Question

An image of a candle (placed infront of a mirror) on a screen is found to be double its size. When the candle is shifted by a distance $5\text{cm}$, then the image becomes triple its size. Find the radius of curvature (in decimeter) of the mirror.

Answer 1

Step 1: Given that:

The size of the image($h_i$) of the candle found on screen= $2\times$ Size of object($h_o$)

When the candle is shifted by $5cm$ , the size of the image($h_i^{\prime }$)= $3\times h_o$

Step 2: Concept and formula used:

1. As the image is formed on the screen, ​the image is real and inverted.
2. Due to the formation of a real image, it is also clear that the mirror is a concave mirror.
3. A concave mirror forms a magnified real image only when the object is brought closer to the mirror.
4. In doing so, the object distance increases due to the increase in x-coordinates.
5. Magnification is given by; $m=\frac{h_i}{h_o}=-\frac{v}{u}$
6. The magnification is negative for a real image and positive for a virtual image.
7. The mirror formula is, $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$
8. Thus, the magnification formula will be given as;

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

Thus,

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

$\frac{1}{v}=\frac{u-f}{uf}$

$v=\frac{fu}{u-f}$

$-v=\frac{fu}{f-u}$

$Thus,$

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

$m=\frac{fu}{u(f-u)}$

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

Step 3: Calculation of the focal length of the mirror:

For the first case,

$m=-\frac{h_i}{h_o}$

$m=-2$

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

$m=-\frac{h_i}{h_o}$

$m=-2$

$\frac{f}{f-u}=-2$.....................(1)

For second case, when the object is shifted,

The object distance becomes, $(u+5)$ , then

$m=-\frac{h_i}{h_o}$

$m=-3$

$\frac{f}{f-(u+5)}=-3$.....................(2)

Dividing equation (1) by equation (2) we get,

$\frac{\frac{f}{f-u}}{\frac{f}{f-(u+5)}}=\frac{-2}{-3}$

$\frac{f-u-5}{f-u}=\frac{2}{3}$

$3\times (f-u-5)=2\times (f-u)$

$3f-3u-15=2f-2u$

$f=u+15$

$u=f-15$

Putting the value of $u$ in equation (1) we get

$\frac{f}{f-(f-15)}=-2$

$\frac{f}{f-f+15}=-2$

$\frac{f}{15}=-2$

$f=-30cm$

Thus, the focal length of the mirror is $30cm$ .

Step 4: Calculation of radius of curvature:
The radius of curvature of a mirror = twice the focal length of the mirror,

That is, $R=2f$

Therefore,The radius of curvature of the mirror will be= $2\times -30cm=-60cm$ = $-6dm$

Thus,

The radius of the curvature of the mirror in the decimeter is $6dm$ .