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 $σ T_{0}^{4} = 460 W m^{- 2}$ (where $σ$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

If the surrounding temperature reduces by a small amount

Δ T_{0} << T_{0}

, then to maintain the same body temperature the same (living) human being needs to radiate

Δ W = 4 σ T_{0}^{3} Δ T_{0}

more energy per unit time

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

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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

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

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Solution

The correct options are
A If the surrounding temperature reduces by a small amount $Δ T_{0} << T_{0}$ , then to maintain the same body temperature the same (living) human being needs to radiate $Δ W = 4 σ T_{0}^{3} Δ T_{0}$ more energy per unit time
C 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
D The amount of energy radiated by the body in 1 second is close to 60 Joules
Given $A = 1 m^{2}$
$T_{b} = T_{o} + 10$ ( $T_{b}$ = temperature of body)
Also $σ T_{o}^{4}$ = $460 W / m^{2}$

$O p t i o n A)$
$W = σ A (T_{b}^{4} - T_{o}^{4})$
$W^{'} = σ A (T_{b}^{4} - (T_{o} - Δ T_{o})^{4})$
Using binomial approximation we get,
$W^{'} = σ A (T_{b}^{4} - (T_{o}^{4} - 4 T_{o}^{3} Δ T_{o}))$ (other terms will be negligible)
Hence $W^{'} = W + 4 σ T_{o}^{3} Δ T_{o}$ (Since $A = 1 m^{2}$ )
$C o r r e c t$

$O p t i o n B)$
We know that $λ T = c o n s t a n t$
Hence if the temperature of a body is increased the wavelength at the peak point will shift to a lower wavelength.
$W r o n g$

$O p t i o n C)$
$W = σ A (T_{b}^{4} - T_{o}^{4})$
Since $A$ is reduced $W$ also has to be reduced.
$C o r r e c t$

$O p t i o n D)$
$W = σ A (T_{b}^{4} - T_{o}^{4})$
Since $σ T_{o}^{4} = 460 W / m^{2}$
$σ = \frac{460}{300^{4}}$
Hence Putting $T_{b} = 310 K$ and $T_{o} = 300 K$
we get $W = 64.46 J / s$ which is close to $60 J / s$
$C o r r e c t$

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Similar questions

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 $σ T_{0}^{4} = 460 W m^{- 2}$ (where $σ$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

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