Question

When observed from the earth the angular diameter of the sun is $0.5$ degree. The diameter of the image of the sun when formed in a concave mirror of focal length of $0.5m$ will be about:
$A.3mm$
$B.4.4mm$
$C.5.6mm$
$D.8.8mm$

Answer 1

We should use the concept that the angle subtended by the common tangents with the sun and its image are vertically opposite in nature. We know that vertically opposite angles are equal to each other. Basic relation between length of arc and radius can be used to find the correct answer.
Formula Used:
We are going to use the following geometrical relation to find the accurate and precise answer.
$D=\theta r$ .

Complete step by step answer:
Angles subtended $(\theta )$ by the common tangents with the sun and its image are equal because they are vertically opposite. So, $\theta ={{0.5}^{o}}$
Focal length will be taken as distance, $r=0.5m$
Let, $D$ be the diameter of the image of the sun when formed in a concave mirror. We have to find $D$.
We will use
$D=\theta r$……………. $(i)$
Putting values we get
$\Rightarrow D={{0.5}^{o}}\times 0.5m$……………….. $(ii)$
We have to convert the degree measure of the angle to radian measure to apply the above mentioned formula. To convert degree measure into radian measure we must multiply it with $\dfrac{\pi }{{{180}^{o}}}$.
Doing this operation in equation $(D)$ we get
$\Rightarrow D=0.5\times \dfrac{\pi }{180}\times 0.5$
$\Rightarrow D=\dfrac{5\times 5\times \pi }{180\times 100}$
$\Rightarrow D=\dfrac{\pi }{36\times 20}$
Putting value of $\pi =3.14$ we get,
$\Rightarrow D=\dfrac{3.14}{36\times 20}$
$\Rightarrow D=4.4\times {{10}^{-3}}m$
$\Rightarrow D=4.4mm$

Hence, the correct option is $(B)$ among the given options.

Note:
In solving these types of problems of optics we must consider the basics of geometry. Nature of mirrors should also be taken care of because properties of concave mirror and convex mirror are quite different from each other. We should also take care whether we have to work on mirrors or lenses. Keeping proper mathematical relations we will get the correct answer most of the time.