Light and Energy: Understanding the Energy Received by the Earth

Scientific Summary

This article analyzes the flux of solar energy towards Earth through the quantized nature of photons. The energy of a photon is given by E = h·f, making blue photons (~10⁻¹⁸ J) more energetic than red ones (~4.6×10⁻¹⁹ J). The solar constant (~1368 W/m²) represents the incident power at the top of the atmosphere. After dividing by 4 and subtracting the albedo (~30 %), the average power absorbed by the Earth is about 239 W/m². Radiative equilibrium (Stefan-Boltzmann law) would yield a temperature of 255 K (-18 °C), but the natural greenhouse effect (water vapor, CO₂) raises this temperature to 288 K (15 °C).

How is the energy of solar photons calculated and how much power does the Earth receive from the Sun?

Sunlight is composed of photons whose energy is given by the formula E = h × f (h = Planck's constant, f = frequency). The shorter the wavelength (towards blue/violet), the higher the energy of the photon: a blue photon carries about 10⁻¹⁸ J, compared to ~4.6×10⁻¹⁹ J for a red photon. To heat a cup of coffee (10 cm³, +50 °C), about 10²¹ blue photons or 10²² red photons are needed. The solar power received by Earth is the solar constant (~1368 W/m² at the top of the atmosphere). On average over the Earth's surface (accounting for geometry and albedo), the absorbed power is about 239 W/m². Without the greenhouse effect, the average temperature would be 255 K (-18 °C); the greenhouse effect (water vapor, CO₂) raises it to 288 K (15 °C).

The Energy of Photons

The energy of photons and their origin

Sunlight warms the Earth. This energy comes from electromagnetic waves and particularly from the photons that strike the surface of our planet. Visible light is part of a small range of electromagnetic vibrations found in the electromagnetic spectrum.

The role of Earth's atmosphere

The Earth's atmosphere only allows a very small portion of this radiation to pass through. Short waves are absorbed in the atmospheric layers and long waves are reflected. Most of the waves that reach us are those of visible light.

The nature of light falls under quantum mechanics: it is both a wave and a particle (one can count the particles contained in the wave).

Wavelengths and colors

The different windows of the electromagnetic spectrum are also characterized by defined frequency ranges.

Frequency and energy of photons

Frequency is the number of electromagnetic oscillations that pass through a given point in one second. It is expressed in hertz (Hz).

Fundamental relationship: The shorter the wavelength, the higher the frequency.

When we receive the sun's rays on our skin, we perceive that the photons carry energy.

Calculating the energy of a photon

The energy of a photon depends on its wavelength: The shorter the wavelength, the more energetic the photon.

How many photons does it take to heat a cup of coffee?

The energy of photons according to their color

This depends on the frequency, hence the color. The frequency of red is 350 THz and that of blue is 750 THz. The formula E = hf gives the energy of a photon.

Calculating the energy of a blue photon

Energy of a blue photon: E = (6.62×10^-34) × (750×10¹²) ≈ 10^-18 joule. Thus, 1 joule ≈ 10¹⁸ blue photons.

Application: heating a cup of coffee

The calorie is the amount of energy needed to raise the temperature of one gram of liquid water from 14.5 to 15.5 °C, which corresponds to 4.1855 joules.

To raise the temperature of 10 cm³ of coffee by 50 °C, about 5 × 10²¹ joules are needed, which is:

Remarks and useful conversions

How much solar power does the Earth receive?

The solar constant: a key measurement

The Earth constantly receives a colossal amount of energy from the Sun. This energy is measured by what is called the solar constant: the amount of solar energy received per unit area, perpendicular to the Sun's rays, at the average Earth-Sun distance (about 150 million kilometers).

The solar constant is approximately 1,361 watts per square meter (W/m²). This means that each square meter directly exposed to the Sun, outside the Earth's atmosphere, receives about 1,361 watts of energy.

The impact of Earth's atmosphere

However, part of this energy is absorbed, reflected, or scattered by the Earth's atmosphere. On average, only about 1,000 W/m² reach the Earth's surface on a clear day when the Sun is at its zenith.

Total power received by the Earth

To estimate the total power received by the Earth, one must consider the surface area exposed to the Sun. The Earth presents a cross-sectional area (or "disk") of πR², where R is the Earth's radius (about 6,371 km).

P = solar constant × πR² ≈ 1,361 W/m² × 1.275 × 10¹⁴ m² ≈ 1.74 × 10¹⁷ watts.

This power is colossal: it is equivalent to about 174 petawatts (PW), or thousands of times the current global energy consumption.

Distribution and use of solar energy

This energy is the source of many natural phenomena, such as winds, ocean currents, the water cycle, and is also exploited by living beings (photosynthesis) and by human technologies (solar panels, solar thermal power plants).

Comparison with human energy needs

This shows the enormous potential of solar energy as a renewable and sustainable energy source.

FAQ: Everything you need to know about light and solar energy

How is the energy of a photon calculated and why does it depend on its color?

The energy of a photon is calculated using the formula E = h × f, where h is Planck's constant (6.62×10⁻³⁴ J·s) and f is the frequency of the electromagnetic wave in hertz. The frequency is inversely proportional to the wavelength: the shorter the wavelength (towards blue, ~450 nm), the higher the frequency, and thus the more energetic the photon. For example, a blue photon (f ≈ 750 THz) carries about 10⁻¹⁸ J, while a red photon (f ≈ 350 THz) carries about 4.6×10⁻¹⁹ J. The energy of a single photon is therefore infinitely small on a human scale, but the sheer number of photons emitted by the Sun (about 10⁴⁵ photons per second) makes the total energy colossal.

How much solar power does the Earth receive and how is it calculated?

The solar power received at the top of the atmosphere is the solar constant, about 1368 W/m². It is calculated from the Sun's luminosity (L ≈ 3.84×10²⁶ W) divided by the surface area of a sphere centered on the Sun with a radius equal to the Earth-Sun distance (1.5×10¹¹ m): solar constant = L / (4πR²). On average over the Earth's surface, this value is divided by 4 (because only half of the Earth is illuminated and the angle of incidence varies), giving 342 W/m². After subtracting the albedo (~30% of light reflected by clouds, ice, and surfaces), the absorbed power by the Earth is about 239 W/m². The Earth re-emits this energy as infrared radiation to maintain thermodynamic equilibrium.

Why is Earth's average temperature 15 °C when the simple calculation gives -18 °C?

The equilibrium temperature of a planet without an atmosphere would be given by the Stefan-Boltzmann law: the absorbed power (239 W/m²) is re-emitted as P = σT⁴, giving T ≈ 255 K (about -18 °C). However, Earth has an atmosphere containing greenhouse gases (water vapor, carbon dioxide, methane). These gases are transparent to visible light (incoming solar radiation) but absorb part of the infrared radiation (heat) re-emitted by the Earth's surface, and then re-emit it in all directions, including back towards the ground. This phenomenon, called the greenhouse effect, traps some of the thermal energy and raises the average surface temperature by about 33 K, bringing it to 288 K (15 °C). Without this natural greenhouse effect, Earth would be a frozen and likely uninhabitable planet.