Write the mathematical expression to calculate the number of image formed by two plane mirrors placed at an angle ' \(\theta\) '

Understanding the law of reflection is crucial when studying how images are formed by plane mirrors. This fundamental principle states that the angle at which light hits a reflective surface (the angle of incidence) is equal to the angle at which it bounces off that surface (the angle of reflection). This means that if a light ray hits a mirror at a 30-degree angle relative to the normal (an imaginary line perpendicular to the point of incidence), it will reflect off the mirror at the same angle.

When dealing with plane mirrors, the images produced are always virtual, upright, and the same size as the object. Due to the law of reflection, the image of the object seems to be behind the mirror at the same distance the object is in front of it. As lights reflects in such predictable manner, we can map out the path light takes and hence the location and number of images formed in setups involving multiple mirrors.

The angles between mirrors have a significant impact on the number of images formed in a system with two plane mirrors. Imagine placing two mirrors at a certain angle \( \theta \) to each other. An object placed between these mirrors will have its light rays reflected back and forth, creating multiple images. The number of images depends on how many times the light can 'fit' into the angular space between the mirrors before completing a full rotation (360 degrees).

For instance, if the mirrors are set at a 90-degree angle, we can fit the light path exactly four times in a full circle (360/90 = 4), resulting in three additional images apart from the real object. The number of potential image positions is calculated by dividing 360 by the angle between the mirrors, but since the actual object is not considered an image, we subtract one from this quotient to get the number of images. It's this relationship between mirror angles and the division of a full circle that drives the formation of multiple images.

In image formation using plane mirrors, it is important to consider the concept of integer values. The formula to calculate the number of images, \( n = \frac{360^\circ}{\theta} - 1 \), will not always yield an integer. Non-integer results are not physically meaningful in the context of image formation because you cannot have a fraction of an image; you either have a full image or none.

Therefore, when the quotient \( \frac{360^\circ}{\theta} \) is not an integer, we must use the floor function, represented by \( \lfloor x \rfloor \), which gives the greatest integer less than or equal to 'x'. This ensures that we only count whole images. The decision to use either formula depends on the result being an integer or not. Thus, we correct the mathematical expression to account for this, ensuring that an accurate, physically possible number of images is given.