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 option is 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

Given $A = 1 m^{2}$

$T_{b} = T_{o} + 10$ ( $T_{b}$ = temperature of body)

Also $σ T_{o}^{4}$ = $460 W / m^{2}$
$\begin{matrix} Option A) Radiation when room temperature is T_{o}, W = σ A (T_{b}^{4} - T_{o}^{4}) Radiation when room temperature is increased by Δ T_{o}, W^{'} = σ A (T_{b}^{4} - {(T_{o} - Δ T_{o})}_{o}^{4}) Using binomial approximation we get, W^{'} = σ A (T_{b}^{4} - (T_{o}^{4} - 4 T_{o}^{3} Δ T_{o})) (other terms will be negligible) Hence W^{'} = W + 4 σ T_{o}^{3} Δ T_{o} (since A = 1 m^{2}) Correct \end{matrix}$
Option 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.

Wrong

Option C

$W = σ A (T_{b}^{4} - T_{o}^{4})$

Since $A$ is reduced $W$ also has to be reduced.

Correct

Option D

$W = σ A T_{b}^{4}$
Since $σ T_{o}^{4} = 460 W / m^{2} a n d T_{b} = 310 K$
$σ T_{b}^{4} > 460 W / m^{2}$
Hence option D is incorrect.

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Similar questions

Q. 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

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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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