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

The required diameter of the moon's image is 6.9cm.

Focal length of the concave mirror, f = − 7.6m  (Given)

Distance between earth and moon taken as object distance, u = −3.8 × 10`5 km  (Given)

Diameter of moon = 3450 km  (Given)

Using the mirror equation - 1/v = 1/u + 1/f

= 1/v + ( - 1/ 3.8 × 10`8)

= -1/7.6

Magnification = m = -v/u = dimage/ dobject

= - 7.6 / -3.8 . 10`8 = dimage / 3450 × 10³

dimage = 3450 × 7.6 × 10³ / 3.8 × 10`8

= - 0.069

= - 6.9

Thus, the required diameter of the moon's image is 6.9cm.

Answer 2

The diameter of the image of the moon formed by a spherical concave mirror of focal length 7.6 m is 6.9 cm.

Explanation:

Given data in the question  

Spherical concave focal length mirror  f = -7.6 cm

Where, f is focal length  

Moon diameter = 3450 km        

u = - 3.8 x 10 ⁵ = Which is quite big compared to f.

So we could consider it as ∞ .

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

                     $\frac{1}{-7.6}=\frac{1}{-3.8 \times 10^{5}}+\frac{1}{v}$

                    $-\frac{1}{7.6}+\frac{1}{3.8 \times 10^{5}}=+\frac{1}{v}$

                    $-\frac{1}{7.6}=+\frac{1}{v}$

Image will at formed at focus, which is inverted

                   $-\frac{1}{7.6}=+\frac{1}{v}$

Here, v = -7.6m

                   $m=\frac{-v}{u}=\frac{d_{\text {image}}}{d_{\text {object}}}$                  

     $\frac{d_{\text {image}}}{3450 \times 10^{3}}=\frac{-(7.6)}{-3.8 \times 10^{8}} m$

             $d_{\text {image}}=\frac{7.6 \times 3450 \times 10^{3}}{3.8 \times 10^{8}}$

              $d_{\text {image}}=\frac{26220 \times 10^{3}}{3.8 \times 10^{8}}$

               $d_{\text {image}}=\frac{6900 \times 10^{3}}{10^{8}}$

               $d_{\text {image}}=6900 \times 10^{-5}$

                          $=0.069 \mathrm{m}$

                          = 6.9 cm.

Therefore, the diameter of the image is 6.9 cm.