Question

Find the total number of images if the two plane mirrors are inclined at an angle of ${90^ \circ }$ and the object is situated between the pane mirrors at an angle of ${30^ \circ }$ from one of them?

Answer 1

Hint : To solve this question, we have to calculate the ratio of the total angle subtended by a circle to the angle of inclination between the plane mirrors. Then applying the condition given in the question on that ratio, we can get the final answer.

Complete step by step answer
According to the question the two plane mirrors are inclined at an angle of ${90^ \circ }$ and the object is situated at an angle of ${30^ \circ }$ from one of them.
We know that the number of images formed for an object situated between two plane mirrors which are inclined at an angle $\theta$ is decided by the ratio $\dfrac{{{{360}^ \circ }}}{\theta }$ .
Case I: When the ratio $\dfrac{{{{360}^ \circ }}}{\theta }$ is even, then the number of images is equal to one less than this ratio, that is $\dfrac{{{{360}^ \circ }}}{\theta } - 1$ .
Case II: When the ratio $\dfrac{{{{360}^ \circ }}}{\theta }$ is odd, then the number of images depends on the position of the object in between the mirrors. If the object is situated symmetrically between the mirrors, then the number of images formed is equal to one less than this ratio, that is, $\dfrac{{{{360}^ \circ }}}{\theta } - 1$ . But if the object is placed asymmetrically between the mirrors, then the number of images formed is equal to this ratio.
Now, in our case the angle between the mirrors is
$\Rightarrow \theta = {90^ \circ }$
So the ratio is given by
$\Rightarrow r = \dfrac{{{{360}^ \circ }}}{{{{90}^ \circ }}} = 4$
So the ratio comes out to be an even number. So as can be seen above, the number of images is given by
$\Rightarrow n = r - 1$
$\therefore n = 4 - 1 = 3$
Hence, the number of images formed is equal to $3$ .

Note
In this question, as the ratio comes out to be an even number, so the position of the object given in the question is irrelevant to the final answer. It is a common misconception to take the number of images formed by two plane mirrors to be equal to the ratio itself. So it is important to remember all the cases discussed above in the solution.