JEE PYQ: Units & Measurements - Question ID 08fd489cf89e (JEE Main 2018)

Core Formula & Concept:

The Planck length, denoted \( \ell_P \), is the natural scale at which quantum gravitational effects become significant. It is constructed by combining three fundamental constants:

• Newton’s gravitational constant \( G \) (units: \( \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \))
• Planck’s constant \( \hbar \) (units: \( \text{J s} = \text{kg m}^2 \text{s}^{-1} \))
• The speed of light \( c \) (units: \( \text{m s}^{-1} \))

The guiding principle is dimensional homogeneity: the combination must yield a length (meters). We therefore seek exponents \( \alpha \), \( \beta \), and \( \gamma \) such that \[ \ell_P = G^\alpha \,\hbar^\beta \,c^\gamma \] has the dimension of length.

Step-by-Step Derivation:

1. Write the dimensional equation

\[ [G^\alpha \,\hbar^\beta \,c^\gamma] = \text{m} \] Substitute the dimensions of each constant: \[ \big(\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}\big)^\alpha \cdot \big(\text{kg}\,\text{m}^2\,\text{s}^{-1}\big)^\beta \cdot \big(\text{m}\,\text{s}^{-1}\big)^\gamma = \text{m}^1\,\text{kg}^0\,\text{s}^0. \]

2. Equate exponents for each base unit

• For meters (m): \[ 3\alpha + 2\beta + \gamma = 1. \]
• For kilograms (kg): \[ -\alpha + \beta = 0 \quad\Longrightarrow\quad \beta = \alpha. \]
• For seconds (s): \[ -2\alpha - \beta - \gamma = 0. \]

3. Solve the linear system

Substitute \( \beta = \alpha \) into the meter and second equations: \[ 3\alpha + 2\alpha + \gamma = 1 \quad\Longrightarrow\quad 5\alpha + \gamma = 1, \] \[ -2\alpha - \alpha - \gamma = 0 \quad\Longrightarrow\quad -3\alpha - \gamma = 0. \] Add the two equations: \[ (5\alpha + \gamma) + (-3\alpha - \gamma) = 1 + 0 \quad\Longrightarrow\quad 2\alpha = 1 \quad\Longrightarrow\quad \alpha = \tfrac12. \] Hence \( \beta = \tfrac12 \) and \( \gamma = -3\alpha = -\tfrac32 \).

4. Construct the Planck length

\[ \ell_P = G^{\tfrac12}\,\hbar^{\tfrac12}\,c^{-\tfrac32} = \sqrt{\frac{G\,\hbar}{c^3}}. \] This matches option D: \[ \left(\frac{G\,\hbar}{c^3}\right)^{\tfrac12}. \] Common Traps & Exam Tip:

Students often confuse the exponents or misapply dimensional analysis. Common mistakes include:

• Forgetting that \( \hbar \) has units of action (energy×time), not just energy.
• Mixing up the signs of exponents when solving the linear system.
• Overlooking the square-root (power \( \tfrac12 \)) in the final expression.

Exam Tip: Always write the dimensional equation explicitly and solve the system of equations step by step. Double-check each exponent against the base units (m, kg, s) to avoid sign errors.