Problem 41 - University Physics with Modern Physics

Question

The 2.728 -K blackbody radiation has its peak wavelength at 1.062 \(\mathrm{mm}\) . What was the peak wavelength at \(t=700,000\) y when the temperature was 3000 \(\mathrm{K} ?\)

Step-by-Step Solution

Answer

The peak wavelength at 3000 K is approximately 966 nm.

Step 1: Identify the formula

The peak wavelength of a blackbody is found using Wien's Law, which states that the peak wavelength \( \lambda_{max} \) is inversely proportional to the temperature \( T \). The formula is given by:\[\lambda_{max} = \frac{b}{T}\]where \( b \) is Wien's displacement constant \( b = 2.897 \times 10^{-3} \text{ m K} \).

Step 2: Calculate the current peak wavelength

Using Wien's Law, calculate the peak wavelength at \( T = 2.728 \text{ K} \):\[\lambda_{max} = \frac{2.897 \times 10^{-3}}{2.728} \approx 1.062 \text{ mm}\]This matches the given current peak wavelength.

Step 3: Calculate past peak wavelength at \( T = 3000 \text{ K} \)

Apply Wien's Law again for the temperature \( T = 3000 \text{ K} \):\[\lambda_{max} = \frac{2.897 \times 10^{-3}}{3000} \approx 9.66 \times 10^{-7} \text{ m} \approx 966 \text{ nm}\]Convert the result to nanometers for better readability, since it is in the visible spectrum.

Key Concepts

Blackbody Radiation

Blackbody radiation refers to the thermal radiation emitted by a body that absorbs all incident radiation, regardless of frequency or angle of incidence. This type of radiation is an idealized concept, as real objects do not absorb all the radiation perfectly. However, blackbody radiation is an essential model in physics to understand how objects emit radiation based on their temperature.

A true blackbody is a perfect emitter and absorber. Its radiation spectrum depends solely on its temperature. As temperature increases, the emitted radiation shifts to higher frequencies or shorter wavelengths. This shift can be understood through Wien's Law, which we will discuss in further sections. Real-life examples of near-blackbody radiation include stars and planets, where the radiation can be approximately modeled by blackbody spectra.

Key characteristics of blackbody radiation include:

• Continuous spectrum: Unlike line spectra, blackbody radiation covers a continuous range of wavelengths.
• Temperature-dependent: The emitted spectrum changes with temperature.
• Emission peak: The radiation has a peak intensity at a specific wavelength that shifts as temperature changes.

Peak Wavelength

The peak wavelength is the wavelength at which the emission of a blackbody radiation is at its maximum. This concept is crucial in determining the color and brightness of stars and other astronomical bodies.

The peak wavelength helps us understand the thermal emission characteristics of an object. As the temperature of the object increases, its peak wavelength shifts to shorter wavelengths, moving through the spectrum from red to blue. This phenomenon is why hotter stars appear bluer than cooler ones.

Using Wien's Law, we can calculate the peak wavelength for a given temperature. For example, if a blackbody has a temperature of 3000 K, the peak wavelength can be calculated using the formula provided in the solution, giving a result in the visible spectrum range. It is important to express this value in nanometers (nm) for better understanding, as it falls within the range visible to the human eye.

Temperature in Physics

In physics, temperature plays a critical role in determining the thermal behavior of materials and systems. It is a measure of the average kinetic energy of particles in a substance. A higher temperature indicates that the particles are moving more quickly.

Temperature affects everything from the physical state of a substance (solid, liquid, gas) to the way it emits radiation. In the study of blackbody radiation, the temperature of an object is directly related to the characteristics of the radiation it emits. This is seen in how changing the temperature alters both the intensity and the peak wavelength of the emitted radiation.

Understanding temperature in physics involves:

• Recognizing that it is a fundamental parameter that influences physical systems.
• Using temperature to predict and calculate the behavior of systems through laws like Wien's Law.
• Relating microscopic motion of particles to macroscopic observables such as pressure and volume, as in the case of gases.

Wien's Displacement Constant

Wien's displacement constant is a pivotal element in Wien's Law, which correlates the temperature of a blackbody with the wavelength at which it emits radiation most intensely. The constant is given by the value \( b = 2.897 \times 10^{-3} \; \text{m K} \). This constant allows for the precise calculation of the peak wavelength based on the blackbody's temperature.

The use of Wien's displacement constant is integral to many fields, from determining temperatures of stars to assessing thermal emissions in engineering applications. For example, by measuring the peak wavelength of a star's emission, we can determine its surface temperature. This method relies heavily on the accuracy of Wien's displacement constant.

In practical applications:

• Wien's constant simplifies the interplay between temperature and wavelength.
• It allows scientists to convert temperature measurements into wavelength data, and vice versa.
• Aids in converting theoretical models into applicable tools for observing and measuring thermal radiation in diverse environments.