The sun is 150,000,000 km from earth; its diameter is 1,400,000 km For a science project on sola

step 1 In this question, a student uses a converging mirror. So it's converging mirror. It has a diamet

The sun is 150,000,000 km from earth; its diameter is...

The sun is $150,000,000 \mathrm{km}$ from earth; its diameter is $1,400,000 \mathrm{km}$ For a science project on solar For a science project on solar power, a student uses a 24 -cm-diameter converging mirror with a focal length of $45 \mathrm{cm}$ to focus sunlight onto an object. This casts an image of the sun on the object. For the most intense heat, the image of the sun should be in focus. a. Where should the object be placed? b. What is the diameter of the image? c. The intensity of the incoming sunlight is $1050 \mathrm{W} / \mathrm{m}^{2} .$ What is the total power of the light captured by the mirror? d. What is the intensity of sunlight in the projected image? Assume that all of the light captured by the mirror is focused into the image.

The sun is $150,000,000 \mathrm{km}$ from earth; its diameter is $1,400,000 \mathrm{km}$ For a science project on solar For a science project on solar power, a student uses a 24 -cm-diameter converging mirror with a focal length of $45 \mathrm{cm}$ to focus sunlight onto an object. This casts an image of the sun on the object. For the most intense heat, the image of the sun should be in focus.
a. Where should the object be placed?
b. What is the diameter of the image?
c. The intensity of the incoming sunlight is $1050 \mathrm{W} / \mathrm{m}^{2} .$ What is the total power of the light captured by the mirror?
d. What is the intensity of sunlight in the projected image? Assume that all of the light captured by the mirror is focused into the image.

Key Concepts

Parallel Rays and Focal Point

Since the sun is extremely distant, the light rays reaching the mirror are nearly parallel. In geometric optics, parallel rays that strike a converging mirror will focus at the mirror’s focal point. This key concept establishes why the object for the most intense heating should be placed at the focal distance from the mirror.

Mirror Equation in Geometric Optics

The mirror equation, which relates the object distance, image distance, and focal length, is central in predicting where an image will form. In situations where the object is at an effectively infinite distance (as with the sun), the equation simplifies to having the image at the focal length. This concept underpins the calculations of image position in mirror systems.

Magnification and Similar Triangles

Magnification in mirror systems is determined by the ratio between the image size and the object size, which can also be derived from similar triangles formed by the geometry of the incoming parallel rays and the converging mirror. This principle allows one to determine the diameter of the sun’s image on the target, based on the ratio of the mirror’s focal length to the sun’s distance.

Area Calculation and Light Collection

Calculating the area of the mirror, which typically involves determining the area of a circle from its diameter, is crucial to determining the total power of the sunlight collected. Multiplying the mirror's area by the given sunlight intensity yields the total light power intercepted by the mirror, a fundamental step in energy and optics problems.

Power, Intensity, and Energy Concentration

Intensity is defined as power per unit area. When a converging mirror focuses sunlight from a large area (its aperture) into a smaller image area, the same power is redistributed over a smaller area, increasing the intensity. Understanding this conservation of energy and how light energy is concentrated is essential for applications such as solar heat concentration.

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