Question

Radiation has been detected from space that is characteristic of an ideal radiator at $T$ = 2.728 K. (This radiation is a relic of the Big Bang at the beginning of the universe.) For this temperature, at what wavelength does the Planck distribution peak? In what part of the electromagnetic spectrum is this wavelength?

Short answer

The peak wavelength is 1.062 mm, in the microwave part of the electromagnetic spectrum.

Step-by-step solution

Understand the Given

We are given the temperature of the radiator as 2.728 K. This temperature relates to cosmic microwave background radiation, which can be analyzed using Wien's displacement law to find the peak wavelength.

Apply Wien's Displacement Law

Wien's displacement law states that the wavelength at which the emission of a black body peaks is inversely proportional to its temperature. The law is expressed as ${\lambda }_{max}=\frac{b}{T}$, where $b$ is Wien's displacement constant $(b=2.897\times {10}^{-3}\text{ m K})$ and $T$ is the temperature in Kelvin.

Calculate the Peak Wavelength

Using Wien's displacement law, substitute the given temperature value $T=2.728$ K into the formula:$${\lambda }_{max}=\frac{2.897\times {10}^{-3}\text{ m K}}{2.728\text{ K}}=1.062\times {10}^{-3}\text{ m}$$Therefore, the peak wavelength ${\lambda }_{max}$ is approximately 1.062 mm.

Determine the Spectrum Region

The calculated wavelength, 1.062 mm, falls within the microwave region of the electromagnetic spectrum. Microwaves are generally considered to span wavelengths from 1 mm to 1 meter.

Key concepts

Cosmic Microwave Background Radiation

The cosmic microwave background (CMB) radiation is an essential clue to our universe's past. Imagine it like the warm glow left over from the Big Bang. This ancient radiation fills the universe and reaches us from all directions. When scientists first detected it, they realized it was a snapshot of the very young universe. The temperature of the CMB is incredibly cold, about 2.728 Kelvin, which is just above absolute zero. Because of this low temperature, cosmic microwave background radiation is an ideal place to apply concepts like Wien's displacement law to understand the peak wavelength of radiation emitted.

• The CMB radiation is smooth and uniform.
• It tells us about the universe just 380,000 years after the Big Bang.
• The study of CMB has helped confirm the Big Bang theory and our understanding of the universe's expansion.

Electromagnetic Spectrum

The electromagnetic spectrum is like a rainbow of energy types. It ranges from high-energy gamma rays to the low-energy radio waves. Each type of radiation is characterized by a different wavelength or frequency. The electromagnetic spectrum shows us the different types of electromagnetic waves, visible and invisible, around us. Within this spectrum, microwaves have longer wavelengths than visible light but shorter than radio waves. When you hear about a wavelength like 1.062 mm, as in the case of cosmic microwave background radiation, it falls right into the microwave region of the electromagnetic spectrum. That’s why the cosmic microwave background is called just that—microwave.

• Visible light is only a small part of the spectrum.
• Different parts of the spectrum have different uses, from cooking (microwaves) to medical imaging (X-rays).
• Understanding different parts of the spectrum is crucial for technologies such as WiFi and broadcasting.

Planck Distribution

The Planck distribution is a formula that describes how the intensity of radiation emitted by a black body varies with wavelength at a given temperature. Named after physicist Max Planck, this distribution is vital in understanding thermal radiation, like that from cosmic microwave background radiation. The distribution shows us not just the peak wavelength, but also how energy is distributed over various wavelengths. It essentially describes the spectrum of radiation emitted by a "perfect" radiator at any temperature. When we apply Wien's displacement law to the Planck distribution, we pinpoint the specific wavelength with the most energy, which is helpful for analyzing cosmic signals.

• The Planck distribution provides insight into quantum mechanics and thermodynamics.
• It's key to technologies that rely on light and heat, such as thermal imaging and laser design.
• Understanding Planck's distribution equips scientists to predict behaviors of stellar objects.