Problem 192 - Master Resource Book in JEE Main Physics

Question

Assuming the sun to be a spherical body of radius \(R\) at a temperature of \(T \mathrm{~K}\) evaluate the total radiant power incident on the earth at a distance \(r\) from the sun. [AIEEE 2006] (a) \(\pi r_{0}^{2} R^{2} \sigma T^{4} / r^{2}\) (b) \(r_{0}^{2} R^{2} \sigma T^{4} / 4 \pi r^{2}\) (c) \(R^{2} \sigma T^{4} / r^{2}\) (d) \(4 \pi r_{0}^{2} R^{2} \sigma T^{4} / r^{2}\)

Step-by-Step Solution

Answer

The correct option is (b) \( \frac{r_0^2 R^2 \sigma T^4}{4 \pi r^2} \).

Step 1: Use Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total power radiated per unit surface area of a black body is proportional to the fourth power of its temperature. Thus, the total power emitted by the sun, which is a black body of radius \( R \) and temperature \( T \), is given by \( P = 4 \pi R^2 \sigma T^4 \), where \( \sigma \) is the Stefan-Boltzmann constant.

Step 2: Calculate Total Radiant Power Incident on Earth

The power emitted is isotropic, meaning that it is spread uniformly in all directions. As the earth is at distance \( r \) from the sun, it intercepts a tiny fraction of this power. The area of interception is the cross-sectional area of Earth, which is \( \pi r_0^2 \) assuming the Earth as a sphere of radius \( r_0 \). The fraction of the sun's power reaching Earth is the ratio of this area to the total surface area of a sphere of radius \( r \), which is \( 4\pi r^2 \). Therefore, the power incident on Earth is \( P = \frac{\pi r_0^2}{4\pi r^2} \times 4 \pi R^2 \sigma T^4 \).

Step 3: Simplify the Expression

Simplify the expression for power incident on Earth as follows: \( P = \pi r_0^2 \frac{4 \pi R^2 \sigma T^4}{4 \pi r^2} = \frac{r_0^2 R^2 \sigma T^4}{r^2} \).

Step 4: Verify and Select the Correct Option

Looking at the options given in the original problem, the expression \( \frac{r_0^2 R^2 \sigma T^4}{r^2} \) matches option (b). Therefore, the correct choice is option (b).

Key Concepts

Black Body Radiation

Black body radiation is a fundamental concept in the field of thermodynamics and quantum physics. A black body is an idealized physical object that absorbs all incident electromagnetic radiation efficiently. When it reaches thermal equilibrium, it emits radiation known as black body radiation.
The radiation emitted by a black body is only dependent on its temperature rather than its shape or size. This emission is uniform across all wavelengths and follows Planck's law, which describes the distribution of electromagnetic radiation from a black body.

• Planck's Law: It helps us understand how radiation emission varies with wavelength and temperature.
• Peak Wavelength: Wien's Displacement Law can be used to calculate the wavelength at which peak emissions occur for a given temperature.
• Intensity: The energy radiated as black body radiation is intense and covers a wide spectrum of wavelengths.

In sum, black body radiation lays the groundwork to understand the physics of stars, planets, and other celestial bodies.

Radiant Power

Radiant power refers to the energy emitted per unit time by a body in the form of radiation. For black bodies, radiant power can be calculated using the Stefan-Boltzmann Law. This law indicates a direct relationship between the temperature of the body and the radiant power it emits.
According to the Stefan-Boltzmann Law, the radiant power emitted by a black body per unit area is proportional to the fourth power of its temperature, described by the equation:\[ P = \sigma T^4 \]where \( \sigma \) is the Stefan-Boltzmann constant:

• Stefan-Boltzmann Constant: Symbolized by \( \sigma \), its value is approximately \( 5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4} \).
• Total Radiant Power: For a spherical body like the sun, the total radiant power is calculated as \( P = 4 \pi R^2 \sigma T^4 \) with \( R \) being the radius of the sphere.

Understanding radiant power aids in calculating energy outputs from stars and other astronomical objects, and it is pivotal for assessing their influence on surrounding bodies, like Earth.

Isotropic Emission

Isotropic emission suggests that the power radiated by a body is uniformly distributed in all directions. This is often assumed in large celestial objects like stars and planets, where radiation spreads out forming concentric spherical waves.
The assumption of isotropic emission allows us to evaluate how these large objects influence surrounding areas. The energy emitted equally in all directions means that, as you move further from the source, the area over which this energy spreads increases, decreasing the intensity per unit area.

• Distribution of Power: The power from a spherical surface is divided over the surface area of a surrounding sphere, which increases with distance.
• Impact on Distance: The energy intercepted by an object like Earth is a small fraction of the total energy emitted by the sun, dependent on its distance (expressed in the equation as \( r \)).

Isotropic emission is crucial in models that estimate how much solar energy reaches Earth and consequently impacts its climate and weather systems.