle of reflection.

• This principle ensures that the path of the light is predictable, hence forming a clear image.
• Plane mirrors also provide images that are laterally inverted, meaning left appears right and vice versa, but the size remains unchanged.

When studying plane mirror reflections in optics, it's essential to remember that these images are virtual and cannot be caught on a screen, but they can be observed directly through the mirror itself. This is what creates the illusion of distance and depth in the reflected image.