JEE PYQ: Units & Measurements - Question ID bbe2455d276c (JEE Main 2020)

Core Formula & Concept:

Thermal conductivity ($k$) is defined through Fourier’s law of heat conduction. The law states that the rate of heat flow ($H$) through a material is proportional to the temperature gradient ($\frac{dT}{dx}$) and the cross-sectional area ($A$) perpendicular to the flow:

$H = -k A \frac{dT}{dx}$

Here:

• $H$ = heat flow rate (energy per unit time), so its dimensions are $\frac{[Energy]}{[Time]} = \frac{ML^2T^{-2}}{T} = ML^2T^{-3}$.
• $A$ = area, dimensions $L^2$.
• $\frac{dT}{dx}$ = temperature gradient, dimensions $\frac{[Temperature]}{[Length]} = \frac{K}{L}$.

Our goal is to isolate $k$ and find its dimensional formula.

Step-by-Step Derivation:

1. Start from Fourier’s law: $H = -k A \frac{dT}{dx}$ 2. Rearrange to solve for $k$: $k = \frac{H}{A \frac{dT}{dx}}$ 3. Substitute the dimensions of each quantity:

• $H \rightarrow ML^2T^{-3}$
• $A \rightarrow L^2$
• $\frac{dT}{dx} \rightarrow \frac{K}{L}$

4. Plug these into the expression for $k$: $[k] = \frac{ML^2T^{-3}}{L^2 \cdot \frac{K}{L}} = \frac{ML^2T^{-3}}{L^2 \cdot K \cdot L^{-1}} = \frac{ML^2T^{-3}}{L^{2-1}K} = \frac{ML^2T^{-3}}{LK}$ 5. Simplify the powers of $L$: $[k] = ML^{2-1}T^{-3}K^{-1} = MLT^{-3}K^{-1}$ 6. Compare with the given options. The dimensional formula $MLT^{-3}K^{-1}$ matches option A.

Common Traps & Exam Tip:

• Confusing temperature gradient with temperature: Students often forget that $\frac{dT}{dx}$ has dimensions $\frac{K}{L}$, not just $K$. This leads to incorrect cancellation of $K$ and wrong options like $MLT^{-3}K$.
• Sign error in Fourier’s law: The negative sign in $H = -k A \frac{dT}{dx}$ is purely conventional and does not affect dimensions. Ignore it during dimensional analysis.
• Miscounting powers of $L$: A frequent arithmetic mistake is miscounting $L$ exponents when dividing $L^2$ by $L^{-1}$. Double-check each step to avoid this.
• Exam Tip: Always write down the dimensions of every physical quantity involved before substituting. This systematic approach minimizes errors and ensures clarity.