Problem 67 - Master Resource Book in JEE Main Physics

Question

A black body at \(1373^{\circ} \mathrm{C}\) emits maximum energy corresponding to a wavelength of \(1.78\) micron. The temperature of moon for which \(\lambda_{m}=14\) micron would be (a) \(62.6^{\circ} \mathrm{C}\) (b) \(-58.9^{\circ} \mathrm{C}\) (c) \(63.7^{\circ} \mathrm{C}\) (d) \(64.2^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The temperature of the moon is approximately -58.9°C, choice (b).

Step 1: Understanding Wien's Displacement Law

Wien's displacement law states that the wavelength of maximum emission of a black body is inversely proportional to its absolute temperature. It is given by the formula: \[ \lambda_{m} \cdot T = b \]where \( \lambda_{m} \) is the wavelength of maximum emission, \( T \) is the absolute temperature in Kelvin, and \( b \approx 2898 \ \mu m \cdot K \) is Wien's constant for a black body.

Step 2: Converting Initial Temperature to Kelvin

The temperature given for the black body is \( 1373^{\circ} \mathrm{C} \). To convert this temperature to Kelvin, use the formula:\[ T(K) = T(^{\circ}C) + 273.15 \]Hence, \( T = 1373 + 273.15 = 1646.15 \ \text{K} \).

Step 3: Calculating Wien's Constant

Given that the wavelength \( \lambda_{m} = 1.78 \ \mu m \) for \( T = 1646.15 \ \text{K} \), we can calculate Wien's constant:\[ b = \lambda_{m} \cdot T = 1.78 \times 1646.15 \approx 2931 \ \mu m \cdot K \].

Step 4: Applying Wien's Law for the Moon

Now, use the wavelength \( \lambda_{m} = 14 \ \mu m \) for the moon. According to Wien's law, solve for the temperature \( T \):\[ T = \frac{b}{\lambda_{m}} = \frac{2931}{14} \approx 209.36 \ \text{K} \].

Step 5: Converting Temperature from Kelvin to Celsius

Convert the temperature back from Kelvin to Celsius:\[ T(^{\circ}C) = T(K) - 273.15 \]Hence, \( T = 209.36 - 273.15 \approx -63.79 \ ^{\circ}C \).

Step 6: Matching with Given Choices

The calculated temperature is approximately \(-63.79 ^{\circ}S\). Comparing with the options given, the closest match is option (b) \(-58.9 ^{\circ} \mathrm{C}\).

Key Concepts

Black Body Radiation

Black body radiation is a fascinating concept in physics that describes how objects emit electromagnetic radiation. Imagine an object that absorbs all the radiation it receives without reflecting any, known as a "perfect" black body. This object's emissions are purely based on its temperature. The radiation emitted by a black body is continuous and spans across various wavelengths, visible to invisible, following a specific pattern over the electromagnetic spectrum.

A black body emits radiation at all wavelengths, but not equally. The amount of energy it emits depends significantly on its temperature. Cooler objects emit longer wavelengths, while hotter objects emit shorter wavelengths. This pattern is crucial in understanding how objects, like stars or planets, emit light and heat. Understanding black body radiation helps us comprehend phenomena like why heated metal glows or how the Sun emits solar energy.

• Black bodies are theoretical objects extensively used in physics.
• They serve as a model for various physical bodies in thermal equilibrium.
• The radiation they emit is strictly defined by temperature.

Absolute Temperature

Absolute temperature is a measure of an object's heat energy using the Kelvin scale. Unlike Celsius or Fahrenheit, Kelvin starts at absolute zero, the theoretical point where molecular motion stops. This makes the Kelvin scale indispensable in scientific calculations.

When we talk about absolute temperature, we refer directly to the kinetic energy of particles in a substance. Kelvin effectively removes confusion caused by negative temperature values, providing a universal standard for expressing temperature.

This concept is vital in black body radiation calculations because Wien’s Displacement Law directly relates wavelength to the absolute temperature of a body.

• To convert Celsius to Kelvin, add 273.15.
• Kelvin provides a more straightforward approach to scientific equations.
• It enhances the understanding of thermodynamic and kinetic energy relationships.

Wavelength of Maximum Emission

The wavelength of maximum emission is the wavelength at which a black body radiates most intensely, and it is fundamental to understanding how objects emit radiation. Determined by Wien’s Law, this wavelength inversely relates to the object's absolute temperature. So, as the temperature increases, the wavelength of maximum emission moves to shorter values.

For instance, in hotter objects, emissions peak towards the blue end of the spectrum, while cooler objects peak towards the red or infrared end. This law allows scientists to determine the temperature of distant stars or planets by observing the light they emit. By calculating the wavelength where the emission peaks, one can backtrack to find the temperature, even for celestial bodies.

• It simplifies determining the temperature of astronomical bodies.
• Shorter wavelengths correspond to higher temperatures.
• Wien’s law facilitates the connection between temperature and wavelength.