Q. 7.52 - An Introduction to Thermal Physics

Question

(a) Estimate (roughly) the total power radiated by your body, neglecting any energy that is returned by your clothes and environment. (Whatever the color of your skin, its emissivity at infrared wavelengths is quite close to $1$ ; almost any nonmetal is a near-perfect blackbody at these wavelengths.)

(b) Compare the total energy radiated by your body in one day (expressed in kilocalories) to the energy in the food you cat. Why is there such a large discrepancy?

(c) The sun has a mass of $2\times {10}^{30} \mathrm{kg}$ and radiates energy at a rate of  $3.9\times {10}^{26}\text{ watts }$. Which puts out more power per units mass-the sun or your body?

Step-by-Step Solution

Answer

(a) . The total power radiated by our body $1 kW$

(b) . $\text{ The energy we would lose in one day is 20,000 cal}$.

(c) . $\text{ The radiation rate per kilogram is 0.0002 W/kg}$

Step 1. Given information

• The total power radiated by our body we use  is given by $\text{ Power }=\sigma eA{T}^4$

Step 2. Calculating the total power radiated by our body we use .

The power radiated is 

$\text{ Power }=\sigma eA{T}^4$

$T=310 K$ , $A = 2m^2$ , $e =1$.

$\text{ Power }=\left(5.67\times {10}^{-8} \mathrm{W}/{\mathrm{m}}^2·{\mathrm{K}}^4\right)\left(2{ \mathrm{m}}^2\right)(310 \mathrm{K}{)}^4$

$=1050 \mathrm{W}$

$\approx 1000 \mathrm{W}$

$\approx 1.0\times {10}^3 \mathrm{W}$

$\approx 1 \mathrm{kW}$

Hence, This is the rate at which we would lose energy, if we were naked in empty space .

Step 3. Calculating the energy we would lose in one day .

The rate, the energy we would lose in one day will be 

$(1050 \mathrm{W})\left(24\mathrm{hr}\right)\left(\frac{3600 \mathrm{s}}{1\mathrm{hr}}\right)=\left(9.0\times {10}^7 \mathrm{J}\right)\left(\frac{1\mathrm{kcal}}{4186 \mathrm{J}}\right)$

$=20,000\mathrm{kcal}$

As we can see that it is  ten times the numbers of calories that an average person conserves in a day about $\left(2000\right)$. The discrepancy is due to the fact that we are not naked in empty space, most of the energy that we radiate is replaced by energy radiated or conducted back to us by our clothes and other surroundings

Step 4. Calculating the radiation rate per kilogram.

Now, the radiation rate per kilogram is 

$\frac{1000 \mathrm{W}}{75 \mathrm{kg}}=14 \mathrm{W}/\mathrm{kg}\text{ in case of seen it is }$

$\frac{3.9\times {10}^{26} \mathrm{W}}{2\times {10}^{30} \mathrm{kg}}=0.0002 \mathrm{W}/\mathrm{kg}$

As it is  70,000 times less, have we get a reason that, how is it possible of although the sun is bright it is also very massive. Although it generates energy by nuclear fusion the reaction in its core actually proceeds extremely slowly giving it a ten billion year life time. So, I on the other hand have to replenish my chemical fuel supply on daily basis.