JEE PYQ: Units & Measurements - Question ID 76214504b6ff (JEE Main 2019)

To derive an expression for time ($t$) in terms of the fundamental constants $G$ (universal gravitational constant), $h$ (Planck constant), and $c$ (speed of light), we use dimensional analysis. This technique relies on the principle that any physically meaningful equation must have consistent dimensions on both sides.

The dimensions of the given constants are:

• $G$: Gravitational constant, with dimensions $[M^{-1}L^3T^{-2}]$.
• $h$: Planck constant, with dimensions $[ML^2T^{-1}]$.
• $c$: Speed of light, with dimensions $[LT^{-1}]$.

Let the time $t$ be proportional to some power of $G$, $h$, and $c$: $t \propto G^x h^y c^z$ where $x$, $y$, and $z$ are unknown exponents to be determined.

Substitute the dimensions of each constant into the equation: $[T] = [M^{-1}L^3T^{-2}]^x \cdot [ML^2T^{-1}]^y \cdot [LT^{-1}]^z$

Expand the right-hand side: $[T] = M^{-x + y} \cdot L^{3x + 2y + z} \cdot T^{-2x - y - z}$

For the equation to hold, the exponents of $M$, $L$, and $T$ on both sides must match. This gives us the following system of equations:

1. For mass ($M$): $-x + y = 0$
2. For length ($L$): $3x + 2y + z = 0$
3. For time ($T$): $-2x - y - z = 1$

Solve the system step-by-step:

1. From the first equation ($-x + y = 0$), we get $y = x$.
2. Substitute $y = x$ into the third equation ($-2x - y - z = 1$): $-2x - x - z = 1 \implies -3x - z = 1 \implies z = -3x - 1$
3. Substitute $y = x$ and $z = -3x - 1$ into the second equation ($3x + 2y + z = 0$): $3x + 2x + (-3x - 1) = 0 \implies 2x - 1 = 0 \implies x = \frac{1}{2}$
4. Now, substitute $x = \frac{1}{2}$ back into $y = x$ and $z = -3x - 1$: $y = \frac{1}{2}, \quad z = -3 \cdot \frac{1}{2} - 1 = -\frac{5}{2}$

Thus, the exponents are $x = \frac{1}{2}$, $y = \frac{1}{2}$, and $z = -\frac{5}{2}$. Substitute these back into the proportionality: $t \propto G^{\frac{1}{2}} h^{\frac{1}{2}} c^{-\frac{5}{2}} = \sqrt{G h} \cdot c^{-\frac{5}{2}} = \sqrt{\frac{G h}{c^5}}$

This matches option C: $\sqrt{\frac{G h}{c^5}}$

Students often make the following mistakes:

1. Incorrect dimensional substitution: Forgetting the dimensions of $G$, $h$, or $c$ leads to wrong exponents. Always double-check the dimensions of each constant.
2. Sign errors in exponents: Misplacing negative signs (e.g., in $z = -3x - 1$) can lead to incorrect results. Carefully solve the system of equations.
3. Confusing proportionality with equality: The question asks for proportionality, not an exact equation. Do not introduce unnecessary constants like $2\pi$ or $\sqrt{2}$.
4. Miscounting exponents: Ensure the exponents of $M$, $L$, and $T$ are balanced correctly. A small arithmetic error can lead to the wrong option.

Exam Tip: When solving dimensional analysis problems, always:

• Write down the dimensions of all given quantities clearly.
• Set up the system of equations carefully.
• Solve step-by-step and verify each substitution.
• Cross-check the final expression with the given options.