(a) A security mirror used in a big showroom has a radius of curvature of 5 m. If a customer is standing at a distance of 20 m from the cash counter, find the position, nature, and size of the image formed in the security mirror.(b) Neha visited a dentist in his clinic. She observed that the dentist was holding an instrument fitted with a mirror. State the nature of this mirror and the reason for its use in the instrument used by dentist.

(a) We know that for a security mirror, a convex mirror is used.

Given:

Object distance, $u$ = $-$20m

Radius of curvature, $R$ = $+$5 m

Then, focal length, $f$ = $\frac {5}{2}$ =+2.5 m   $(\because f=\frac {R}{2})$

To find: The position, nature, $(v)$ and size of the image $(h')$.

Solution:

From the mirror formula, we know that-

$\frac {1}{v}+\frac {1}{u}=\frac {1}{f}$

Substituting the given values we get-

$\frac {1}{v}+\frac {1}{(-20)}=\frac {1}{2.5}$

$\frac {1}{v}-\frac {1}{20}=\frac {10}{25}$

$\frac {1}{v}=\frac {10}{25}+\frac {1}{20}$

$\frac {1}{v}=\frac {40+5}{100}$

$\frac {1}{v}=\frac {45}{100}$

$v=\frac {100}{45}$

$v=\frac {20}{9}$

$v=+2.2m$

Thus, the position or distance of the image, $v$ is 2.2 m from the mirror, and the positive sign implies that it forms on the right side of the image. 

Therefore, the nature of the image will be virtual and erect, and its size will be diminished, as the convex mirror always formed a virtual, erect, and diminished image regardless of the object distance from the mirror.

Now, from magnification formula we know that-

$m=\frac {v}{u}$

$m=\frac {2.2}{-20}$

$m=-\frac {22}{200}$

$m=-0.11$

Thus, the height of the image, $h'$ is 0.11 times smaller than the object.

(b) The nature of this mirror is concave mirror and the reason for its use in the instrument used by dentist is to obtain an erect and magnified image of teeth.

Explanation

The dentist holds a small concave mirror in such a way that the tooth lies within the focus of the mirror, and then a magnified image of the tooth is seen by the dentist in the concave mirror. Since the tooth looks much larger, it becomes easier to check for a defect in the tooth.