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Note: This exercise must be done on a clear day and/or during a 20‐minute period when there are no clouds.

Almost all of the Earth’s available energy comes from the sun (geothermal and nuclear energy do not). Fossil fuels such as oil and coal are a result of ancient photosynthesis. Renewable biomass fuel, as well as our food supply, is also a product of photosynthesis. Wind and waves are driven by heat from the sun, and solar panels directly collect the sun’s radiant energy. Consequently, the planet’s available energy and our food supply are both restricted by the amount of energy received from the sun. Humans and other animals depend on the plant kingdom for food, which is completely dependent on solar radiation.

In this exercise you and your partners will determine how much radiant energy the earth receives from the sun at the earthʹs surface. From this and some additional information you will be able to draw a lot of interesting conclusions such as how much of this energy is used in photosynthesis, how much of total solar energy is used by plants humans can eat; how many people the earth could support if all we did was feed them, whether there is enough solar energy to run our economy in the United States, and how much of the solar energy coming from the sun gets all the way to the earthʹs surface. To measure the energy coming from the sun you and your partners will determine the amount of the sunʹs energy that is absorbed by a quantity of water or copper, which will increase in temperature as the sun heats it.

Pre‐lab reading

Textbook section 8.2

Equipment

Alternate equipment

Predictions/preliminary questions

  1. To determine how much solar energy reaches the earth’s surface you can let sunlight heat a substance such as water or metal, then calculate how much heat has been gained by the substance and from that obtain an estimate of the amount of solar energy reaching the earth. Suppose the substance you are heating is water.
    • a) What would you need to measure to be able to determine the radiant heat absorbed by the water?
    • b) How would you calculate the amount of heat gained?
    • a) To determine the power supplied by the sun what must you measure in addition to the heat gained?
    • b) After measuring the quantity needed in part a) how do you calculate the power supplied by the sun?
    • a) To determine the power supplied by the sun per square meter what must you measure?
    • b) After measuring the quantity in part a) how do you calculate the power per square meter?
  4. Once you know the power supplied by the sun per square meter how would you calculate the power supplied to the whole earth?

Procedure

In this exercise, you will put a parabolic mirror in the sun. It will focus the sun’s rays on a container of water or an piece of copper that lies at the focal point of the mirror. Refer to the Historical Notes at the end of this lab to learn more about the use of parabolic mirrors in the past.

You will use the focused rays to heat water or copper, much as a solar power plant uses mirrors to heat water in order to drive a turbine. Based on the increase in the heat energy of the water or metal , you will find how much energy Earth receives from the sun. From this calculation, you and your partners can estimate how much of this energy is used in photosynthesis; how much of that is used by plants we can eat; and how large a human population Earth can support.

  1. Determine the cross‐sectional area of the parabolic mirror. In other words, if you look at the mirror head‐on, what is the apparent area of the circle you see? If you have difficulties with this, ask your instructor for assistance.
  2. If you are using a water sample, fill the receiver at the focal point with water. The temperature of the water should be close to the temperature outside. You and your partners should determine the mass of this water using a styro‐foam cup and the triple‐beam balance.
  3. Find a location outside that has direct sunlight and little or no wind. Try to stay away from buildings or walls because they can reflect additional sunlight toward the mirror. If the wind picks up, shield the set‐up as best as you can, but don’t block any sunlight!
  4. Be sure to aim your parabolic mirror directly at the sun. Be very careful about this. All members of your group should check the container – the one with water or a metal plate ‐ to be certain it is pointing directly at the sun. You will probably have to adjust the rectangular container (if you are using that) during the time it is receiving solar energy. (The parabolic mirror or its container might also need adjustment during collection time.) Also be sure that you shield the mirror from the sunlight until you start the stopwatch.
  5. Record the initial temperature of the water or metal. Record the temperature once every three minutes for 15 minutes (or, if you are using water, until the water starts to boil). Be careful ‐‐ the water, metal and receiver might become very hot. After your last measurement, make sure you have recorded the mass of the water or copper. If you were using water, pour it out. Gather your equipment and return to the lab.

