Assume the intensity of sunlight is 1.00 kW / m^2 at a particular location. A highly reflecting

step 1 We have power equals intensity times area equals intensity times pi r squared. So we find r equa

Assume the intensity of sunlight is 1.00 kW / m^2 at a...

Assume the intensity of sunlight is 1.00 $\mathrm{kW} / \mathrm{m}^{2}$ at a particular location. A highly reflecting concave mirror is to be pointed toward the Sun to produce a power of at least 350 $\mathrm{W}$ at the image. (a) Find the required radius $R_{a}$ of the circular face area of the mirror. (b) Now suppose the light intensity is to be at least 120 $\mathrm{kW} / \mathrm{m}^{2}$ at the image. Find the required relationship between $R_{a}$ and the radius of curvature $R$ of the mirror. The disk of the Sun subtends an angle of $0.533^{\circ}$ at the Earth.

Key Concepts

Concave Mirror Focusing

A concave mirror converges incoming parallel rays (from a distant source such as the Sun) to a focal point by reflecting them. The radius of curvature of the mirror is a key parameter that determines its focusing properties, including the size and intensity of the image produced. Understanding how a concave mirror works is essential when designing an optical system intended to concentrate light onto a small region.

Angular Size of the Source and Concentration Limits

The finite angular size of the light source, such as the Sun, imposes a limit on the ability of any optical system to concentrate light. Because the source does not emit perfectly parallel rays but instead subtends an angle at the collection point, there is a minimum achievable spot size at the focus. This concept is critical when deriving relationships between the mirror’s dimensions and its curvature to meet specific concentration requirements.

Intensity and Power

Intensity is defined as the power per unit area. In problems involving energy collection, such as capturing sunlight, the total power collected can be determined by multiplying the source's intensity by the area over which it is collected. This relationship is fundamental in calculating how much energy a surface can gather from a uniform energy flux.

Mirror Geometry and Aperture

The mirror's geometric configuration, specifically its circular face, dictates the area that intercepts the incident light. Determining the required radius of the mirror’s aperture involves equating the product of the mirror’s area and the incoming light intensity to the desired power level. This design aspect is crucial for focusing and redirecting light, ensuring adequate energy is concentrated at the image point.

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