Multiple Choice

A human body has a surface area of approximately $$1 m^2$$. The normal body temperature is $$10 K$$ above the surrounding room temperature $$T_0$$. Take the room temperature to he $$T_0 = 300 K$$. For $$T_0 = 300 K$$ the value of $$\sigma T_0^4 = 460 Wm^{-2}$$ (where $$\sigma$$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

AIf the surrounding temperature reduces by a small amount $$\Delta T_0 << T_0$$, then to maintain the same body temperature the same (living) human being needs to radiate $$\Delta W = 4\sigma T_0^3\Delta T_0$$ more energy per unit time
Correct Answer
BIf the body temperature rise significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths
CReducing 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
Correct Answer
DThe amount of energy radiated by the body in 1 second is close to 60 Joules
Correct Answer

Solution

Given $$A = 1 m^2$$
$$T_b = T_o+10$$ ($$T_b$$ = temperature of body)
Also $$\sigma\ T_o^4$$= $$ 460\ W/m^2$$

$$Option\ A)$$
$$ W= \sigma\ A\ (T_b^4-T_o^4)$$
$$W'= \sigma\ A\ (T_b^4-(T_o-\Delta T_o)^4)$$
Using binomial approximation we get,
$$W'= \sigma\ A\ (T_b^4-(T_o^4-4T_o^3\Delta T_o))$$ (other terms will be negligible)
Hence $$ W' = W+4\sigma T_o^3 \Delta T_o$$ (Since $$A= 1 m^2$$)
$$Correct$$

$$Option\ B)$$
We know that $$\lambda T=constant$$
Hence if the temperature of a body is increased the wavelength at the peak point will shift to a lower wavelength.
$$Wrong$$

$$Option\ C)$$
$$ W= \sigma\ A\ (T_b^4-T_o^4)$$
Since $$A$$ is reduced $$W$$ also has to be reduced.
$$Correct$$

$$Option\ D)$$
$$ W= \sigma\ A\ (T_b^4-T_o^4)$$
Since $$\sigma\ T_o^4= 460\ W/m^2$$
$$\sigma = \dfrac{460}{300^4}$$
Hence Putting $$T_b= 310 K$$ and $$T_o= 300 K$$
we get $$W= 64.46 J/s$$ which is close to $$60 J/s$$
$$Correct$$

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