Question

Find the diameter of the image of the moon formed by a spherical concave mirror of focal length 7.6 m. The diameter of the moon is 3450 km and the distance between the earth and the moon is 3.8 × 10⁵ km.

Answer 1

Given,
Focal length of the concave mirror, f = − 7.6 m
Distance between earth and moon taken as object distance, u = −3.8 × 10⁵ km
Diameter of moon = 3450 km
Using mirror equation:
$\Rightarrow \frac{1}{v}=\frac{1}{u}+\frac{1}{f}$
$\therefore \frac{1}{v}+\left(-\frac{1}{3.8\times {10}^8}\right)=\left(-\frac{1}{7.6}\right)$

As we know, the distance of moon from earth is very large as compared to focal length it can be taken as ∞.
Therefore, image of the moon will be formed at focus, which is inverted.
$\Rightarrow \frac{1}{v}=-\frac{1}{7.6}$
⇒ v = −7.6 m

We know that magnification is given by:
$m=-\frac{v}{u}=\frac{d_{\mathrm{image}}}{d_{\mathrm{object}}}$
$\Rightarrow \frac{-(-7.6)}{(-3.8\times {10}^8)}=\frac{d_{image}}{3450\times {10}^3}$
$d_{\mathrm{image}}=-\frac{3450\times 7.6\times {10}^3}{3.8\times {10}^8}$
= $-$0.069 m = $-$6.9 cm
Hence, the required diameter of the image of moon is 6.9 cm.