Problem 151 - Master Resource Book in JEE Main Physics

Question

Assertion The SI unit of Stefan's constant is \(\mathrm{Wm}^{-2} \mathrm{~K}^{-4}\). Reason This follows from Stefan's Law, \(\therefore\) $$ \begin{aligned} E &=\alpha T^{4} \\ \alpha &=\frac{E}{T^{4}}=\frac{\mathrm{Wm}^{-2}}{\mathrm{~K}^{4}} \end{aligned} $$

Step-by-Step Solution

Answer

The SI unit for Stefan's constant is correctly derived as \( \mathrm{Wm}^{-2} \mathrm{~K}^{-4} \).

Step 1: Identifying the Assertion

The assertion claims that the SI unit of Stefan's constant, often denoted as \( \sigma \), is \( \mathrm{Wm}^{-2} \mathrm{~K}^{-4} \).

Step 2: Understanding Stefan's Law

Stefan's Law states that the total energy radiated per unit surface area of a black body per unit time (energy per unit area per unit time, \( E \)) is proportional to the fourth power of the black body's thermodynamic temperature \( T \). This law is expressed mathematically as \( E = \sigma T^4 \).

Step 3: Deriving the Units from Stefan's Law

According to Stefan's Law, \( E = \sigma T^4 \). Rearranging the formula gives \( \sigma = \frac{E}{T^4} \). Therefore, the unit of \( \sigma \) can be derived from the units of \( E \) and \( T \).

Step 4: Analyzing Units for Energy and Temperature

In the equation \( \sigma = \frac{E}{T^4} \), \( E \) has units of \( \mathrm{Wm}^{-2} \), where \( \mathrm{W} \) represents watts (power) and \( m \) represents meters (area). \( T \) has units of \( \mathrm{K} \) (kelvin). Thus, \( T^4 \) has units of \( \mathrm{K}^4 \).

Step 5: Final Unit Calculation for Stefan's Constant

Substituting the units into \( \sigma = \frac{E}{T^4} \) gives us \( \sigma = \frac{\mathrm{Wm}^{-2}}{\mathrm{K}^4} \). Thus, the unit for Stefan's constant is indeed \( \mathrm{Wm}^{-2} \mathrm{~K}^{-4} \).

Key Concepts

Stefan's Constant

Stefan's constant, symbolized as \( \sigma \), plays a crucial role in the realm of thermodynamics and black body radiation. Stefan's Law, formally known as the Stefan-Boltzmann Law, uses this constant to express the relationship between the radiant energy emitted by a black body and its temperature. More specifically, Stefan's constant helps to define how much energy a perfect black body emits per square meter of its surface area at a certain temperature.One can picture it as a key factor that bridges the gap between temperature and energy emission according to the formula: - \( E = \sigma T^4 \) - Where \( E \) is the energy emitted per unit area per unit time, and \( T \) is the absolute temperature of the black body in Kelvin. The proportionality factor, \( \sigma \), ensures that this relationship is quantitatively correct, making it indispensable for calculating energy outputs in physics.

SI units in Physics

Understanding units in physics, particularly in the International System of Units (SI), is essential for precise measurement and communication. In the context of Stefan's constant, it's important to breakdown its derived SI unit: \( \mathrm{Wm}^{-2} \mathrm{K}^{-4} \). Here's how it unfolds:

• Watts (\( \mathrm{W} \)): Reflect power, or the rate of energy transfer.
• Meters squared (\( \mathrm{m}^{-2} \)): Denotes the area over which this power is distributed.
• Kelvin to the fourth power (\( \mathrm{K}^{-4} \)): Relates to temperature, encapsulating how energy emission scales with the fourth power of temperature.

Temperature in Kelvin is crucial since it facilitates the absolute scale necessary for precise thermal calculations. By ensuring accurate dimensional analysis, we guarantee that our equations and constants remain consistent across varying scientific contexts.

Black Body Radiation

Black body radiation is a fundamental concept in physics that explains how objects emit electromagnetic radiation. A "black body" is a theoretical object that absorbs all incident radiation without reflecting any light, making it an ideal emitter. Why is black body radiation important? Here's why:

• It provides a model to understand other types of radiative objects, as real objects can usually be approximated as black bodies.
• Black bodies emit radiation at all frequencies, and the intensity of emission is dependent on temperature. This helps in understanding thermal emissions from stars and other astronomical bodies.
• The behavior of a black body as described by Stefan-Boltzmann Law shows how energy radiated is proportional to the fourth power of the body's temperature, which is crucial in fields like astrophysics and climate science.

In practice, while no perfect black bodies exist in nature, many objects approximate them closely, allowing these theoretical insights to be applied in real-world situations.