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 be `T_0 = 300 K`. For `T_0 = 300 K`, the value of `sigmaT_0^4=460 Wm^(-2)` (where `sigma` is the Stefan-Boltzmann constant). Which of the following options is/are correct?

Options

A. The amount of energy radiated by the body in/second is close to 60 joules.
B. If the surrounding temperature reduces by a small amount `Delta T_(0) lt lt 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 times.
C. Reducing the exposed surface area of the body (e.g., by curbing up) allows humans to maintain the same body temperature while reducing the energy lost by radiation.
D. If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths.

Answer

A, B, C

Explanation

Energy emitted per second from the black body of surface area A at temperature TK, with surrounding temperature `T_(0)` is
(a) `E=sigma A(T^(4)-T_(0)^(4)) =sigma (T^(4)-T_(0)^(4))` `(.: A= 1m^(2))`
`=sigma[(T_(0) + 10)^(4) - T_(0)^(4))] =sigma T_(0)^(4)[(1+(10)/(T_(0)))^(4) - 1]`
`=sigma T_(0)^(4)[1 +(4xx10)/(T_(0)) + ... -1] ~~ sigma T_(0)^(4) xx(40)/(T_(0))`
`460xx(40)/(300)=61.33 ~~ 60` joule
Energy radiated in 1 second `~~ 60` joule [Thus option (a) is true ].
(b) When surrounding temperature reduces by a small amount `Delta T_(0)` , keeping the body temperature T constant, the amount of heat energy radiated per second per unit are is `E =sigma[T^(4) - T_(0)^(4)]`
`dE =sigma[0 - 4T_(0)^(3) Delta T_(0)]` `[But Delta T_(0) =-Delta T_(0)]`
So `dE =sigma 4T_(0)^(3) Delta T` [Thus option (b) is true]
(c) Since energy radiated per second from a body is directly proportional to the surface area of the body. By curbing, the surface area decreases, so energy radiation decreases. Thus option (c) is true.
(d) When the body temperature rises, the peak in the spectorum of electromagnetic radiation emitted by the body shifts towards lower, wavelength side (ac cording to Wien's law). Thus option (d) is wrong.