Problem 56 - Master Resource Book in JEE Main Physics

Question

The rate of radiation of a black body at \(0^{\circ} \mathrm{C}\) is \(E\) watt. The rate of radiation of this body at \(273^{\circ} \mathrm{C}\) will be (a) \(16 \bar{E}\) (b) \(8 E\) (c) \(4 \underline{E}\) (d) \(E\)

Step-by-Step Solution

Answer

The rate of radiation at \(273^{\circ} \mathrm{C}\) is \(16E\).

Step 1: Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. Mathematically, this can be expressed as \( P = \sigma A T^4 \), where \( P \) is the power radiated, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the area of the black body, and \( T \) is the absolute temperature in kelvin.

Step 2: Convert Temperatures to Kelvin

We need to convert the given temperatures from Celsius to Kelvin. The absolute temperature in Kelvin is obtained by adding 273.15 to the Celsius temperature.- For \(0^{\circ} \mathrm{C}\): \[ T_1 = 0 + 273.15 = 273.15 \text{ K} \]- For \(273^{\circ} \mathrm{C}\): \[ T_2 = 273 + 273.15 = 546.15 \text{ K} \]

Step 3: Apply the Stefan-Boltzmann Law to Both Cases

From the Stefan-Boltzmann Law, the radiation rates at these temperatures are:1. At \(0^{\circ} \mathrm{C}\): \[ E = \sigma A (273.15)^4 \]2. At \(273^{\circ} \mathrm{C}\): \[ E_2 = \sigma A (546.15)^4 \]

Step 4: Calculate the Ratio of the Radiation Powers

Find the ratio of the radiation rates at the two temperatures:\[ \frac{E_2}{E} = \frac{\sigma A (546.15)^4}{\sigma A (273.15)^4} = \left( \frac{546.15}{273.15} \right)^4 \]Calculating the ratio:\[ \frac{546.15}{273.15} = 2 \]Thus,\[ \frac{E_2}{E} = 2^4 = 16 \]

Step 5: Determine the New Rate of Radiation

The radiation rate at \(273^{\circ} \mathrm{C}\) is 16 times the rate at \(0^{\circ} \mathrm{C}\). Therefore, \( E_2 = 16E \).

Key Concepts

Black Body Radiation

A black body is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It neither reflects nor transmits any light, making it a perfect absorber.
In actuality, a true black body does not exist, but plenty of materials come close, allowing us to apply the concept in real-world physics problems and theoretical discussions.
Black body radiation is the thermal electromagnetic radiation emitted from a black body. This radiation has a characteristic spectrum that solely depends on the temperature of the body.

• The radiation emanates from the body in a continuous spectrum.
• At higher temperatures, the peak of the emission spectrum shifts to shorter wavelengths.
• This behaviour is in line with Wien's Displacement Law.

Understanding black body radiation allows scientists and people like you and me to dive deeper into complex topics such as stellar optics and the study of cosmic microwave background radiation.

Absolute Temperature

Absolute temperature is a temperature measured on a scale starting from absolute zero, the point where all thermal motion ceases. The Kelvin scale is the most commonly used scale for absolute temperature in scientific measurements.
Absolute temperature plays a crucial role in physics, especially when dealing with thermodynamic processes. It is distinguished from Celsius and Fahrenheit by its absolute nature, where no negative values exist. Thus, zero on the Kelvin scale is the coldest possible temperature, any material can theoretically reach known as absolute zero.

• Absolute zero is 0 Kelvin, equivalent to -273.15 °C.
• To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.
• Kelvin is commonly used in scientific calculations, such as those involving the gas laws and thermodynamics.

In the context of the Stefan-Boltzmann Law, the power radiated is closely tied to the fourth power of the absolute temperature, demonstrating how pivotal it is in achieving precise scientific comprehension.

Power Radiated

The power radiated by an object is the total energy emitted per unit time. In the context of the Stefan-Boltzmann Law, the power radiated by a black body is defined as proportional to the fourth power of its absolute temperature. This fundamental relationship provides insight into how temperature influences energy emission.
The equation used to determine the power radiated by a black body is:
\[ P = \sigma A T^4 \]
Where:

• \( P \) is the power radiated.
• \( \sigma \) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4})\).
• \( A \) is the surface area of the black body.
• \( T \) is the absolute temperature in Kelvin.

By analyzing the power radiated at different temperatures, researchers can infer the temperature of distant objects, like stars, by measuring the radiation they emit. This is pivotal in astronomical observations and helps us understand the universe better.