A boy is trying to start a fire by focusing sunlight on a...

A boy is trying to start a fire by focusing sunlight on a piece of paper using an equiconvex lens of focal length $10 \mathrm{~cm}$. The diameter of the sun is $1.39 \times 10^{9} \mathrm{~m}$ and its mean distance from the earth is $1.5 \times 10^{11} \mathrm{~m}$. What is the diameter of the sun's image on the paper? (a) $6.5 \times 10^{-5} \mathrm{~m}$ (b) $12.4 \times 10^{-4} \mathrm{~m}$ (c) $9.2 \times 10^{-4} \mathrm{~m}$ (d) $6.5 \times 10^{-4} \mathrm{~m}$

A boy is trying to start a fire by focusing sunlight on a piece of paper using an equiconvex lens of focal length $10 \mathrm{~cm}$. The diameter of the sun is $1.39 \times 10^{9} \mathrm{~m}$ and its mean distance from the earth is $1.5 \times 10^{11} \mathrm{~m}$. What is the diameter of the sun's image on the paper?
(a) $6.5 \times 10^{-5} \mathrm{~m}$
(b) $12.4 \times 10^{-4} \mathrm{~m}$
(c) $9.2 \times 10^{-4} \mathrm{~m}$
(d) $6.5 \times 10^{-4} \mathrm{~m}$

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Key Concepts

Convex Lens

A convex lens is a converging optical element that bends incoming parallel light rays to a focal point. It is essential in forming real, inverted images when the object is placed beyond the focal length, as is the case in focusing sunlight to produce an image on a piece of paper.

Focal Length

The focal length is the distance from the center of the lens to its focal point, where parallel rays converge. It is a critical parameter in determining the size and placement of the image, as it scales the angular size of the incoming light to the actual image size on the screen.

Angular Size and Small-Angle Approximation

Angular size is the angle that an object subtends at the lens. For objects that are very far away, like the sun, this angle is small and can be approximated by the ratio of the object’s physical size to its distance. Multiplying this angle by the focal length of the lens gives an estimate of the image size on the screen.

Image Formation Using Similar Triangles

The process of image formation in a lens system can be analyzed using similar triangles. The ratio of the image size to the focal length is equivalent to the ratio of the object's size to its distance. This relationship helps to determine the dimensions of the image produced by the lens.

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