Describe how the rate of emission of radiation depends on the surface temperature and surface area of an object

2.3.3 Radiation

Objective

Describe how the rate of emission of radiation depends on the surface temperature and surface area of an object.

Key Concepts

• All objects with a temperature above absolute zero emit electromagnetic radiation.
• The amount of energy emitted per unit time is called the radiative power or rate of emission.
• For a perfect black‑body the radiative power is given by the Stefan‑Boltzmann law.

Stefan‑Boltzmann Law

The total power \$P\$ radiated by a black‑body is proportional to its surface area \$A\$ and to the fourth power of its absolute temperature \$T\$:

\$P = \sigma A T^{4}\$

where \$\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\$ is the Stefan‑Boltzmann constant.

Real Surfaces – Emissivity

Real objects are not perfect black‑bodies. Their ability to emit radiation is described by the emissivity \$e\$, a dimensionless number between 0 and 1.

\$P = e\,\sigma A T^{4}\$

Typical emissivity values:

• Polished metal: \$e \approx 0.03\$ – 0.10
• Matte black paint: \$e \approx 0.95\$ – 0.98
• Human skin: \$e \approx 0.98\$

Dependence on Surface Area

From the equation \$P = e\sigma A T^{4}\$, if the temperature \$T\$ and emissivity \$e\$ are kept constant, the radiative power is directly proportional to the surface area:

\$P \propto A\$

Doubling the exposed area doubles the rate of emission.

Dependence on Surface Temperature

The temperature appears to the fourth power, giving a very strong dependence:

\$P \propto T^{4}\$

Examples:

1. If the temperature is increased from \$300\ \text{K}\$ to \$600\ \text{K}\$ (a factor of 2), the power increases by \$2^{4}=16\$ times.
2. A small rise of \$10\%\$ in temperature (\$T \to 1.10T\$) raises the power by \$(1.10)^{4}\approx 1.46\$, i.e., a \$46\%\$ increase.

Combined Effect

Both area and temperature act simultaneously. The overall rate of emission can be compared using the ratio:

\$\frac{P{2}}{P{1}} = \frac{e{2}}{e{1}}\;\frac{A{2}}{A{1}}\;\left(\frac{T{2}}{T{1}}\right)^{4}\$

This expression is useful for solving exam questions that involve changes in area, temperature, or surface coating.

Summary Table

Worked Example

Calculate the radiative power emitted by a \$0.5\ \text{m} \times 0.5\ \text{m}\$ black plate (\$e=0.95\$) at \$400\ \text{K}\$.

1. Surface area: \$A = 0.5 \times 0.5 = 0.25\ \text{m}^{2}\$
2. Insert values into the Stefan‑Boltzmann equation:

   \$P = 0.95 \times 5.67 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4} \times 0.25\ \text{m}^{2} \times (400\ \text{K})^{4}\$

3. Compute \$(400)^{4} = 2.56 \times 10^{10}\$
4. \$P = 0.95 \times 5.67 \times 10^{-8} \times 0.25 \times 2.56 \times 10^{10}\$

   \$P \approx 0.95 \times 5.67 \times 0.25 \times 2.56 \times 10^{2}\$

   \$P \approx 0.95 \times 5.67 \times 0.64 \times 10^{2}\$

   \$P \approx 0.95 \times 3.6288 \times 10^{2}\$

   \$P \approx 345\ \text{W}\$

The plate radiates roughly \$345\ \text{W}\$ of power.

Common Exam Questions

• Predict how the power changes when the temperature of a filament is increased.
• Compare the radiative power of two objects of different sizes but the same temperature.
• Explain why a polished metal spoon feels cooler than a matte black spoon at the same temperature.

Key Take‑aways

• The rate of emission of radiation follows \$P = e\sigma A T^{4}\$.
• Doubling the surface area doubles the power; doubling the temperature increases the power by a factor of 16.
• Emissivity determines how close a real surface is to an ideal black‑body.