Question

Calculate the range of temperatures for which the peak emission of the blackbody radiation from a hot filament occurs within the visible range of the electromagnetic spectrum. Take the visible spectrum as extending from \(380 \mathrm{nm}\) to \(780 \mathrm{nm}\). What is the total intensity of the radiation from the filament at these two temperatures?

Short answer

Answer: The range of temperatures for which the peak emission of the blackbody radiation from a hot filament occurs within the visible range is approximately 3718 K to 7632 K. The total intensities of radiation at these temperatures are approximately 2.468 * 10^7 W m^(-2) at the lower temperature and 4.502 * 10^8 W m^(-2) at the higher temperature.

Step-by-step solution

Understand Wien's Displacement Law

Wien's displacement law states that the peak wavelength (λ_max) of the blackbody radiation is inversely proportional to the temperature (T) of the blackbody. Mathematically, it is represented as λ_max * T = b, where b is Wien's displacement constant, and approximately equals 2.898 * 10^(-3) m K.

Calculate the temperature range for visible wavelengths

To find the temperatures corresponding to the visible range, we need to use Wien's displacement law. First, let's find the temperature that corresponds to the lower limit of the visible range (380 nm). λ_max, lower = 380 * 10^(-9) m Using Wien's displacement law: T_lower = b / λ_max, lower = (2.898 * 10^(-3) m K) / (380 * 10^(-9) m) ≈ 7631.58 K Now let's find the temperature that corresponds to the upper limit of the visible range (780 nm). λ_max, upper = 780 * 10^(-9) m Using Wien's displacement law: T_upper = b / λ_max, upper = (2.898 * 10^(-3) m K) / (780 * 10^(-9) m) ≈ 3717.95 K Thus, the range of temperatures for which the peak emission of the blackbody radiation from a hot filament occurs within the visible range is approximately 3718 K to 7632 K.

Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann law relates the temperature of a blackbody to its total intensity of radiation. Mathematically, it is represented as I = σ * T^4, where I is the total intensity of the radiation, T is the temperature of the blackbody, and σ is the Stefan-Boltzmann constant, which is approximately 5.670 * 10^(-8) W m^(-2) K^(-4).

Calculate the total intensity of radiation at the extreme temperatures

Now let's find the total intensity of radiation from the hot filament at both T_lower (3718 K) and T_upper (7632 K). Using the Stefan-Boltzmann law: I_lower = σ * T_lower^4 = (5.670 * 10^(-8) W m^(-2) K^(-4)) * (3717.95 K)^4 ≈ 2.468 * 10^7 W m^(-2) I_upper = σ * T_upper^4 = (5.670 * 10^(-8) W m^(-2) K^(-4)) * (7631.58 K)^4 ≈ 4.502 * 10^8 W m^(-2) So, the total intensity of the radiation from the filament at the temperatures corresponding to the extreme values of the visible range is approximately 2.468 * 10^7 W m^(-2) at the lower temperature and 4.502 * 10^8 W m^(-2) at the higher temperature.

Key concepts

Wien's Displacement Law

Wien’s Displacement Law is an essential concept when it comes to understanding blackbody radiation, which is radiation emitted by objects due to their temperature. It provides a simple yet crucial relationship between the temperature of an object and the peak wavelength of emitted radiation. By stating that as a blackbody gets hotter, the wavelength at which it emits the most radiation gets shorter, we can predict the color of the radiation without having to measure the entire spectrum.

Applying this to practical problems, we can determine the temperature range at which objects, like a filament, emit light in the visible spectrum. For example, the temperatures at which a filament's peak emission is visible range from approximately 3718 K to 7632 K. This calculation relies on Wien's constant, approximately equal to 2.898 x 10^-3 m K, and the boundaries of the visible spectrum (380 nm to 780 nm).

Understanding this concept not only helps in theoretical physics but is also crucial in fields like astronomy, where it is used to determine the temperatures of stars based on their color, and in designing energy-efficient lighting, where the objective is to produce the maximum visible light with minimal heat emission.

Visible Spectrum

The visible spectrum is the segment of the electromagnetic spectrum that is visible to the human eye. Ranging from about 380 nm to 780 nm in wavelength, it encompasses all the colors that we can see, from violet at the shorter wavelengths to red at the longer wavelengths. Different objects and materials can emit or reflect light at various wavelengths within this range, and that's why we perceive different colors.

When considering blackbody radiation, the visible spectrum tells us the range of electromagnetic waves that can be seen as light, rather than just felt as heat. In the context of the calculation, the visible spectrum provides the critical limits within which we apply Wien's Displacement Law to find out if the filament shines bright with visible light or glows with infrared heat that we cannot see but can feel as warmth.

For students, recognizing the limits of the visible spectrum is important not only in physics but also in understanding how various devices, like cameras and telescopes, are designed to detect different types of light beyond what the human eye can perceive.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is another fundamental concept, which relates to a blackbody's temperature and its total electromagnetic radiation emission. To put it simply, this law tells us that hotter objects not only emit light at shorter wavelengths, as Wien's Displacement Law shows, but they also emit more energy overall.

This law is expressed through the equation I = \( \sigma T^4 \), where:

• I is the total intensity of the blackbody's radiation in watts per square meter (W/m²),
• T is the temperature in kelvins (K),
• and \( \sigma \) is the Stefan-Boltzmann constant, about 5.670 x 10^-8 W m^-2 K^-4.

By using it, we can deduce the energy emitted by a filament at any given temperature, such as the temperatures corresponding to the visible spectrum calculated previously.

It's an incredibly useful tool in many areas of science and engineering, helping to understand phenomena such as energy transfer, radiation processes in the atmosphere, and the thermal performance of materials.