Deriving the Formula for Images Formed Between Two Mirrors

Mirrors are fascinating optical tools whose behavior can be quantified and understood mathematically. When two mirrors are placed at an angle a to each other, they can form multiple images of an object placed in between. The formula that gives the number of images formed is n 360°/a - 1, where n is the number of images. This article delves into the derivation of this formula and explores the geometric principles behind it.

Understanding the Geometry

When two mirrors are placed at an angle a to each other, the reflections create images of the object. Each image acts as a new full circle, contributing to the total number of images formed. This is because the reflections and images of the mirrors themselves create additional angles of reflection around the object.

Full Circle Consideration

The key idea in the derivation is to consider the full circle around the object. The total angle around the object is 360 degrees. This full circle is divided into sectors by the angle a between the mirrors. Each sector represents a region where reflections occur and additional images are formed.

Angles Between Mirrors

Since the mirrors are inclined at an angle a, the number of reflections or images created is determined by how many times the angle a can fit into 360 degrees. The number of times a fits into 360 degrees is given by:

[ frac{360}{a} ]

Counting Images

However, the original object itself does not count as an image. Therefore, to find the total number of distinct images formed, we subtract 1 from this quotient:

[ n frac{360}{a} - 1 ]

Conclusion

Thus, the formula ( n frac{360°}{a} - 1 ) gives the number of images formed by two mirrors inclined at an angle a. This formula accounts for all reflections except the original object. It is typically used when the angle ( a ) divides 360 degrees evenly, and ( a ) is measured in degrees.

Additional Insights

The formula ( n frac{360°}{a} - 1 ) holds true if ( frac{360°}{a} ) is an integer. If the division ( frac{360°}{a} ) is not an integer, then it is not possible to form an exact number of images based on this formula. However, it can be useful to pose the question in a different way: if you wish to produce ( n ) images in the reflections of angled mirrors, the mirrors can be angled precisely at ( frac{360°}{n 1} ) degrees. By drawing this out, you can see that the mirrors and their images divide the space into ( n 1 ) zones. One of these zones contains the object, and the other ( n ) zones contain images of the object.

Conclusion and Summary

The formula ( n frac{360°}{a} - 1 ) encapsulates the complex interplay of angles and reflections that occur when two mirrors are placed at an angle. This derivation is a fundamental tool in understanding and predicting the behavior of mirrors in various optical setups. By grasping the geometric principles behind this formula, you can unlock a deeper understanding of the world of reflections and images.