Question

The dimensional of Stefan’s constant are
(A) \[\left[ {{M^0}{L^1}{T^{ - 3}}{K^{ - 4}}} \right]\]
(B) \[\left[ {{M^1}{L^1}{T^{ - 3}}{K^{ - 3}}} \right]\]
(C) \[\left[ {{M^1}{L^2}{T^{ - 3}}{K^{ - 4}}} \right]\]
(D) \[\left[ {{M^1}{L^0}{T^{ - 3}}{K^{ - 4}}} \right]\]

Answer 1

Hint: A physical quantity which is the total intensity radiated over all wavelengths as the temperature increases is known as the Stefan-Boltzmann constant. It is denoted by (sigma). According to Stefan law, the radiated power density from a black body is proportional to its absolute temperature, T raised to the fourth power. A black body is an object that absorbs all the radiant energy reaching on its surface from any direction and at any angle. In this question, write the formula for the Stefan-Boltzmann constant and write the dimension of each term from the formula to find the dimension of Stefan’s constant.

Complete step by step answer:
Power radiated from a body is given as
\[P = \sigma Ae{T^4}\]
Where
\[\sigma \] is Stefan’s constant,
\[e\] is the emissivity
\[A\] is the surface area, and
\[T\] is the temperature
Since Stefan’s constant (\[\sigma \]) is a constant quantity hence it does not have a dimension, whereas
The dimension of Power (P) \[ = \left[ {M{L^2}{T^{ - 3}}} \right]\]
The dimension of Area (A) \[ = \left[ {{L^2}} \right]\]
The dimension of Temperature (T) \[ = \left[ K \right]\]
Whereas emissivity\[e\]is a constant quantity, hence the dimensional of Stefan’s constant will be
 \[
  \sigma = \dfrac{P}{{Ae{T^4}}} = \dfrac{{\left[ {M{L^2}{T^{ - 3}}} \right]}}{{\left[ {{L^2}} \right]\left[ {{K^4}} \right]}} \\
   = \left[ {{M^1}{L^0}{T^{ - 3}}{K^{ - 4}}} \right] \\
 \]
Hence option (D) is correct.

Note: It is to be noted down here that while writing the dimension formula, only SI units of the measuring quantities (not CGS, or any other units) should be used and should be bifurcated further.