Question

Stefan’s constant has the unit as:
\[{\text{A}}{\text{. J}}{{\text{S}}^{ - 1}}{m^{ - 2}}{k^4}\]
\[{\text{B}}{\text{. Kg}}{{\text{s}}^{ - 3}}{k^4}\]
\[{\text{C}}{\text{. W}}{m^{ - 2}}{k^{ - 4}}\]
\[{\text{D}}{\text{. N}}{\text{.}}m.{s^{ - 2}}{k^{ - 4}}\]

Answer 1

- Hint – For a perfect black body, the energy radiated per unit area per unit time is given by, $E = \sigma {T^4}$ , where $\sigma $ is Stefan’s constant.
Formula used - $E = \sigma {T^4}$ , $\dfrac{P}{A} = \sigma {T^4}$

Complete step-by-step solution -

We have to tell the unit of Stefan’s constant.
So, as we know for a perfect black body, the energy radiated per unit area per unit time is given by, $E = \sigma {T^4}$.
Now, here in the above formula, $\sigma $ is the Stefan’s constant and T is the temperature in Kelvin scale.
Now, as we know that energy per unit time is power (P).
So, power radiated per unit area is given by, $\dfrac{P}{A} = \sigma {T^4}$ .
Or, we can also write as, $\sigma = \dfrac{P}{{A{T^4}}}$
Now substituting the unit of P, A and T as $W,{m^2},{k^4}$ respectively.
So, we will get the unit of Stefan’s constant as-
$\sigma = \dfrac{W}{{{m^2}{k^4}}} = W{m^{ - 2}}{k^{ - 4}}$
Therefore, the unit of Stefan’s constant is option \[{\text{C}}{\text{. W}}{m^{ - 2}}{k^{ - 4}}\] .

Note- Whenever it is asked to find the unit of any constant then write the formula associated with that constant, as mentioned in the solution, which is $E = \sigma {T^4}$ . Then, as we know, energy per unit time is power, so we can write $\dfrac{P}{A} = \sigma {T^4}$ or, $\sigma = \dfrac{P}{{A{T^4}}}$. Now, we know the unit of power (P) is W, the unit of area (A) is $m_2$ and temperature has unit K. Putting these to find the unit of Stefan’s constant.