Problem 74 - Master Resource Book in JEE Main Physics

Question

A black body is at a temperature of \(2880 \mathrm{~K}\). The energy of radiation emitted by this object with wavelength between \(499 \mathrm{~nm}\) and \(500 \mathrm{~nm}\) is \(U_{1}\), between 999 and \(1000 \mathrm{~nm}\) is \(U_{2}\) and between \(1499 \mathrm{~nm}\) and \(1500 \mathrm{~nm}\) is \(U_{3}\). The Wien's constant \(b=2.88 \times 10^{6} \mathrm{nmK}\). Then (a) \(U_{1}=0\) (b) \(U_{3}=0\) (c) \(U_{1}>U_{2}\) (d) \(U_{2}>U_{1}\)

Step-by-Step Solution

Answer

Options (b) and (d) are correct; \( U_3 = 0 \) and \( U_2 > U_1 \).

Step 1: Determine Peak Wavelength

Wien's Displacement Law states \( \lambda_{max} = \frac{b}{T} \), where \( \lambda_{max} \) is the peak wavelength, \( b \) is the Wien's constant, and \( T \) is the temperature. Substituting the given values, we have \( \lambda_{max} = \frac{2.88 \times 10^6 \ nm \cdot K}{2880 \ K} = 1000 \ nm \). This means the peak radiation wavelength for the black body is 1000 nm.

Step 2: Compare Given Wavelengths with Peak Wavelength

The peak wavelength \(1000 \ nm\) suggests that most radiation is emitted around this wavelength. The segments are 499-500 nm, 999-1000 nm, and 1499-1500 nm. Given the peak at 1000 nm, the segment 999-1000 nm is closest to the peak, implying \( U_2 \) should be significant.

Step 3: Analyze Segment Contributions

At 499-500 nm, since it's far from the peak at 1000 nm, \( U_1 \) will be very small. For 1499-1500 nm, again it's far, making \( U_3 \) insignificant. Hence, \( U_1 \) and \( U_3 \) are expected to be approximately zero compared to \( U_2 \).

Step 4: Conclusion Based on Wien's Law

Since \( U_3 = 0 \) due to exceeding Wien's spectrum efficiency and \( U_1 \) is also near zero, the contribution comparison is \( U_2 > U_1 = 0 \). Hence, the highest energy emission segment is 999-1000 nm. Option (b) "\( U_3 = 0 \)" and option (d) "\( U_2 > U_1 \)" are correct based on Wien's Law.

Key Concepts

Black Body Radiation

Black body radiation refers to the phenomenon where an idealized object, known as a black body, emits radiation due to its temperature. This type of radiation depends purely on the body's temperature, not on its shape, size, or material composition. A black body absorbs all incoming radiation without reflecting any, making it a perfect emitter.

As it heats up, it emits energy across different wavelengths. This radiation spectrum can be predictable and follows a specific distribution based on quantum mechanics. It's important because it's fundamental to understanding how objects radiate energy as they become hotter.

Energy Emission

Energy emission in the context of black body radiation involves the release of energy in the form of electromagnetic waves. The energy emitted spans a spectrum of wavelengths, from short to long. The intensity or amount of energy emission varies across these wavelengths.

The distribution of energy across wavelengths is not uniform. Generally, energy emission increases to a peak level and then decreases, forming a bell-shaped curve when plotted. The behavior of this emission is dictated by Planck's Law, which describes how energy is distributed across different wavelengths for a given temperature.

Understanding energy emission is crucial in physics, as it helps to predict how much and what type of energy is released by objects at various temperatures.

Peak Wavelength

The peak wavelength is the specific wavelength at which the maximum amount of radiation is emitted by a black body at a given temperature. Wien's Displacement Law is the principle that relates the temperature of a black body to its peak wavelength.

According to Wien's Law, the peak wavelength \( \lambda_{max} \) is determined by the formula \( \lambda_{max} = \frac{b}{T} \), where \( b \) is the Wien's constant and \( T \) is the absolute temperature in Kelvin. This formula highlights the inverse relationship between temperature and peak wavelength: as temperature increases, the peak wavelength decreases.

This concept is essential because it allows us to understand why hotter objects emit more energy at shorter wavelengths, which tends to shift the radiation towards visible or even ultraviolet light as the temperature rises.

Temperature and Radiation

Temperature plays a critical role in determining the characteristics of radiation emitted by any object. For black bodies, higher temperatures lead to greater radiation intensities and shorter peak wavelengths.

As temperature rises, not only does the amount of emitted radiation increase but the spectrum of emitted radiation shifts. This shift means that more energy is emitted at shorter wavelengths. For instance, a black body at a moderate temperature might emit mainly infrared radiation, while a much hotter black body could emit visible light or even higher energy ultraviolet radiation.

This temperature dependence makes understanding black body radiation crucial in fields like astronomy and climate science, as it helps in analyzing stars' temperatures and explaining natural phenomena related to heat and light emission.