Question

A candle flame $3cm$ high is placed at a distance of $3m$ from a wall$.$ How far from the wall must a concave mirror be placed so that it may form a $9cm$ high image of the flame on the same wall$?$ Also find the focal length of the mirror$.$

Answer 1

Concave mirror is placed at, say distance $d$ metres from the candle$.$

So the distance of wall from the mirror $=3+d$ metres

Object distance $u=-d$ metres

Image distance $v=-(3+d)$ metres

Since we are able to obtain the image on the wall, then the image must be real and inverted.

Given:

Height of the image $h^{\prime }=-9cm$

(negative sign because real and inverted image)

Height of the object $h=+3cm$

Magnification $m=\frac{h^{\prime }}{h}=\frac{-9}{+3}=-3$

We know that, magnification is also expressed as:

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

$\Rightarrow -3=-\frac{-(3+d)}{-d}$

$\Rightarrow 3=\frac{3+d}{d}$

$\Rightarrow 3d=3+d$

$\Rightarrow d=1.5m$

Thus, the concave mirror is to be placed at a distance of $3+1.5=4.5m$ from the wall.

Now,

$u=-1.5m$

$v=-4.5m$

Focal length $f=?$

By mirror formula,

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

$\frac{1}{f}=\frac{1}{-4.5}+\frac{1}{-1.5}$

$\frac{1}{f}=-\left(\frac{1.5+4.5}{4.5\times 1.5}\right)=-\frac{6}{4.5\times 1.5}=-\frac{4}{4.5}$

$f=-\frac{4.5}{4}=-1.125m$

Answer:

• The concave mirror is to be placed at a distance of $4.5m$ from the wall.
  
• The focal length of the mirror is $f=-1.125m.$