Problem 185 - Master Resource Book in JEE Main Physics

Question

A body with area \(A\) at maintained temperature \(T\) and emissivity \(e=0.6\) is kept inside a spherical black body. What will be the maximum energy radiated per second? (a) \(0.60 \sigma A T^{4}\) (b) \(0.80 \sigma A T^{4}\) (c) \(1.00 \sigma A T^{4}\) (d) \(0.40 \sigma A T^{4}\)

Step-by-Step Solution

Answer

Option (a) \( 0.60 \sigma A T^{4} \) is the maximum energy radiated per second.

Step 1: Understanding the Stefan-Boltzmann Law

The Stefan-Boltzmann Law gives the power radiated per unit area of a black body as \( j = \sigma T^4 \), where \( \sigma \) is the Stefan-Boltzmann constant. For a body with emissivity \( e \), the power radiated is \( j = e \sigma T^4 \).

Step 2: Calculating the Total Emitted Power

The total energy radiated per second by a body with area \( A \) and emissivity \( e \) at temperature \( T \) is given by the formula \( P = e \sigma A T^4 \). Here, the emissivity \( e \) is 0.6.

Step 3: Insert Known Values

Insert the given values into the formula: \( P = 0.6 \sigma A T^4 \). This represents the power, or energy radiated per second, from the body inside the spherical black body.

Step 4: Selecting the Correct Option

Given the options, the correct answer corresponds to \( 0.60 \sigma A T^4 \), which is option (a).

Key Concepts

Emissivity

Emissivity is a measure of how efficiently a surface emits thermal radiation compared to an ideal black body, which has an emissivity of 1. Real surfaces have an emissivity value between 0 and 1. A surface with low emissivity reflects more thermal radiation, while a surface with high emissivity emits more thermal radiation. Some key points about emissivity:

• A perfect black body has an emissivity of 1.
• A perfect reflector, which emits no radiation, has an emissivity of 0.
• Emissivity depends on factors such as material composition, surface texture, and temperature.

The concept of emissivity is crucial in the Stefan-Boltzmann Law, where it modifies the power radiated by a real object to reflect its efficiency relative to a black body. For example, in our exercise, the object with an emissivity of 0.6 emits 60% of the thermal radiation that a perfect black body would emit at the same temperature.

Radiation

Radiation is a mode of energy transfer that occurs through electromagnetic waves. Unlike conduction and convection, radiation does not require a medium, meaning it can occur through a vacuum. Some important aspects of radiation include:

• All objects emit some level of thermal radiation if their temperature is above absolute zero.
• The amount of emitted radiation depends on the temperature and surface properties of the object.
• Radiation can occur in any direction and over any distance, making it a key process in applications such as heating, cooling, and climate science.

In the context of our exercise, radiation specifically refers to the thermal energy emitted by the body due to its temperature and emissivity properties. The Stefan-Boltzmann Law quantitatively describes the power emitted in terms of temperature and emissivity.

Black Body Radiation

Black body radiation is the theoretical concept where a perfect black body absorbs all incidental energy, reflects none, and re-emits energy at maximum efficiency for a given temperature. It serves as an ideal standard to measure real-world radiation against. The spectral distribution of this radiation depends only on the body's temperature. Key characteristics of black body radiation:

• The energy emitted covers the entire electromagnetic spectrum, with a specific dependence on wavelength according to Planck’s Law.
• It provides a benchmark for evaluating the emissive properties of different materials.
• Black body radiation was crucial in the development of quantum mechanics, helping to explain observed deviations in classical models.

In our exercise, the black body acts as a reference environment that perfectly absorbs and re-emits energy, allowing us to evaluate the energy emitted by the real object within it using the modified Stefan-Boltzmann Law.