A camper is trying to start a fire by focusing sunlight onto...

A camper is trying to start a fire by focusing sunlight onto a piece of paper. The diameter of the sun is $1.39 \times 10^{9} \mathrm{m},$ and its mean distance from the earth is $1.50 \times 10^{11} \mathrm{m} .$ The camper is using a converging lens whose focal length is 10.0 $\mathrm{cm} .$ (a) What is the area of the sun's image on the paper? (b) If 0.530 $\mathrm{W}$ of sunlight passes through the lens, what is the intensity of the sunlight at the paper?

Key Concepts

Intensity and Power Distribution

Intensity describes the power per unit area carried by light. When analyzing optical systems like lenses, it is often useful to calculate the intensity of light at the image plane by dividing the total power transmitted through the lens by the area over which the light is distributed. This concept is crucial in understanding how much energy is concentrated on a target, such as a piece of paper being used to start a fire.

Angular Size and Magnification

Angular size refers to the angle an object subtends at the eye or lens, and it is a useful way to characterize objects that are far away. The magnification of a lens for distant objects is determined by the ratio of the image size to the actual size of the object, which can be derived from their angular sizes. For distant sources such as the sun, the small-angle approximation simplifies the calculation of the image size formed by the lens.

Lens Imaging and Focal Length

This concept involves understanding how a converging lens forms an image. The focal length is the distance from the lens at which parallel rays (such as those coming from a distant object) converge to a point. For objects at a very large distance, the image is formed approximately at the focal plane, making the focal length a critical parameter in determining the size and location of the image.

Similar Triangles in Geometric Optics

In geometric optics, similar triangles are used to relate the sizes and distances of objects and their images. When light from a distant object passes through a lens, the ratios of the object’s size-to-distance and the image’s size-to-focal length are approximately equal. This provides a method to calculate the image size based on known quantities like the object’s actual size, the distance to the object, and the lens’s focal length.

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