Question

A solar furnace uses a large parabolic mirror (mirrors several stories high have been constructed) to focus the light of the Sun to heat a target. A large solar furnace can melt metals. Is it possible to attain temperatures exceeding \(6000 \mathrm{~K}\) (the temperature of the photo sphere of the Sun) in a solar furnace? How, or why not?

Short answer

Explain your answer. Answer: No, it is not possible for a solar furnace to attain temperatures exceeding 6000 K. This is because the energy per unit area from the sunlight (1361 W/m²) is much lower than the energy per unit area radiated by a blackbody at 6000 K (7.34 × 10^7 W/m²). The solar furnace can only concentrate the sunlight but can't produce energy beyond the energy it receives from the sunlight, which is limited by the solar constant (1361 W/m²).

Step-by-step solution

Find the energy per unit area from sunlight

We need to find the energy per unit area from the sunlight. The sunlight's energy received on Earth's surface is given by the solar constant, which is approximately 1361 W/m² (watts per square meter). Therefore, the energy per unit area \(E_{sun}\) from sunlight is: \(E_{sun} = 1361 \: \mathrm{W/m^2}\)

Calculate the energy radiated per unit area from a black body

Now, we will calculate the energy radiated per unit area \(E_{blackbody}\) using the Stefan-Boltzmann law. The Stefan-Boltzmann law states that the power radiated by a surface per unit area is given by: \(E_{blackbody} = \sigma T^4\) where \(\sigma\) is the Stefan-Boltzmann constant (\(\sigma \approx 5.67 \times 10^{-8} \mathrm{W\cdot m^{-2} \cdot K^{-4}}\)) and \(T\) is the temperature in Kelvin. Since the temperature of the photosphere of the Sun is given as \(6000 \:\mathrm{K}\), we can plug this value into the formula to find the energy radiated per unit area: \(E_{blackbody} = \sigma (6000 \:\mathrm{K})^4\) \(E_{blackbody} \approx 7.34 \times 10^7 \:\mathrm{W/m^2}\)

Compare the energy per unit area

Now, we will compare the energy per unit area of the sunlight and the blackbody at 6000 K: \(E_{sun} = 1361 \: \mathrm{W/m^2}\) \(E_{blackbody} \approx 7.34 \times 10^7 \:\mathrm{W/m^2}\) As we can see, the energy per unit area from the photosphere of the Sun (\(7.34 \times 10^7 \:\mathrm{W/m^2}\)) is much greater than the energy per unit area from the sunlight (\(1361 \: \mathrm{W/m^2}\)). The solar furnace can only concentrate the sunlight up to 1361 W/m², which is much lower than the energy per unit area radiated by a blackbody at 6000 K.

Conclusion

It is not possible to attain temperatures exceeding 6000 K in a solar furnace. This is because the energy per unit area from the sunlight is much lower than the energy per unit area radiated by a blackbody at 6000 K. The solar furnace can only concentrate the sunlight but it can't produce energy beyond the energy it receives from the sunlight, which is limited by the solar constant (1361 W/m²).

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann law is a fundamental principle in physics that provides a direct relationship between the temperature of an object and the amount of thermal radiation it emits per unit area. Specifically, it states that the total radiant heat energy emitted from a surface is proportional to the fourth power of its absolute temperature. This relationship is mathematically represented as:

\[E_{blackbody} = \[sigma\] T^4\]

where \(E_{blackbody}\) is the energy radiated per unit area, \(\(sigma\)\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \textrm{W/m}^2 \textrm{K}^{-4}\)), and \(T\) is the absolute temperature in Kelvin. This means, for example, that if the temperature of an object doubles, the energy emitted per unit area increases by a factor of 16 (since \(2^4 = 16\)). As such, it plays a crucial role in understanding why we cannot reach temperatures higher than the Sun's photosphere with a solar furnace on Earth.

Solar Constant

The solar constant represents the amount of incoming solar electromagnetic radiation per unit area, measured at the outer boundary of Earth's atmosphere, on a surface perpendicular to the Sun's rays. The solar constant has an average value of approximately \(1361 \mathrm{W/m^2}\). It is called a 'constant' because it remains relatively stable, but it can vary slightly due to changes in Earth's distance to the Sun and solar activity.

Understanding the solar constant is essential when considering the operation of a solar furnace, as it defines the maximum potential energy available from sunlight that can be focused onto a point or target. Even with techniques to concentrate sunlight, a solar furnace cannot amplify the energy beyond the limit set by the solar constant.

Blackbody Radiation

Blackbody radiation refers to the theoretical radiation emitted by a perfect black body - an idealized physical body that absorbs all incident electromagnetic radiation regardless of frequency or angle of incidence. A black body in thermal equilibrium has an emissivity of \(1\), meaning it is a perfect emitter of radiation.

The concept of blackbody radiation is significant for two reasons. First, it is a good approximation for many real objects, including the photosphere of the Sun. Second, the spectrum of blackbody radiation is determined solely by the body's temperature. Consequently, the Sun, with a photosphere temperature around 6000 K, radiates energy across a vast spectrum, with the peak emission in visible light, which is why the Sun appears bright to our eyes.

Energy Per Unit Area

Energy per unit area is a measure of how much energy is transmitted or received over a specific area. It is especially important when analyzing the performance of devices like solar furnaces, which focus energy onto a particular target.

When comparing the energy per unit area of the sunlight (solar constant) with the energy per unit area radiated by a blackbody (like the Sun's photosphere), it becomes clear why a solar furnace on Earth cannot exceed the Sun's photosphere temperature. The theoretical maximum energy that can be harnessed and concentrated does not match the intense energy per unit area that is radiated by the Sun's photosphere.

Photosphere Temperature

The photosphere temperature refers to the temperature of the visible surface of the Sun, which is about 6000 Kelvin. This temperature is important because it determines the spectrum and amount of solar radiation that reaches Earth. Through the concept of blackbody radiation, the high temperature of the photosphere accounts for the vast amount of energy per unit area that the Sun emits - around \(7.34 \times 10^7 \mathrm{W/m^2}\).

This immense energy, originating from nuclear fusion reactions in the Sun's core, translates into the radiant energy that solar furnaces aim to capture and concentrate. However, as previously discussed, the limitations imposed by the solar constant and the nature of the energy per unit area mean that we cannot achieve the photosphere's temperature here on Earth with current solar furnace technology.