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You have been hired as an engineer on a NASA project to design a microwave spectrometer for an orbital mission to measure the cosmic background radiation, which has a black body spectrum with an effective temperature of \(2.725 \mathrm{K}\). (a) The spectrometer is to scan the sky between wavelengths of \(0.50 \mathrm{mm}\) and \(5.0 \mathrm{mm},\) and at each wavelength it accepts radiation in a wavelength range of \(3.0 \times 10^{-4} \mathrm{mm} .\) What maximum and minimum radiation intensity do you expect to find in this region? \((b)\) The photon detector in the spectrometer is in the form of a disk of diameter \(0.86 \mathrm{cm} .\) How many photons per second will the spectrometer record at its maximum and minimum intensities?

Step by step solution

01

- Identify the relevant formula

Use Planck's Law to find the spectral radiance of a black body: \[B(λ, T) = \frac{2hc^2}{λ^5} \frac{1}{e^{(hc/λkT)} - 1}\]

02

- Convert units if necessary

Convert the temperature to Kelvin, if necessary. Here, it is already given as \(T = 2.725 \mathrm{K} \).

03

- Calculate maximum radiation intensity at 0.50 mm

Substitute \( λ = 0.50 \mathrm{mm} \), \(T = 2.725 \mathrm{K}\), and other constants into Planck’s Law: \[B(0.50 \mathrm{mm}, 2.725 \mathrm{K}) = \frac{2hc^2}{(0.50 \times 10^{-3} \mathrm{m})^5} \frac{1}{e^{(hc/(0.50 \times 10^{-3} \mathrm{m} k 2.725 \mathrm{K}))} - 1}\]

04

- Calculate minimum radiation intensity at 5.0 mm

Substitute \( λ = 5.0 \mathrm{mm} \), \(T = 2.725 \mathrm{K}\), and other constants into Planck’s Law: \[B(5.0 \mathrm{mm}, 2.725 \mathrm{K}) = \frac{2hc^2}{(5.0 \times 10^{-3} \mathrm{m})^5} \frac{1}{e^{(hc/(5.0 \times 10^{-3} \mathrm{m} k 2.725 \mathrm{K}))} - 1}\]

05

- Calculate photon flux

The power received per unit area per unit wavelength from a black body is given by the formula using Planck’s Law. Use it to calculate the photon number flux by dividing the intensity by the energy of a single photon: \[\Phi = \frac{B(λ, T) \Delta λ A}{hc/λ} = \frac{B(λ, T) \Delta λ A λ}{hc}\]

06

- Calculate maximum photon flux

Substitute maximum intensity values and area (using disk diameter) into photon flux formula: \[A = \pi (\frac{d}{2})^2\]Compute for \( λ = 0.5 \mathrm{mm}\): \[\Phi_{max} = \frac{B(0.50 \mathrm{mm}, 2.725 \mathrm{K}) \times 3.0 \times 10^{-4} \mathrm{mm} \times A \times 0.50 \mathrm{mm}}{hc}\]

07

- Calculate minimum photon flux

Compute for \( λ = 5.0 \mathrm{mm}\): \[\Phi_{min} = \frac{B(5.0 \mathrm{mm}, 2.725 \mathrm{K}) \times 3.0 \times 10^{-4} \mathrm{mm} \times A \times 5.0 \mathrm{mm}}{hc}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosmic Background Radiation

Cosmic background radiation, also known as the Cosmic Microwave Background (CMB), is the thermal radiation left over from the Big Bang. It provides a snapshot of the Universe when it was just 380,000 years old. This radiation is remarkably uniform in all directions, but tiny fluctuations give clues about the early stages of the Universe. The CMB is crucial for cosmology as it supports the Big Bang theory and helps us understand the Universe's composition and evolution.

Planck's Law

Planck's Law describes the spectral radiance of a black body at a given temperature and wavelength. It is fundamental in understanding emissions from black bodies. The formula is expressed as:
\[ B(λ, T) = \frac{2hc^2}{λ^5} \frac{1}{e^{\frac{hc}{λkT}} - 1} \]
Where:

• \(h\) is Planck's constant
• \(c\) is the speed of light
• \(λ\) is the wavelength
• \(k\) is Boltzmann constant
• \(T\) is the temperature in Kelvin

Planck's law explains how the energy emitted by a black body is distributed across different wavelengths.

Black Body Spectrum

A black body is an idealized object that absorbs all incident electromagnetic radiation, regardless of frequency or angle. The black body spectrum refers to the range of wavelengths emitted by a black body. It is characterized by its temperature and follows Planck's law. The spectrum shows how much radiation is emitted at different wavelengths. Higher temperatures shift the peak of the spectrum to shorter wavelengths, demonstrating the relationship between temperature and emitted radiation.

Spectral Radiance

Spectral radiance measures the power emitted by a black body per unit area per unit solid angle per unit wavelength. It is a crucial concept in physics and astronomy since it helps quantify the intensity of radiation across different wavelengths. Using Planck's Law, spectral radiance in terms of wavelength is given by:
\[ B(λ, T) = \frac{2hc^2}{λ^5} \frac{1}{e^{\frac{hc}{λkT}} - 1} \]
This formula allows the calculation of intensity at specified temperatures, essential for applications like designing microwave spectrometers.

Photon Flux Calculation

Photon flux is the number of photons passing through a unit area per unit time. It can be determined using the spectral radiance calculated via Planck's Law. For a given wavelength and area of a detector, the photon number flux \( \Phi \) can be calculated using:
\[ \Phi = \frac{B(λ, T) Δλ A λ}{hc} \]
Where \( \Deltaλ \) is the wavelength range, \(A\) is the detector area, and the other symbols retain their usual meanings. This formula helps in estimating how many photons will hit the detector per second, crucial for designing and evaluating the spectrometer.