Problem 82 - Master Resource Book in JEE Main Physics

Question

When the temperature of a black body increases, it is observed that the wavelength corresponding of maximum energy changes from \(0.26 \mu \mathrm{m}\) to \(0.13 \mu \mathrm{m}\) to a body at the respective temperature. Then ratio of the emissivities \(\frac{E_{2}}{E_{1}}\) is (a) \(16 / 1\) (b) \(4 / 1\) (c) \(1 / 4\) (d) \(1 / 16\)

Step-by-Step Solution

Answer

The ratio of the emissivities \(\frac{E_2}{E_1}\) is 16/1, option (a).

Step 1: Understanding Wien's Displacement Law

Wien's Displacement Law states that the wavelength of maximum emission of a black body (\(\lambda_{\text{max}}\)) is inversely proportional to its temperature (T). The formula is given by:\[\lambda_{\text{max}} \cdot T = b\]where \(b\) is a constant (Wien's constant).In this problem, we have two different maximum wavelengths, \(\lambda_1 = 0.26 \ \mu \text{m}\) and \(\lambda_2 = 0.13 \ \mu \text{m}\), corresponding to different temperatures \(T_1\) and \(T_2\). We need to find the ratio of emissivities \(\frac{E_{2}}{E_{1}}\).

Step 2: Relating Temperature and Wavelengths

Using Wien's Displacement Law, we write:\[\lambda_1 \cdot T_1 = \lambda_2 \cdot T_2 = b\]Taking the ratio of these two equations:\[\frac{T_2}{T_1} = \frac{\lambda_1}{\lambda_2}\]

Step 3: Calculate the Temperature Ratio

Insert the given wavelengths into the formula obtained from Step 2:\[\frac{T_2}{T_1} = \frac{0.26 \ \mu \text{m}}{0.13 \ \mu \text{m}} = 2\]So the temperature \(T_2\) is double the temperature \(T_1\) (\(T_2 = 2 \times T_1\)).

Step 4: Using the Stefan-Boltzmann Law for Emissivity

The Stefan-Boltzmann Law relates the emissivity of a black body to its temperature by:\[E = \sigma T^4\]Therefore, to find the ratio of the emissivities:\[\frac{E_2}{E_1} = \left(\frac{T_2}{T_1}\right)^4 = (2)^4 = 16\]Thus, \(\frac{E_2}{E_1} = \frac{16}{1}\).

Step 5: Identify the Correct Answer

The result from our calculation is \(\frac{E_2}{E_1} = \frac{16}{1}\).Therefore, the correct answer is (a) 16/1.

Key Concepts

Black Body Radiation

Black body radiation describes the phenomenon where an idealized physical body, known as a black body, absorbs all incoming radiation and re-emits energy as thermal radiation. This concept is fundamental in understanding how objects emit heat and light.A black body does not reflect any incident light, but instead, it emits radiation based solely on its temperature. This radiation spans a range of wavelengths, including visible light, infrared, and ultraviolet.Wien's Displacement Law, which this exercise extensively uses, is derived from the concept of black body radiation. It demonstrates that there is a peak wavelength (\(\lambda_{\text{max}}\)) at which the emission of a black body is at its maximum. This specific wavelength shifts as the temperature changes. As the temperature rises, the peak wavelength decreases, causing the black body to emit more light in the visible spectrum, which is often observed as a glowing of the object as the temperature increases.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law provides a relationship between the total energy radiated by a black body and its temperature. This is a pivotal principle in physics used to calculate the energy an object can emit based on its temperature.The law states that the total emissive power (\(E\)) is proportional to the fourth power of the black body's absolute temperature (\(T\)). Mathematically, it's expressed as:\[ E = \sigma T^4 \]where \(\sigma\) is the Stefan-Boltzmann constant. This formula is crucial because it shows how much more energy a black body will radiate when its temperature increases, even by a small amount. For instance, as seen in the exercise, when the temperature doubles, the energy emitted increases sixteenfold, illustrating the power of exponential relationships.

Emissivity Ratio Calculation

Emissivity is a measure of how effectively a surface emits thermal radiation compared to an ideal black body. It is a critical concept when considering thermal emissions from real-world objects.In our exercise, we focused on finding the ratio of emissivities from two different temperatures. The ratio of emissivities is important in applications where heat transfer and emission properties of materials are considered.Using the Stefan-Boltzmann Law, as the temperature rises, the emissivity's change factor becomes apparent through the formula:\[ \frac{E_2}{E_1} = \left(\frac{T_2}{T_1}\right)^4 \]In this problem, the temperature ratio was found to be 2, meaning \(T_2\) is double \(T_1\). This results in the emissivity ratio being \(16\), indicating a much higher emission of thermal radiation at the higher temperature. Understanding this concept is crucial for solving problems involving thermal dynamics and radiation processes.