A mirror forms an image which is 30 cm from an object and twice its height.(a) Where must the mirror be situated?(b) What is the radius of curvature?(c) Is the mirror convex or concave?

(a) Given:

Image distance $-$ Object distance, $(v-u)$ = 30 cm

Then, $v=30+u$

Magnification, $m$ = 2

To find: Position of the mirror.

Solution:

From the magnification formula, we know that-

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

Substituting the given values in the magnification formula we get-

$2=\frac{-v}{u}$

$2u=-v$     

$-2u=-(-v)$      (multiply both sides with a negative sign)

$v=-2u$  .............................. (i)              

By putting the value of $'v'$ in the equation (i), we get-

$30+u=-2u$     

$30=-2u-u$     

$-3u=30$   

$u=-\frac{30}{3}$

$u=-10cm$

Thus, the object is situated at a distance of 10 cm from the lens. Hence, the mirror must be situated at a distance of 10 cm from the object.

Now, putting the value of $u$ in the equation (i) we get the image distance as- $v=−2\times {−10}=20cm$

(b) We have,

Image distance, $v$ = 20 cm

Object distance, $u$ = $-$10 cm

To find: Radius of curvature, $R$.

Solution:

From the mirror formula, we know that-

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

Substituting the given values in the mirror formula we get-

$\frac{1}{f}=\frac{1}{20}+\frac{1}{(-10)}$

$\frac{1}{f}=\frac{1}{20}-\frac{1}{10}$

$\frac{1}{f}=\frac{1-2}{20}$

$\frac{1}{f}=\frac{-1}{20}$

$-f=20$

$-(-f)=-20$                     (multiply both sides with a negative sign)

$f=-20$ 

Thus, the focal length of the mirror is 20 cm.

Now, we know that-

$R=2f$, where R = Radius of curvature, and f = focal length

$R=2\times {-20}$

$R=-40cm$

Thus, the radius of curvature of the mirror is 40 cm.

(c) Since the focal length has a negative value, the given mirror is concave.

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Updated on: 10-Oct-2022

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