Question

An object is placed between two plane mirrors inclined at some angle to each other. If the number of images formed is \[7\] then the angle of inclination is:
A. \[{32^ \circ }\]
B. \[{53^ \circ }\]
C. \[{45^ \circ }\]
D. \[{60^ \circ }\]

Answer 1

Hint:Mirrors that are angled. When an object's image is viewed through two plane mirrors that are inclined to each other, multiple images are created. The number of pictures produced is determined by the angle formed by the two mirrors.

Formula used:
$n = \dfrac{{360}}{\theta } - 1$
Here, $n$ is the number of images formed and $\theta $ is the angle between the mirrors.

Complete step by step answer:
We know that if $\theta $ is the angle between the mirrors, then \[\dfrac{{360}}{\theta }\] is the number of pictures formed. If n is an even number, then the image's number is \[\left( {n - 1} \right)\] . (This is true for both symmetrical and asymmetrical item positions.)
Now, as we know from the above information, for inclined mirrors,
$n = \dfrac{{360}}{\theta } - 1$
Here, $n$ is the number of images formed and $\theta $ is the angle between the mirrors.

It is given here that; $n$ = $7$
Therefore, putting it into the formula, we will get the desired angle;
\[\Rightarrow 7 = \dfrac{{{{360}^ \circ }}}{\theta } - 1 \\
\Rightarrow 7 + 1 = \dfrac{{{{360}^ \circ }}}{\theta } \\
\Rightarrow 8 = \dfrac{{{{360}^ \circ }}}{\theta } \\
\therefore \theta = \dfrac{{{{360}^ \circ }}}{8} = {45^ \circ } \]
Thus, the angle of inclination of the mirror for the formation of \[7\] images is ${45^ \circ }$

So, the correct option is C.

Additional Information: If \[n\] is an even number, the image's number is n for the object's symmetrical position and \[\left( {n - 1} \right)\] for the object's unsymmetrical position. If \[n\] is a fraction, then the number of pictures is the same as the integer component

Note:When the angle between the two mirrors is \[{180^ \circ }\] , they serve as one mirror, allowing only one image to be seen. The object, as well as the mirrors themselves, became pictured in one another as the angle between them reduced. As a result, as the angle between the mirrors is reduced, one can see image within image, image within image, and so on. This allows one to see a large number of images. Infinite pictures are expected to appear if the angle between the mirrors is eventually reduced to zero.