40. A boy is trying to start a fire by focusing sunlight on a piece of paper using an equiconvex lens of focal length 10 cm. The diameter of the Sun is 1.39 × 10<sup>9</sup> m and its mean distance from the earth is 1.5 × 10<sup>11</sup> m. What is the diameter of the Sun’s image on the paper ?
(a) \( 9.2 \times 10^{-4} \ \text{m} \)
(b) \( 6.5 \times 10^{-4} \ \text{m} \)
(c) \( 6.5 \times 10^{-5} \ \text{m} \)
(d) \( 12.4 \times 10^{-4} \ \text{m} \)
Answer : (a) \( 9.2 \times 10^{-4} \ \text{m} \)
\( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)
\( \frac{1}{10\, \text{cm}} = \frac{1}{v} - \frac{1}{1.5 \times 10^{11}} \)
\( \frac{1}{v} = \frac{1}{10\, \text{cm}} + \frac{1}{1.5 \times 10^{11} \times 100} \)
\( v = 10\, \text{cm} \)
\( \frac{h_2}{\frac{1.39 \times 10^9}{2}} = \frac{10\, \text{cm}}{1.5 \times 10^{11}\, \text{m}} \)
\( h_2 = \frac{10}{1.5 \times 10^{13}} \times \frac{1.39 \times 10^9}{2} \times 2 \quad \text{(For diameter)} \)
\( = \frac{14}{15} \times 10^{-3} \)
\( = 0.9 \times 10^{-3} \)
\( = 9 \times 10^{-4}\, \text{m} \)
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