Question:

A human body has a surface area of approximately 1m². The normal body temperature is 10K above the surrounding room temperature T₀. Take the room temperature to be T₀=300K. For T₀=300K the value of σT⁴₀=460Wm⁻²(where σ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

The amount of energy radiated by the body in 1 second is close to 60 Joules

If the surrounding temperature reduces by a small amount ΔT₀<<T₀, then to maintain the same body temperature the same (living) human being needs to radiate ΔW=4σT³₀ΔT₀ more energy per unit time

If the body temperature rise significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation

Solution:

Given A=1m² Tb=T₀+10 (Tb= temperature of body) Also σT⁴₀=460 W/m²
Option A) W=σA(T⁴b−T⁴₀)
W′=σA(T⁴b−(T₀−ΔT₀)⁴)
Using binomial approximation we get,
W′=σA(T⁴b−(T⁴₀−4T³₀ΔT₀)) (other terms will be negligible)
Hence W′=W+4σT³₀ΔT₀ (Since A=1m²)
Correct
Option B) We know that λT=constant
Hence if the temperature of a body is increased the wavelength at the peak point will shift to a lower wavelength.
Wrong
Option C) W=σA(T⁴b−T⁴₀)
Since A is reduced W also has to be reduced.
Correct
Option D) W=σA(T⁴b−T⁴₀)
Since σT⁴₀=460 W/m² σ=460/300⁴
Hence Putting Tb=310K and T₀=300K we get W=64.46J/s which is close to 60J/s
Correct