Question

A circular wire of diameter 30 mm is placed 25 cm away from a concave mirror of focal length 10 cm. If area of image is $X\pi$ $m{m}^2$, what is the value of x? The centre of the wire is on the axis of the mirror, with its plane normal to the axis.

Answer 1

Focal length of concave mirror $f=-10cm$
Object distance $u=-25cm$

Object height $=$ diameter of circular wire $=3cm=h_o$

Object height $h_o=3cm$

Mirror formula: $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

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

$=\frac{-1}{10}+\frac{1}{25}$

$=\frac{(-5+2)}{50}$

$=\frac{-3}{50}$

$v=\frac{-50}{3}cm=-16.7cm$

Magnification $m=\frac{-v}{u}=\frac{-16.7}{25}=0.668=\frac{h_i}{h_o}=\frac{h_i}{3}$

$\Rightarrow hi=3\times 0.668=2.00=4cm\simeq 2cm=$ diameter of image
Area of the image = $Area=\pi \frac{d^2}{4}$

$Area=\pi \frac{d^2}{4}=\pi \times \frac{2^2}{4}=\pi =3.14$

Value of $X$ is 1

$X=1$