Problem 43 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

The peak wavelength at 3000 K is approximately 969 nm.

Step 1: Identify Given Values

We are given the peak wavelength of blackbody radiation at a temperature of 2.728 K, which is 1.062 mm. We need to find the peak wavelength at a temperature of 3000 K.

Step 2: Recall Wein's Law

Wein's Law states that the peak wavelength \( \lambda_{max} \) of blackbody radiation is inversely proportional to the temperature \( T \), represented as \( \lambda_{max} T = b \), where \( b \) is Wein's constant (\( 2.897 \times 10^{-3} \ \mathrm{m \cdot K} \)).

Step 3: Calculate the Constant Using Current Values

Using the given temperature and peak wavelength at 2.728 K, we substitute into Wein's Law: \( \lambda_{max} = 1.062 \ \mathrm{mm} = 1.062 \times 10^{-3} \ \mathrm{m} \) and \( T = 2.728 \ \mathrm{K} \). This gives us \( b = \lambda_{max} \times T = 1.062 \times 10^{-3} \times 2.728 \).

Step 4: Determine the Peak Wavelength at 3000 K

Now, we use the constant \( b \) and the new temperature \( T = 3000 \ \mathrm{K} \) to find \( \lambda_{max} \) by rearranging Wein’s Law: \( \lambda_{max} = \frac{b}{T} = \frac{1.062 \times 10^{-3} \times 2.728}{3000} \).

Step 5: Compute the Value

Calculating the above expression yields the peak wavelength \( \lambda_{max} \) at 3000 K. The result is approximately 969 nm, or 0.969 micrometers.

Key Concepts

Understanding Blackbody Radiation

Blackbody radiation is a fundamental concept in physics that describes how idealized objects, known as blackbodies, emit radiation. A blackbody absorbs all incident light, meaning it does not reflect or transmit any light, leading to the perfect emission of thermal radiation.

• This emission is entirely dependent on the body's temperature.
• At higher temperatures, a blackbody emits more radiation and its color changes, moving from red to blue in the visible spectrum as it gets hotter.
• The radiation emitted covers a spectrum of wavelengths, including infrared and visible light.

Understanding blackbody radiation helps explain why objects like the Sun appear yellowish or why a hot stove glows red.

Peak Wavelength and Its Significance

The peak wavelength of blackbody radiation is where the emission of radiation is most intense, and it shifts according to the temperature of the body.
We can visualize this through a spectrum where different colors represent various wavelengths. A shift in peak wavelength indicates a change in color.

• At lower temperatures, the peak wavelength is longer, usually in the infrared range.
• As the temperature increases, the peak wavelength moves towards shorter wavelengths, entering the visible spectrum.
• For example, the Sun emits most radiation in the visible light range, centered around green, which is why the Sun appears bright and white to our eyes.

Peak wavelength is critical in applications like astronomy, where it helps determine the temperature of stars based on the color of their emitted light.

The Temperature Dependence of Wavelength

The relationship between temperature and peak wavelength is elegantly described by Wein's Law, which forms the backbone of understanding thermal radiation. Wein's Law is presented as \( \lambda_{max} T = b \), showing that the peak wavelength \( \lambda_{max} \) is inversely proportional to the temperature \( T \).

• When the temperature of a blackbody increases, the peak wavelength decreases.
• This inverse relationship means hotter objects will have peak emissions at shorter wavelengths.
• The constant \( b \), known as Wein's constant, helps us quantify the relationship across different temperatures.

This concept is crucial in various scientific fields, including climate science, where understanding how Earth's temperature change affects radiation patterns can be significantly impactful.

Physics Problem-Solving with Wein's Law

Applying Wein's Law to solve physics problems involves a clear understanding of the problem setup and a sequence of logical steps. Let's break it down:

• First, gather the known values, such as current temperature and peak wavelength.
• Next, use these values to calculate necessary constants, like \( b \) in Wein's Law.
• Finally, manipulate the law to find unknowns, such as the peak wavelength at a different temperature.

This structured approach not only helps in tackling physics problems involving temperature and radiation but also builds a strong foundation for solving more complex issues in thermodynamics and astronomy. Practice and familiarity with these techniques allow for efficient and accurate solutions.