Problem 104 - Fundamentals of Physics

Question

Two plane mirrors are placed parallel to each other and \(40 \mathrm{~cm}\) apart. An object is placed \(10 \mathrm{~cm}\) from one mirror. Determine the (a) smallest, (b) second smallest, (c) third smallest (occurs twice), and (d) fourth smallest distance between the object and image of the object.

Step-by-Step Solution

Answer

(a) 10 cm, (b) 30 cm, (c) 50 cm, (d) 70 cm.

Step 1: Understand the Scenario

We have two mirrors placed parallel to each other and an object between them. This setup creates multiple images due to the reflections between the mirrors. The key to solving this problem is to calculate the distances of these images from the original object.

Step 2: Calculate First Image Distance

The first image of the object appears to be behind the first mirror, at a distance equal to that of the object from the mirror. If the object is 10 cm from the first mirror, then the first image is also 10 cm on the other side of this mirror. Therefore, the distance from the object to the first image is 10 cm.

Step 3: Calculate Second Image Distance

The second image forms in the opposite mirror, which is 40 cm away from the first mirror. Since the first image was 10 cm behind the first mirror, the second image will appear at 40 cm - 10 cm = 30 cm from the second mirror towards the object. Therefore, the distance between the object and this second image is 30 cm.

Step 4: Calculate Third Image Distance

The third image is a reflection of the first image in the second mirror. The first image was 10 cm behind the first mirror, making it 40 + 10 = 50 cm from the second mirror. Thus, the third image is 50 cm behind the second mirror, toward the first mirror, resulting in the object being 50 cm away from this third image.

Step 5: Calculate Fourth Image Distance

The fourth image is a reflection of the second image back in the first mirror. Since the second image is 30 cm behind the second mirror, it is actually 30 cm + 40 cm (the separation between mirrors) = 70 cm behind the first mirror, leading to a distance of 70 cm between the object and this fourth image.

Step 6: List and Sort Distances

We have obtained the following distances: 10 cm, 30 cm, 50 cm, and 70 cm. Sorting these gives 10 cm, 30 cm, 50 cm, and 70 cm in ascending order.

Step 7: Identify the Solutions

According to the sorted list, the smallest distance is 10 cm, the second smallest is 30 cm, the third smallest is 50 cm (occurring twice due to symmetry in mirror positions), and the fourth smallest is 70 cm.

Key Concepts

Understanding Plane Mirrors

A plane mirror is a flat, reflective surface that causes light rays to reflect at the same angle as they arrive. Plane mirrors are used widely, including in daily life, such as in bathroom mirrors and decorative room mirrors. When light from an object hits a plane mirror, it reflects in such a way that the angle of incidence equals the angle of reflection. The image that appears as a result is virtual, which means it cannot be projected onto a screen. Instead, the image seems to exist "inside" or "behind" the mirror. An important quality of images seen in plane mirrors is that they are of the same size as the object. Moreover, this image is a mirror image, meaning it has left-right reversal — what you raise with your right hand, appears as the left hand in the image.

Exploring Image Formation with Mirrors

The phenomenon of image formation in plane mirrors is fascinating. Images form because light from an object reflects off the mirror surface and into our eyes. When two plane mirrors are involved, like in our exercise, multiple images can form because light keeps bouncing back and forth between the mirrors. If an object is placed between two plane mirrors, one mirror will form the first image by reflecting the object. This image can act like a "new object" for the other mirror, leading to the creation of a second image, and the process can repeat. Each subsequent image is formed at a position equal to the distance of the preceding image from the last mirror it was reflected off of. As a result, distances can quickly become complex, requiring step-by-step calculations.

Delving into Reflection Principles

The core principle of reflection states that when a light ray hits a surface, it bounces back. The law of reflection is fundamental to understanding any mirror's function. Essentially, this law says the angle of incidence (the angle the incoming light ray makes with a line perpendicular to the surface) is equal to the angle of reflection (the angle the reflected light ray makes with the same perpendicular line). This principle allows us to clearly track how light travels and forms images in mirrors, and is crucial in solving problems that involve multiple reflectors, like our parallel mirror setup.

Analyzing Parallel Mirrors

Parallel mirrors, such as the ones in our scenario, create interesting image patterns. Since the mirrors are parallel, images reflect back and forth between them multiple times. This results in a series of images forming at particular intervals. The trickiest part often involves calculating the positions of these images. Understanding that each image can be considered a virtual "object" for the next mirror can help. It’s crucial to measure the distances for each step carefully. When dealing with parallel mirrors, the calculation follows a recursive pattern due to continual reflections. Looking at the exercise, each reflection results in an image forming further away, respecting the symmetry and distance between the mirrors. These calculated distances represent various positions where you can "see" the object’s images due to these reflections.