Alternate equipment: A rectangular box containing an aluminum plate to absorb solar energy can be used to collect solar energy. The principles are the same as when using water. It works best to cool the plate 10oC to 20oC below the outside temperature before starting. (Be certain the plate is dry before you start using it to collect solar energy.) Heat the box until it is 10oC to 20oC above the outside temperature. You need to align the container so the sun’s rays are perpendicular to it. (You can tell the rays are perpendicular when the white plastic peg in the middle casts no shadow.)The temperature can be obtained by connecting leads from the terminals on the side of the box to an ohmmeter. Ask your TA for a conversion chart to change resistance into temperature. The rest of the process uses the same principles as used with the water.

Analysis

The ʺsolar inputʺ is the amount of power that an area receives from the sun when directly overhead. Using your measurements, you and your partners can determine the heat energy absorbed by the water or copper based on the temperature increase. We will assume the receiver on the parabolic mirror is a perfect insulator, meaning that it gains no heat from nor loses any to the environment. Computation of the solar input involves solving an equation that equates the thermal energy gained by the copper or water equal to the radiant energy received from the sun.

  1. You and your partners should have already calculated the cross‐sectional area of your parabolic mirror. As stated previously, the solar input is the amount of power that an area receives when the sun is directly overhead. When the sun is at an angle (which is always true in Minnesota and all places outside the Tropics of Capricorn and Cancer), an area receives a fraction of the full energy. Since you aimed the mirror directly at the sun, we do not need to make a correction for this.
  2. Next, you and your lab partners should determine the thermal energy gained by the water or copper due to the solar input. Use the equation:
thermal energy transferred = (mass of water or copper) x (specific heat of water or copper) x change in temperature\text{thermal~energy~transferred~=~(mass~of~water~or~copper)~x~(specific~heat~of~water~or~copper)~x~change~in~temperature}
(1)

The change in the temperature is the final temperature of the water or copper minus the initial temperature of the water or copper. The specific heat of water in SI units is 4186 Joules/kg • degrees C. The specific heat of copper is 387 Joules/kgC Be sure to use SI units in your calculations.

  1. You and your partners can now determine the value of the solar input. As mentioned already, you can find the solar input by setting the thermal energy gained by the water or copper equal to the radiant energy which came in from the sun:
Energy received from sun = thermal energy gained by the water or copper\text{Energy~received~from~sun~=~thermal~energy~gained~by~the~water~or~copper}
(2)

Remember, however, that the solar input is a measure of the sun’s power, not energy. Since power is energy over time, we should change our equation to:

Power from the sun = thermalenergyaddedtowaterorcopperTimeinthesun{\rm Power~from~the~sun~=~\frac{thermal energy added to water or copper}{Time in the sun}}
(3)
  1. Power from the sun for the mirror:
Solar input = PowerfromtheSunArea receiving sunlight (in square meters){\rm Solar~input~=~\frac{Power from the Sun}{Area~receiving~sunlight~(in~square~meters)}}
(4)
Solar input = thermal energy addedto the water or copper(mirror area)×(time){\rm Solar~input~=~\frac{thermal~energy~addedto~the~water~or~copper}{(mirror~area)\times (time)}}
(5)
Solar input = mcΔT(mirror area)×(timeinsun){\rm Solar~input~=~\frac{mc\Delta T}{(mirror~area)\times (time in sun)}}
(6)

The solar input is the amount of power that an area receives from the sun when it is directly over that area. If you used only SI units, your result for the solar input is the amount of power (in watts) received by one square meter.

  1. To determine the total solar power received by Earth, you and your partners should multiply your result for the solar input by Earth’s cross‐sectional area. (Use the radius of Earth, 6.38 x 106 m, to calculate its cross‐sectional area.)
Solar power recieved by Earth =Solar Input)(Earth′s cross−sectional Area){\rm Solar~power~recieved~by~Earth~=\frac{Solar~Input)}{(Earth's~cross-sectional~Area)}}
(7)

  1. Determine the solar power used in photosynthesis. It has been estimated that about 0.0757% of the solar power that reaches the Earth is captured by plants and utilized in the photosynthesis process. Knowing this, how much solar power is used by plants to make their food?

  2. Now you can estimate the amount of food could be produced with this much sunlight, if it were all used to produce food. In The Hungry Planet, George Borgstrom, a professor of Food Science and Geography at Michigan State University, states that 10% of the solar power used by plants in photosynthesis can be used by humans as food. Determine how many watts of power this equals.

  3. Knowing that one watt equals 0.24 cal/sec and one food Calorie equals 1000 calories, calculate how many food Calories could be produced on the Earth in one day.

