Question

At the surface of the Earth, the Sun delivers an estimated \(1.00 \mathrm{~kW} / \mathrm{m}^{2}\) of energy. Suppose sunlight hits a \(10.0 \mathrm{~m}\) by \(30.0 \mathrm{~m}\) roof at an angle of \(90.0^{\circ}\) a) Estimate the total power incident on the roof. b) Find the radiation pressure on the roof.

Short answer

Answer: The total power incident on the roof is 300.0 kW and the radiation pressure on the roof is 1.00 x 10^-6 N/m^2.

Step-by-step solution

Find the effective area of sunlight on the roof

The angle at which sunlight hits the roof is \(90.0^{\circ}\). This means that sunlight falls directly on the surface of the roof and all of its area is exposed to sunlight. Therefore, the effective area of sunlight on the roof is equal to the roof area. The roof area can be calculated as the product of its length and width: Area \(= 10.0 \, \text{m} \times 30.0 \, \text{m} = 300 \, \text{m}^2\)

Calculate the total power incident on the roof

Now that we have the effective area of sunlight on the roof, we can calculate the total power incident using the given power per unit area: Total power \(= 300.0\, \text{m}^2 \times 1.00\, \cfrac{\text{kW}}{\text{m}^2} = 300.0\, \text{kW}\) So, the total power incident on the roof is \(300.0\, \text{kW}\).

Calculate the radiation pressure on the roof

To find the radiation pressure on the roof, we'll need the following formula: Radiation pressure \(= \cfrac{P}{c}\) where \(P\) is the total power incident on the roof, and \(c\) is the speed of light. The speed of light is approximately \(3.00 \times 10^8\, \cfrac{\text{m}}{\text{s}}\). Converting the total power to watts, we have \(P = 300.0 \times 10^3 \, \text{W}\). Now, we can calculate the radiation pressure: Radiation pressure \(= \cfrac{300.0 \times 10^3 \, \text{W}}{3.00 \times 10^8 \, \cfrac{\text{m}}{\text{s}}} = 1.00 \times 10^{-6} \, \cfrac{\text{N}}{\text{m}^2}\) So, the radiation pressure on the roof is \(1.00 \times 10^{-6} \, \cfrac{\text{N}}{\text{m}^2}\).

Key concepts

Solar Power Calculation

Understanding how to calculate solar power is crucial for a variety of applications, ranging from the design of solar panels to the estimation of energy production. The aforementioned exercise starts by examining the amount of energy delivered by the Sun to the surface of the Earth, expressed as power per unit area. To put this into perspective, imagine that each square meter of Earth's surface is receiving a lamp of 1 kilowatt when exposed to direct sunlight.

The solution proceeds by determining the total area exposed to the sunlight, which is simply the product of the length and width of the roof. It's essential to note that the angle of incidence plays a pivotal role in these calculations. Since the sunlight is assumed to hit at a right angle, or a 90-degree angle, the entire area of the roof is effectively utilized. The 'effective area' concept is important here, as any deviation from a 90-degree angle would result in a need to adjust the calculations to account for a reduced effective area.

Furthermore, considering that solar panels are not 100% efficient, this calculation can be adjusted to estimate actual electricity generation. This involves multiplying the power incident by the efficiency of the solar panels, which typically ranges from 15% to 20% for most commercially available panels. This inclusion makes the theoretical exercise more practical and relevant for real-world applications.

Radiation Pressure

The sensational concept of radiation pressure might seem more at home in science fiction, but it is in fact a very real physical phenomenon. It arises from the momentum of photons, the particles of light, which exert a force when they interact with a surface. This pressure is incredibly small, often negligible under ordinary circumstances on Earth, but it is significant in certain contexts, such as in the study of comet tails or the development of spacecraft propulsion systems.

In the context of the exercise, we discuss radiation pressure in relation to sunlight hitting a roof. The step by step solution provides a clear method to calculate it using the formula for radiation pressure, which involves the total power incident on the roof and the speed of light. Solar sails in space use this concept to propel themselves – the collective pressure of solar radiation on large, reflective surfaces creates enough force to maneuver a lightweight spacecraft. However, on Earth, the modest value of solar radiation pressure calculated in the exercise hints at why we don't feel pushed around by sunlight in our daily lives. It is a testament to the nuanced elegance of physics that such a small force can have critical applications in technology and space exploration.

Power Incident on Surface

When we discuss power incident on a surface, we are referring to how much energy is received by a given area per unit time. This concept is not only relevant for solar energy calculations but is also essential in understanding various physical phenomena, such as the heating effects of sunlight and the behavior of electromagnetic waves.

The power incident on the roof, as worked out in the exercise, depends directly on the intensity of sunlight and the area over which it's spread. The intensity, typically measured in watts per square meter, when multiplied by the area of the surface, gives you the total power in watts or kilowatts. This simple calculation has profound implications for renewable energy; it allows for the estimation of how much power can be harnessed from solar panels, as well as understanding heat gain in buildings and optimizing the design for passive solar heating or cooling.

To enhance the understanding of the power incident on a surface, one should also consider factors like the surface albedo, which is the measure of how much light is reflected without being absorbed. Darker surfaces tend to absorb more light and therefore capture more power, while lighter surfaces reflect more. Taking into account the diurnal and seasonal changes in the Sun's path could also improve the accuracy of such calculations for real-life applications.