Question

Two plane mirrors intersect at right angles. A laser beam strikes the first of them at a point 11.5 cm from their point of intersection, as shown in the figure .
their point of intersection, as shown in the figure .

For what angle of incidence at the first mirror will this ray strike the midpoint of the second mirror(which is 28.0 cm long) after reflecting from the first mirror?

Answer 1

Concepts and reason

The concepts needed to solve this problem are the law of reflection, trigonometry ratio of tangent and the concept of alternative angles.

First, draw a needful ray diagram of the situation. Use the concept of alternative angle and the formula for trigonometry ratio of tangent and solve for required angle using the law of reflection.

Fundamentals

The formula for trigonometry ratio of tangent is,

${\rm{tan}}\theta = \frac{{{\rm{opposite side of the angle}}}}{{{\rm{adjacent side of the angle}}}}$

Here, $\theta$ is the angle.

One of the laws of reflection is that the angle of incidence is equal to the angle of reflection.

The needful ray diagram of the situation is as follows:

In the diagram, ${i_1}$ is the incident angle at first incident, $r$ is angle of refraction at first incident, and ${i_2}$ is the glancing angle of incident of light at midpoint of second mirror.

The formula for trigonometry ratio of tangent is,

${\rm{tan}}\theta = \frac{{{\rm{opposite side of the angle}}}}{{{\rm{adjacent side of the angle}}}}$

Substitute ${i_2}$ for $\theta$ , 11.5 cm for opposite side of angle, and 14.0 cm for adjacent side of the angle.

$\begin{array}{c}\\\tan {i_2} = \frac{{11.5{\rm{ cm}}}}{{14.0{\rm{ cm}}}}\\\\ = {\rm{39}}{\rm{.4}}^\circ \\\end{array}$

In the diagram, the angle ${i_2}$ and the angle r are alternative angles.

${i_2} = r = 39.4^\circ$

From the law of reflection, the angle of incidence is equal to the angle of reflection.

${i_1} = r = 39.4^\circ$

Ans:

The required value of angle of incidence is $39.4^\circ$