JEE PYQ: Units & Measurements - Question ID c09d20ae4445 (JEE Main 2022)

Core Formula & Concept:

In dimensional analysis, every physical equation must be dimensionally consistent. This means:

• The argument of a logarithmic function (like $\log_e$) must be dimensionless.
• If $P$ is dimensionless, then the entire right-hand side of the equation must also be dimensionless.

Given the expression: $P = \frac{\alpha}{\beta} \log_e\left(\frac{kt}{\beta x}\right)$ we know:

• $k$ (Boltzmann constant) has dimensions $[M L^2 T^{-2} \Theta^{-1}]$, where $\Theta$ is temperature.
• $t$ is temperature, so $[t] = \Theta$.
• $x$ is distance, so $[x] = L$.
• $\beta$ is a constant whose dimensions we do not yet know.

Our goal is to find the dimensions of $\alpha$ such that $P$ remains dimensionless.

Step-by-Step Derivation:

Step 1: Ensure the argument of the logarithm is dimensionless.

The term inside the logarithm is $\frac{kt}{\beta x}$. For the logarithm to be defined, this must be dimensionless: $\left[\frac{kt}{\beta x}\right] = 1$ Substitute the dimensions of each quantity: $\frac{[k][t]}{[\beta][x]} = 1 \implies \frac{[M L^2 T^{-2} \Theta^{-1}] \cdot \Theta}{[\beta] \cdot L} = 1$ Simplify: $\frac{M L^2 T^{-2}}{[\beta] \cdot L} = 1 \implies \frac{M L T^{-2}}{[\beta]} = 1$ Thus, the dimensions of $\beta$ are: $[\beta] = M L T^{-2}$

Step 2: Ensure the entire expression for $P$ is dimensionless.

The expression for $P$ is: $P = \frac{\alpha}{\beta} \log_e\left(\frac{kt}{\beta x}\right)$ Since $\log_e\left(\frac{kt}{\beta x}\right)$ is dimensionless, the term $\frac{\alpha}{\beta}$ must also be dimensionless for $P$ to be dimensionless: $\left[\frac{\alpha}{\beta}\right] = 1 \implies [\alpha] = [\beta]$ From Step 1, we found $[\beta] = M L T^{-2}$. Therefore: $[\alpha] = M L T^{-2}$

Step 3: Match with the given options.

The dimensions of $\alpha$ are $[M L T^{-2}]$, which corresponds to option C.

Common Traps & Exam Tip:

1. Ignoring the dimensionless requirement of the logarithm: Many students forget that the argument of $\log_e$ must be dimensionless. This leads to incorrect dimensional analysis of $\beta$ and, consequently, $\alpha$.
2. Misidentifying the dimensions of the Boltzmann constant: The Boltzmann constant $k$ has dimensions $[M L^2 T^{-2} \Theta^{-1}]$, not just energy. Forgetting the temperature dimension ($\Theta$) can lead to errors.
3. Assuming $\alpha$ or $\beta$ are dimensionless: Some students incorrectly assume that constants like $\alpha$ or $\beta$ are dimensionless, which violates the dimensional consistency of the equation.
4. Overcomplicating the problem: This question is purely about dimensional analysis. Avoid introducing unnecessary physics (e.g., thermodynamics) unless explicitly required.

Exam Tip: Always start by ensuring the argument of logarithmic, exponential, or trigonometric functions is dimensionless. This is a fundamental rule in dimensional analysis and often the key to solving such problems.