Problem 43 - Algebra and Trigonometry

Question

Radiation Energy The total radiation energy \(E\) emitted by a heated surface per unit area varies as the fourth power of its absolute temperature \(T\) . The temperature is 6000 \(\mathrm{K}\) at the surface of the sun and 300 \(\mathrm{K}\) at the surface of the earth. (a) How many times more radiation energy per unit area is produced by the sun than by the earth? (b) The radius of the earth is 3960 mi and the radius of the sun is \(435,000\) mi. How many times more total radiation does the sun emit than the earth?

Step-by-Step Solution

Answer

(a) 160,000 times more; (b) 1,930,880,000 times more.

Step 1: Identify Stefan-Boltzmann Law

The problem describes a relationship between radiation energy emitted and temperature, which follows the Stefan-Boltzmann Law: \[ E = \sigma T^4 \]where \sigma is the Stefan-Boltzmann constant, and \(T\) is the absolute temperature in Kelvin.

Step 2: Calculate the Ratio for Part (a)

To find how many times more radiation energy per unit area is produced by the sun than by the earth, compute the ratio of their Stefan-Boltzmann equations:\[ \frac{E_{sun}}{E_{earth}} = \frac{T_{sun}^4}{T_{earth}^4} \]Substituting the given temperatures:\[ \frac{6000^4}{300^4} = \frac{(6 \times 10^3)^4}{(3 \times 10^2)^4} = (\frac{6}{0.3})^4 = 20^4 \]\[ (20)^4 = 160,000 \]

Step 3: Conclude Part (a) Result

The computation shows that the sun emits 160,000 times more radiation energy per unit area than the earth.

Step 4: Understand Part (b) - Total Radiation

Total radiation energy depends not only on radiation per unit area but also on the surface area of the emitting body. Calculate the surface area using the formula for the surface area of a sphere: \[ A = 4\pi r^2 \]where \(r\) is the radius of the sphere. Compute total radiation using:\[ E_{total} = E \times A \]

Step 5: Calculate Surface Areas

Find the surface area of the Earth and Sun:\[ A_{earth} = 4\pi (3960)^2 \]\[ A_{sun} = 4\pi (435,000)^2 \]

Step 6: Calculate Total Emitted Radiation Ratio for Part (b)

Now calculate the total radiation ratio:\[ \frac{E_{total, sun}}{E_{total, earth}} = \frac{E_{sun} \times A_{sun}}{E_{earth} \times A_{earth}} \]Substitute the areas and use the fact that \(\frac{E_{sun}}{E_{earth}} = 160,000\):\[ (160,000) \times \frac{(435,000)^2}{(3960)^2} \]\[ = 160,000 \times \frac{189,225,000,000}{15,681,600} \]\[ = 160,000 \times 12,068 \]\[ = 1,930,880,000 \]

Step 7: Conclude Part (b) Result

The sun emits approximately 1,930,880,000 times more total radiation than the earth due to both higher emission per unit area and its vastly larger surface area.

Key Concepts

Radiation Energy

Radiation energy is the total energy emitted from a surface in the form of electromagnetic waves. This energy, particularly in the context of thermal radiation, is radiant heat transferring through space. For any heated object, this energy is directly tied to its **absolute temperature**, meaning the temperature in Kelvin (K). The Stefan-Boltzmann Law expresses this relationship. It states that the power emitted per unit area of a black body is directly proportional to the fourth power of its temperature. This can be represented as:

• \( E = \sigma T^4 \)

where \(E\) is the radiation energy per unit area, \(T\) is the absolute temperature, and \(\sigma\) is the Stefan-Boltzmann constant. This law helps us understand why stars, like our sun, have such incredibly high emissions of radiation energy.

Surface Area of a Sphere

The surface area of a sphere is an important concept when analyzing radiation because the total emitted energy depends not only on energy per unit area but also on the entire surface area itself. For a sphere, the surface area \(A\) can be calculated using the formula:

• \( A = 4\pi r^2 \)

where \(r\) is the radius of the sphere. In simple terms, the bigger the sphere, and thus the bigger the radius, the more surface there is to emit radiation. With this understanding, it becomes clear why the total radiant energy from the sun is much greater than that of the earth. Despite having an impressive surface area of its own, the earth is dwarfed by the sun's gigantic surface area, due to its much larger radius.

Absolute Temperature

Absolute temperature, measured in Kelvin, plays a crucial role in determining radiation energy. The Kelvin scale is an absolute scale with its zero point at absolute zero, the theoretical temperature where molecular motion ceases. This concept is particularly important in physics and chemistry where precise temperature measurements are critical. In applications like the Stefan-Boltzmann Law, using Kelvin is necessary because temperature ratios are calculated in absolute terms. For instance, the surface of the sun has a temperature of 6000 K, far exceeding the earth's 300 K. When these temperatures are raised to the fourth power in accordance with the Stefan-Boltzmann equation \((T^4)\), the temperature difference is magnified significantly, showing why the sun emits much more radiation energy per unit area.

Thermal Radiation

Thermal radiation refers to the emission of electromagnetic waves from all matter that has a temperature greater than absolute zero. It's how heat travels through a vacuum and is one mode of heat transfer alongside conduction and convection. The sun, a prime example of a thermal radiator, emits a spectrum of electromagnetic radiation due to the immense energy produced by nuclear fusion occurring in its core. This thermal radiation reaches Earth, providing the heat necessary to sustain life. Understanding thermal radiation helps explain phenomena such as why we feel warm stepping outside on a sunny day, or how terrestrial planets can differ in temperature based on their distance from the sun and their surface characteristics. In essence, thermal radiation encompasses everything from the warmth produced by a heater in a cozy room to the intense energy radiated from a star thousands of lightyears away.