Question

A very small electric furnace is used for heating metals. The hole nearly acts as a black body. The area of the hole is \[200m{m^2}\] To keep a metal at ${727^ \circ }C$ , heat energy flowing through this hole per sec, in joules is:
[ $\sigma = 5.67 \times {10^{ - 8}}W{m^{ - 2}}{k^{ - 4}}$ ]
(A) $22.68$
(B) $2.268$
(C) $1.134$
(D) $11.34$

Answer 1

Hint: To solve this type of problem we must know about emission phenomena. We should know about the Stefan-Boltzmann law, All bodies radiate energy $W$ depending on temperature, $T$ .
Stefan-Boltzmann law states that the total radiant heat power emitted from the surface of heated matter is proportional to the fourth power of its absolute temperature. This law applies only on black bodies.
Blackbody: theoretical surface that absorbs all incident heat radiation.
Formula used:
 $E = \sigma A{T^4}$
Here,
 $E = $ The energy radiant heat energy emitted in one second (That is, the power from the unit area).
The Greek letter sigma $\sigma $ represents a constant of proportionality, called the Stefan-Boltzmann constant. This constant has the value, $\sigma = 5.67 \times {10^{ - 8}}W{m^{ - 2}}{k^{ - 4}}$ .
 $T$, is the absolute temperature (in kelvins).

Complete step by step solution:
 As maintained above the hole acts as a blackbody.
Hence we can apply Stefan-Boltzmann law.
Given,
 $\sigma = 5.67 \times {10^{ - 8}}W{m^{ - 2}}{k^{ - 4}}$
 $T = {727^ \circ }C = 727 + 273 = 1000K$
 $A = 200m{m^2} = 2 \times {10^{ - 4}}{m^2}$
Hence, keeping value above mentioned equation,
 $E = 5.67 \times {10^{ - 8}} \times 2 \times {10^{ - 4}} \times {1000^4}$
  $ \Rightarrow E = 11.34J.{s^{ - 1}}$
Hence, we find that heat energy flowing through this hole per sec, in joules is, $E = 11.34J.{s^{ - 1}}$ .

So, (d) option is right.

Additional information:
The Greek letter sigma $\sigma $ represents a constant of proportionality, called the Stefan-Boltzmann constant.
It calculated by experimental observation and it calculated by following formula: $\dfrac{{2{\pi ^5}{k_B}^4}}{{15{h^3}{c^2}}}$
Here,
 \[{k_B} = \] Boltzmann constant.
 $h = $ Plank’s constant.
 $c = $ Speed of light in vacuum.

Note:
Always notice in question what is asked as in this question we have to find heat energy flowing through this hole per sec, in joule. So, use $E = \sigma A{T^4}$ But many times heat energy flowing through this hole per unit area per sec, in joules can be asked .So, must use, $E = \sigma {T^4}$ .