  4. Lastly, estimate the maximum population of Earth. The current opinion is that the normal food requirement of a full‐grown person is about 3,000 food Calories per day. Since most humans eat animal products as well as plants, their intake includes the energy needed to feed the animal products they consume. So, although the normal food requirement is about 3,000 food Calories per day, this figure increases to about 10,000 food Calories a day when the conversion of plants to animal products is taken into account. If you use a 10,000‐Calorie intake as a standard, what is the theoretical maximum number of humans that Earth can support?

  5. The amount of solar radiant energy reaching the top of the earthʹs atmosphere from space was first measured by the first space satellites in the middle of the last century and is about 1360 W/m2. How does this number compare with the radiant power per meter squared which you measured? Do you expect it to be bigger or smaller? Explain the factors which would lead it to be different.

  6. In the US economy, you can deduce from Figure 1 that the power consumption is about 8.5 kilowatts per person on average in recent times. If the solar energy were coming in to the surface of the earth at the rate which you measured, assuming that the US population is 300 million, and assuming that 1/5 of the solar energy could be converted to an easily used form such as electrical energy, how much area of the surface of the US would be needed to collect enough solar energy to power the economy?

United States power consumption per capita in Watts from 1960 to 2023 Data source: [World Bank Group](https://data.worldbank.org/indicator/EG.USE.PCAP.KG.OE?locations=AF-US)

Figure 1:United States power consumption per capita in Watts from 1960 to 2023. Data source: World Bank Group

Conclusions

  1. In the experiment, you assumed that all the energy in the sunlight which was incident on the mirror was transformed into thermal energy in the water or copper. What effects might make this assumption invalid? Which of them is the most important? Can you make any quantitative estimate of how big they might be and how much they could affect the result? Do you expect these effects will make your result for the energy coming in from the sun too big or too small? How could you alter the experiment to make the effects less important?
  2. How does your result compare with the current population of Earth, approximately six billion? Based on current rates of growth, the global population has a doubling time of about 40 years. What are the implications of that figure based on your results? In how many years will the number of humans reach your maximum population estimate?
  3. What if all humans became vegetarian? Go back to the final step of your analysis, and assume that all our food came from photosynthesis. This would mean that each person would need about 3,000 food Calories per day, not 10,000 Calories. If everyone was a vegetarian, how many people could inhabit the planet? How does this compare to your previous result? In about how many years will the population reach your new estimate?
  4. Check the website http://www.oksolar.com/images/daily_solar_radiation1.gif about energy from the sun. It discusses the sun input and shows a map of the minimum solar energy received in the United States per day. How does your result compare with the website values? Redo steps 5 through 9 using the numbers form the website. What are your new answers?

Historical notes

The use of curved mirrors to heat water or some object dates back millennia. In fact, the ancient Greeks, Romans, and Chinese fashioned mirrors that could focus the sun’s energy onto an object and cause it to catch fire in mere seconds. Parabolic mirrors, like the one you and your lab partners used, focus the sun’s rays to a very small point. This was discovered by the mathematician Dositheius, who built the first parabolic mirror in the third century B.C. Plutarch wrote that, when the temple at Delphi was sacked and the sacred flame extinguished, it was relit with the “pure and unpolluted flame from the sun” using “concave vessels of brass.” The ancient Chinese used concave mirrors for similar purposes ‐ a book of ceremonies states that “Directors of the Sun Fire have the duty of receiving, with a concave mirror, brilliant fire from the sun…to prepare brilliant torches for sacrifice.”

Later, it was imagined that large parabolic mirrors could be used as weapons. One apparently apocryphal legend has it that Archimedes used parabolic mirrors to set fire to the vessels of Romans at Syracuse in 212 B.C. In the thirteenth century, Roger Bacon, a monk, suggested that a parabolic mirror could be used as a weapon of mass destruction in the crusades. In the seventeenth century, Giambattista Porta imagined a mirror that so destructive that it would “cast forth terrible fire” and set ablaze "the enemy’s ships, gates, bridges, and the like.

A parabolic mirror uses the sun's rays to boil a kettle of water in Tibet.

Figure 2:A parabolic mirror uses the sun’s rays to boil a kettle of water in Tibet.

Experiments
Lab 5 - Heat and Insulation
Experiments
Lab 7: Mechanical from Thermal Energy: a simple thermal engine and the 2nd law of thermodynamics