Grey bodies as approximations to black bodies

Black bodies, have a characteristic emission spectrum with a maximum intensity at a specific wavelength ($\lambda_{max}$) for a given temperature (T). This relationship is described by Wien's displacement law:

$\lambda_{max}T = 2.9 \times 10^{-3} \text{ m K}$

The total power output of a black body is related to its surface area (A) and absolute temperature (T) through the Stefan-Boltzmann law:

$P = \sigma AT^4$

In reality, objects can behave similarly to black bodies without being perfect. These approximations are called grey bodies. At a given temperature, a grey body emits less energy per second than a perfect black body with the same dimensions.

Worked example - calculating the power emitted by a grey body

Calculate the power emitted by a brick wall with a surface area of 10 m² at a temperature of 300 K. The emissivity of brick is 0.90.

Step 1: Formula

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

Step 2: Identify known values

• e = 0.90
• $\sigma$ = 5.67 $\times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$
• A = 10 $\text{ m}^2$
• T = 300 K

Step 3: Substitution and correct evaluation

$P_e = 0.90 \times 5.67 \times 10^{-8} \times 10 \times 300^4$

$P_e = 4133.43 \text{ W}$

Typical emissivity values

The table below shows typical emissivity values at visible wavelengths for some common substances:

It's important to note that emissivity values are wavelength-dependent. Surprisingly, snow, which appears white and reflective, is an effective emitter (and absorber) at infrared wavelengths.