The emissivity of the human skin is 97.0 percent. Use 35.0 °C for the skin temperature and approximate the human body by a rectangular block with a height of 1.63 m, a width of 33.5 cm and a length of 31.0 cm. Calculate the power emitted by the human body. What is the wavelength of the peak in the spectral distribution for this temperature?

E = 1.143 KW

(λ) max = 9.4 μm = 9.4 x 10⁻⁶ m

Explanation:

a)

The total emissive power of human body can be given by Stefan-Boltzman Law:

E = AεσΤ⁴

where,

E = Total emitted power

ε = emissivity = 97% = 0.97

σ = Stefan Boltzman Constant = 5.67 x 10⁻⁸ W/m². K⁴

T = Absolute Temperature = 35°C + 273 = 308 K

A = Total Surface Area of rectangular approximation = 2 (1.63 m) (0.335 m) + 2 (1.63 m) (0.31 m) + 2 (0.335 m) (0.31 m)

A = 2.31 m²

Therefore,

E = (2.31 m²) (0.97) (5.67 x 10⁻⁸ W/m². K⁴) (308 K) ⁴

E = 1143.52 W = 1.143 KW

b)

The peak wavelength or the maximum wavelength can be found out by using wein's displacement law:

[ (λ) max][T] = 2.8978 x 10⁻³ m. K

(λ) max = 2897.8 μm. K/308 K

(λ) max = 9.4 μm = 9.4 x 10⁻⁶ m

P = 1145.75W

λ = 4.0*10⁻²⁸m

Explanation:

Temperature T = 35°C = 308.15K

h = 1.63m

width (w) = 33.5cm = 0.335m

Length (L) = 31cm = 0.31m

Surface area of the human body =

2hw + 2hL + 2Lw

2 * (1.63*0.335) + 2 * (1.63*0.31) + 2 * (0.31*0.335)

Area = 1.0921 + 1.0106 + 0.2077

Area = 2.3104m²

Using Stefan-Boltzmann equation,

E = εσΤ⁴

Ε = 0.97 * 5.67*10⁻⁸ * (308.15) ⁴

E = 495.91 w/m²

Power = energy * surface area

Power = 495.91 * 2.3104

P = 1145.75W

b)

Applying Energy-Wavelength equation

E = hc / λ

λ = hc / E

λ = (6.626*10⁻³⁴ * 3.0⁸) / 495.91

λ = 4.0*10⁻²⁸m