Chapter 17, Problem 46A

To determine

To calculate: The angle of reflection of light ray off the second mirror.

Answer to Problem 46A

The angle of reflection of the light ray off the second mirror is $15°$ .

Explanation of Solution

Given:

The angle between the two given mirror is $45°$ .

Angle of incidence is $i=30°$ .

Formula used:

The famous mirror formula can be expressed as

  $\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$  ......( (1)

Here, $f,v$ and $u$ are focal length of the concave mirror, image distance and object distance respectively.

Magnification is calculated as

  $m=\frac{heightofimage}{heightofobject}$  ......(2)

Calculation:

Sketch a diagram showing two mirror and the rays of light:

  

The angle of incidence and the angle of reflection are measured with respect to the normal which is perpendicular to the surface.

Extend the normal lines to the two mirrors so that they can meet.

  

In figure above, one can see a quadrilateral ABCD. Since the two normal make right angles with the surface of the mirrors, $\angle DAB$ and $\angle BCD$ are $90°$ . The sum of the angles in a quadrilateral is $360°$ . Using all these information, one can find the angle D.

  $\begin{array}{l}\angle D=360°-90°-90°-45°\\ \angle D=135°\end{array}$

Now, consider the triangle with angle with angles $30°,135°$ ,and ${\theta }_i$ . One can find the angle of incidence ${\theta }_i$ using the fact that the sum of the angles of a triangle is $180°$ .

The angle of incidence on the second mirror is $15°$ . By the law of reflection, the angle of reflection is equal to the angle of incidence for a ray of light falling on a mirror.

Thus, the angle of reflection of the light ray off the second mirror is $15°$ .

Conclusion:

Hence, the angle of reflection of the light ray off the second mirror is $15°$ .