According to Stefan’s law of radiation, a black body radiates energy σT4 from its unit surface area every second where T is the surface temperature of the black body and σ = 5.67 × 10-8 W/m2K4 is known as Stefan’s constant. A nuclear weapon may be thought of as a ball of radius 0.5 m. When denoted, it reaches temperature of 106K and can be treated as a black body.
(a) estimate the power it radiates
(b) if surrounding has water at 30oC, how much water can 10% of the energy produced evaporate in 1 sec?
(c) if all this energy U is in the form of radiation, corresponding momentum is p = U/c. How much momentum per unit time does it impart on unit area at a distance of 1 km?
(a) E = σT4
Total energy = energy radiated from surface area A per second
P = σAT4
σ = 5.67 × 10-8 W/m2K4
R = 0.5 m
T = 106 K
Substituting the values we get,
P = 1.8 × 1017 J/s
(b) P = 18 × 1016 Watt
10% of this energy is used in evaporation of water
E = (10/100)(18)(1016) Watt = 1.8 × 1016 J/s
Energy required to evaporate water in 1 sec using 10% of energy is
$m S_{w}\left(T_{2}-T_{1}\right)+m L=m\left(S_{w}\left(T_{2}-T_{1}\right)+L\right)$
m = 7 109 kg
(c) Momentum per unit time p’ = U/C = 0.6 × 109
P’ = 6 × 108 kg.m/s2
P per unit time at a distance 1 km is
$\frac{6 \times 10^{8}}{4 \pi R^{2}}$
Substituting the values we get P per sec at 1 km on m2 = 47.8 N/m2