JEE PYQ: Units & Measurements - Question ID ac264705f760 (JEE Main 2020)

Core Formula & Concept:

The solar constant is defined as the amount of solar energy received per unit area per unit time on the Earth’s surface. In dimensional analysis, we express every physical quantity in terms of the fundamental dimensions:

• Mass → $M$
• Length → $L$
• Time → $T$

Energy itself has the dimension $ML^2T^{-2}$ (work = force × distance, and force = mass × acceleration). Area has the dimension $L^2$, and time has the dimension $T$.

Step-by-Step Derivation:

1. Write the definition of the solar constant $S$: $S = \frac{\text{Energy received}}{\text{Area} \times \text{Time}}$ 2. Substitute the dimensions of each term: $\text{Energy} \rightarrow ML^2T^{-2}$ $\text{Area} \rightarrow L^2$ $\text{Time} \rightarrow T$ 3. Plug these into the expression for $S$: $S = \frac{ML^2T^{-2}}{L^2 \cdot T}$ 4. Simplify the fraction: $S = \frac{ML^2T^{-2}}{L^2T} = M \cdot L^{2-2} \cdot T^{-2-1} = M \cdot L^0 \cdot T^{-3}$ 5. Therefore, the dimension of the solar constant is: $[S] = ML^0T^{-3}$

Common Traps & Exam Tip:

Students often confuse the solar constant with energy itself, leading them to choose option D ($ML^2T^{-2}$). Another frequent mistake is forgetting to subtract exponents when dividing dimensions, resulting in incorrect options like $MLT^{-2}$ (option A) or $M^2L^0T^{-1}$ (option C). Always double-check that the denominator’s dimensions are correctly subtracted from the numerator’s dimensions.