Question

The wavelength at which the blackbody emissive power reaches its maximum value at $300\mathrm{K}$ is (a) $5.1\mu \mathrm{m}$ (b) $9.7\mu \mathrm{m}$ (c) $15.5\mu \mathrm{m}$ (d) $38.0\mu \mathrm{m}$ (e) $73.1\mu \mathrm{m}$

Short answer

Answer: (b) 9.7 μm

Step-by-step solution

Recall Wien's Displacement Law

Wien's Displacement Law states that the wavelength at which the blackbody emissive power reaches its maximum value is inversely proportional to the temperature. Mathematically, it can be written as: ${\lambda }_{max}T=b$ where ${\lambda }_{max}$ is the wavelength at which the emissive power is maximum, $T$ is the temperature in Kelvin, and $b$ is Wien's constant, which is equal to $2.898\times {10}^{-3}\mathrm{m}.\mathrm{K}$.

Calculate the wavelength at maximum emissive power

To find the wavelength at maximum emissive power, we will use Wien's Displacement Law with the given temperature: ${\lambda }_{max}=\frac{b}{T}$ Plug in the known values: ${\lambda }_{max}=\frac{2.898\times {10}^{-3}\mathrm{m}.\mathrm{K}}{300\mathrm{K}}$ Calculate the result: ${\lambda }_{max}=9.66\times {10}^{-6}\mathrm{m}$

Convert the result to micrometers and compare it with the given options

Since the options are given in micrometers ($\mu$m), we need to convert our result to micrometers: ${\lambda }_{max}=9.66\times {10}^{-6}\mathrm{m}\times \frac{{10}^6\mu \mathrm{m}}{1\mathrm{m}}=9.66\mu \mathrm{m}$ Approximate the result and compare it with the given options: ${\lambda }_{max}\approx 9.7\mu \mathrm{m}$ The closest value to our calculated result is option (b) $9.7\mu \mathrm{m}$. Therefore, the correct answer is: (b) $9.7\mu \mathrm{m}$

Key concepts

Blackbody Radiation

Blackbody radiation is the type of thermal radiation emitted by an idealized object called a blackbody. A blackbody is a perfect emitter and absorber of radiation; it does not reflect or transmit any radiation. Instead, it absorbs all the incoming radiation and emits radiation uniformly at all wavelengths. This makes it an excellent theoretical model for understanding how objects emit radiation based on their temperature.

Blackbody radiation is crucial for various applications, including understanding cosmic microwave background radiation, designing thermal imaging cameras, and improving the efficiency of photovoltaic cells. By characterizing how radiation varies with temperature, scientists can make predictions about the behavior of different materials and systems.

One of the key features of blackbody radiation is that it follows a specific distribution, known as Planck's distribution, which shows how the emissive power is spread across different wavelengths. This distribution helps in determining the peak wavelength at which the emissive power of radiation is maximum, a concept central to other topics like Wien's Displacement Law.

Thermal Radiation

Thermal radiation refers to the emission of electromagnetic waves from all matter that has a temperature above absolute zero. This type of radiation is generated by the thermal motion of charged particles in matter. Because every object at a temperature above zero emits thermal radiation, it's integral to understanding heat transfer in various systems.

Thermal radiation is one of the three primary mechanisms of heat transfer, alongside conduction and convection. Unlike conduction and convection, thermal radiation can occur in a vacuum, as it does not require a medium to transfer heat. This phenomenon is responsible for energy transfer in the form of electromagnetic waves across space. A familiar example is the warmth felt from sunlight, which is solar radiation reaching Earth.

The amount and wavelength of radiation emitted depend on the temperature and properties of the material. For instance, hotter objects emit shorter wavelength radiation compared to cooler objects. This principle is encapsulated by Wien's Displacement Law, which describes how the peak wavelength is inversely related to temperature.

Emissive Power

Emissive power is the measure of the radiation energy emitted by a surface per unit area. This power is an important aspect of understanding how effectively a surface radiates energy. In the context of blackbody radiation, the emissive power helps describe how much energy is emitted by a blackbody at a given temperature.

The concept of emissive power is governed by Planck's Law, which provides a formula for the spectral distribution of radiation emitted by a blackbody. This law gives insight into the variations in emissive power based on temperature and wavelength, showing how these factors affect the energy output.

Emissive power is essential in practical applications like designing thermal insulators, improving energy efficiency in buildings, and developing various electronic devices. By leveraging this concept, engineers and scientists can optimize the way materials use and dissipate energy through radiation.

Temperature Conversion

Temperature conversion is the process of changing the measurement of temperature from one unit to another. This concept is crucial in scientific calculations because different equations require temperature in specific units, primarily Kelvin, Celsius, or Fahrenheit.

In the realm of blackbody radiation and thermal physics, Kelvin is often used because it is an absolute temperature scale that starts at absolute zero, where no thermal motion occurs. This makes it ideal for calculations involving thermodynamics and radiation, as many formulas, including Wien's Displacement Law, require temperature input in Kelvin.

Students often encounter temperature conversion in exercises dealing with real-world problems, where they have to switch between different temperature scales to perform correct calculations. Common conversion formulas are:

• From Celsius to Kelvin: $$T(K)=T(°C)+273.15$$
• From Kelvin to Celsius: $$T(°C)=T(K)-273.15$$

By understanding and applying these conversion techniques, you ensure accuracy and consistency in thermal physics problems.