Question

If \(\sigma\) denotes Stefan constant and S denotes heat capacity, then the dimensional formula of \(\frac{\text{S}}{\sigma}\) is

• A. [M\textsuperscript{0}L\textsuperscript{2}T\textsuperscript{-1}K\textsuperscript{3}]
• B. [M\textsuperscript{0}L\textsuperscript{2}TK\textsuperscript{3}]
• C. [ML\textsuperscript{2}T\textsuperscript{-1}K\textsuperscript{-4}]
• D. [M\textsuperscript{0}L\textsuperscript{-1}T\textsuperscript{-3}K\textsuperscript{-3}]

Hint:

Remember the fundamental dimensional formulas: \textbf{Energy (Q):} [ML\textsuperscript{2}T\textsuperscript{-2}] \textbf{Power (P):} [ML\textsuperscript{2}T\textsuperscript{-3}] \textbf{Temperature (T):} [K] \textbf{Area (A):} [L\textsuperscript{2}] \textbf{Time (t):} [T] \textbf{Stefan-Boltzmann Constant (\(\sigma\)):} Derived from P = \(\sigma\)AT\textsuperscript{4} as [ML\textsuperscript{0}T\textsuperscript{-3}K\textsuperscript{-4}]. \textbf{Heat Capacity (S):} Derived from Q = S\(\Delta\)T as [ML\textsuperscript{2}T\textsuperscript{-2}K\textsuperscript{-1}].

Answer 1

The Correct Option is B

Step 1: Determine the dimensional formula of Stefan-Boltzmann constant (\(\sigma\)).
Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of the black body's absolute temperature (T). So, \( \frac{\text{P}}{\text{A}} = \sigma\text{T\textsuperscript{4}} \) Where:
P = Power (Energy/Time) = [ML\textsuperscript{2}T\textsuperscript{-3}] A = Area = [L\textsuperscript{2}]
T = Temperature = [K]
\(\sigma\) = Stefan-Boltzmann constant
Therefore, \( \sigma = \frac{\text{P}}{\text{AT\textsuperscript{4}}} = \frac{\text{[ML\textsuperscript{2}T\textsuperscript{-3}]}}{\text{[L\textsuperscript{2}][K\textsuperscript{4}]}} = \text{[ML\textsuperscript{0}T\textsuperscript{-3}K\textsuperscript{-4}]} \) Step 2: Determine the dimensional formula of Heat Capacity (S).
Heat capacity (S) is the amount of heat energy required to raise the temperature of a substance by one degree Celsius (or Kelvin).
\( \text{S} = \frac{\text{Q}}{\Delta\text{T}} \) Where:
Q = Heat Energy = [ML\textsuperscript{2}T\textsuperscript{-2}] \(\Delta\)T = Change in Temperature = [K] Therefore, \( \text{S} = \frac{\text{[ML\textsuperscript{2}T\textsuperscript{-2}]}}{\text{[K]}} = \text{[ML\textsuperscript{2}T\textsuperscript{-2}K\textsuperscript{-1}]} \) Step 3: Calculate the dimensional formula of \(\frac{\text{S}}{\sigma}\). \[ \frac{\text{S}}{\sigma} = \frac{\text{[ML\textsuperscript{2}T\textsuperscript{-2}K\textsuperscript{-1}]}}{\text{[MT\textsuperscript{-3}K\textsuperscript{-4}]}} \] Now, simplify the powers of M, L, T, and K:
For M: M\textsuperscript{1-1} = M\textsuperscript{0}
For L: L\textsuperscript{2-0} = L\textsuperscript{2}
For T: T\textsuperscript{-2 - (-3)} = T\textsuperscript{-2 + 3} = T\textsuperscript{1}
For K: K\textsuperscript{-1 - (-4)} = K\textsuperscript{-1 + 4} = K\textsuperscript{3}
So, the dimensional formula of \(\frac{\text{S}}{\sigma}\) is [M\textsuperscript{0}L\textsuperscript{2}TK\textsuperscript{3}]. Step 4: Select the correct option.
The calculated dimensional formula [M\textsuperscript{0}L\textsuperscript{2}TK\textsuperscript{3}] matches option (2).