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 × 10^{5} km.

#### Solution

Given,

Focal length of the concave mirror,* f *= − 7.6 m

Distance between earth and moon taken as object distance, *u* = −3.8 × 10^{5} 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:

\[m = - \frac{v}{u} = \frac{d_{image}}{d_{object}}\]

\[\Rightarrow \frac{- ( - 7 . 6)}{( - 3 . 8 \times {10}^8 )} = \frac{d_{image}}{3450 \times {10}^3}\]

\[d_{image} = - \frac{3450 \times 7 . 6 \times {10}^3}{3 . 8 \times {10}^8}\] = \[-\] 0.069 m = -6.9 cm

Hence, the required diameter of the image of moon is 6.9 cm.