(II) The illuminance of direct sunlight on Earth is about 10⁵ lm/m². Estimate the luminous flux and luminous intensity of the Sun.

Verified step by step guidance

1. Step 1: Understand the given value, which is the illuminance of direct sunlight on Earth, approximately 100,000 lumens per square meter (lm/m²).
2. Step 2: Calculate the luminous flux (Φ) of the Sun. To do this, use the formula for illuminance (E) which is E = Φ / A, where A is the area over which the flux is spread. Consider the area as the surface area of a sphere with a radius equal to the average distance from the Sun to the Earth (approximately 150 million kilometers or 1.5 x 10^11 meters).
3. Step 3: Rearrange the formula to solve for Φ: Φ = E * A. Substitute the values for E (100,000 lm/m²) and A (the surface area of a sphere, which is 4πr², where r is the radius of the sphere).
4. Step 4: Calculate the luminous intensity (I) of the Sun. Luminous intensity is defined as the luminous flux emitted per unit solid angle along a particular direction. Use the formula I = Φ / Ω, where Ω is the solid angle in steradians (sr). For a sphere, the total solid angle is 4π steradians.
5. Step 5: Substitute the value of Φ calculated in step 3 into the formula for I and simplify to find the luminous intensity of the Sun in candelas (cd), which is the unit of luminous intensity.

Key Concepts

Illuminance

Illuminance is the measure of how much luminous flux is spread over a given area. It is quantified in lux (lm/m²) and indicates the intensity of light that illuminates a surface. In this context, the illuminance of direct sunlight on Earth is approximately 10⁵ lm/m², which serves as a basis for estimating other light-related quantities.

Luminous Flux

Luminous flux refers to the total amount of visible light emitted by a source per unit time, measured in lumens (lm). It quantifies the perceived power of light and is essential for understanding how much light is available for illumination. To estimate the luminous flux of the Sun, one can multiply the illuminance by the area over which it is distributed.

Luminous Intensity

Luminous intensity is the measure of the amount of light emitted by a source in a particular direction, expressed in candelas (cd). It is defined as the luminous flux per unit solid angle. To estimate the luminous intensity of the Sun, one can use the relationship between luminous flux and the solid angle over which the light is emitted, considering the Sun's apparent size in the sky.