Question

Taking the Earth's temperature. A physical law, called the Stefan-Boltzmann (SB) law, relates the transfer rate of electromagnetic radiation of a body (called blackbody radiation to its temperature. The SB law is a relative of the Wein law, the principle that explains why an object that is hot enough will emit visible light, as when a burning ember in a fire glows red, or glows blue if is even hotter. An object having a nonzero temperature will emit electromagnetic radiation. Conversely, an object that absorbs energy via radiation increases its temperature. The SB law says that the electromagnetic power \(P\) is related to the temperature \(T\) by $$ P=\left(5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}\right) T^{4} $$ (a) If the incoming solar power hitting the Earth is \(P=342 \mathrm{~W} \mathrm{~m}^{-2}\), what would you predict should be the Earth's temperature? (b) Now, turn around this argument around. The Earth's actual average temperature is \(T=288 \mathrm{~K}\) or \(+15^{\circ} \mathrm{C}\). Use the SB law to compute the power of radiated heat from the Earth's core.

Short answer

a) Earth's temperature is approximately 278.6 K. b) The radiated power from Earth's core is approximately 390.08 Wm^{-2}.

Step-by-step solution

Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the electromagnetic power emitted by a blackbody, denoted as \(P\), is proportional to the fourth power of the temperature \(T\). The law is given by the formula: \[ P = \big(5.67 \times 10^{-8} \text {Wm}^{-2} \text {K}^{-4} \big) T^{4} \]

Solving for Temperature

We are given the incoming solar power hitting the Earth as \(P = 342 \text{ Wm}^{-2}\). Plugging in the given power into the Stefan-Boltzmann Law, we need to solve for \(T\): \[ 342 = \big(5.67 \times 10^{-8} \text {Wm}^{-2} \text {K}^{-4} \big) T^{4} \]

Isolate the Temperature Variable

Rearrange the equation to solve for \(T\): \[ T^{4} = \frac {342} {5.67 \times 10^{-8}} \]

Calculate the Temperature

Calculate \(T\) by taking the fourth root of the right-hand side: \[ T = \bigg(\frac {342} {5.67 \times 10^{-8}} \bigg)^{1/4} \] Computing this, we get \[ T \thickapprox 278.6 \text {K} \]

Use Earth's Actual Temperature for Radiated Heat

Given that Earth's average temperature is \(T = 288 \text{K}\), use the Stefan-Boltzmann Law to find the radiated power \(P\): \[ P = \big(5.67 \times 10^{-8} \text {Wm}^{-2} \text {K}^{-4} \big) (288)^{4} \]

Calculate Radiated Power

Substitute \(T = 288 \text{K}\) into the equation and solve for \(P\): \[ P \thickapprox \big(5.67 \times 10^{-8} \big) (288)^{4} \] With the values computed, we find \[ P \thickapprox 390.08 \text {Wm}^{-2} \]

Key concepts

Blackbody Radiation

Blackbody radiation refers to the way an idealized physical body absorbs and emits all frequencies of electromagnetic radiation. When a body reaches thermal equilibrium, the radiation it emits is called blackbody radiation. This concept is crucial in understanding how objects emit heat and light based on their temperature. For example, a blackbody heated to around 5000K will emit white light, while cooler bodies emit primarily infrared radiation. This principle helps explain the Stefan-Boltzmann Law and why bodies at different temperatures emit different types and amounts of electromagnetic radiation.

Electromagnetic Radiation

Electromagnetic radiation encompasses various types of waves, including visible light, radio waves, X-rays, and more. When an object has a temperature above absolute zero, it emits electromagnetic radiation. This energy travels through space in the form of waves. The physical laws governing this emission are key to understanding blackbody radiation and calculating the temperature of objects like the Earth. The different types of electromagnetic radiation are determined by their wavelength and frequency. For example:
- Visible light has wavelengths between 400 and 700 nanometers
- Infrared radiation has longer wavelengths than visible light, and
- Ultraviolet radiation has shorter wavelengths than visible light. Understanding these differences helps explain the varying energy emissions of objects, whether they are as cool as the Earth's surface or as hot as the sun.

Temperature Calculation

Using the Stefan-Boltzmann Law, we can calculate the temperature of objects by knowing their emitted electromagnetic power. In the exercise, the incoming solar power hitting Earth is given as 342 W/m². By using the formula:

\[ P = (5.67 \times 10^{-8} \text {Wm}^{-2} \text {K}^{-4}) T^{4} \]

we solve for the temperature, T.

First, isolate the temperature variable:

\[ T^{4} = \frac {342} {5.67 \times 10^{-8}} \]

Then, compute the temperature by taking the fourth root of the right-hand side:

\[ T = \bigg(\frac {342} {5.67 \times 10^{-8}} \bigg)^{1/4} \]

From this, we obtain:

\[ T \thickapprox 278.6 \text {K} \]

This calculation tells us the predicted temperature of Earth based on the incoming solar power. Inversely, by knowing Earth’s average temperature (288K), we can determine the power of radiated heat using the same law:

\[ P \thickapprox (5.67 \times 10^{-8}) (288)^{4} \thickapprox 390.08 \text {Wm}^{-2} \]

This showcases the relationship between temperature and radiative power.

Wein's Law

Wein's Law helps explain the relationship between the temperature of an object and the peak wavelength of the emitted radiation. It's given by the formula:

\[ \lambda_{max} = \frac{b}{T} \]

where \( \lambda_{max} \) is the wavelength at which the emission is at its peak,
\( b \) is a constant (2.897 × 10^{-3} m K), and
\( T \) is the temperature in Kelvin.

This law reveals that as an object's temperature increases, the peak wavelength of its emitted radiation shifts to shorter wavelengths. For example, the sun, with a temperature of around 5778K, emits most of its energy in the visible spectrum, making it appear white. Cooler objects like the Earth emit primarily in the infrared spectrum. Understanding this shift helps explain phenomena like why hot objects glow visibly and why infrared cameras can detect heat emissions from cooler objects based on their shift in wavelength according to Wein's Law.