JEE PYQ: Units & Measurements - Question ID 0aa41a0f679b (JEE Main 2019)

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($A$), thermodynamic temperature ($Θ$), amount of substance ($N$), and luminous intensity ($J$). Here, we focus on the dimensions of inductance ($ℓ$), resistance ($r$), capacitance ($C$), and voltage ($V$).

Key formulas and their dimensional representations in SI units are:

• Inductance ($ℓ$): From the definition of induced emf, $ε = -ℓ \frac{dI}{dt}$, we derive the dimension of inductance as $[ℓ] = \frac{[V][T]}{[A]} = \frac{ML^2T^{-3}A^{-1} \cdot T}{A} = ML^2T^{-2}A^{-2}$.
• Resistance ($r$): From Ohm’s law, $V = Ir$, so $[r] = \frac{[V]}{[A]} = ML^2T^{-3}A^{-2}$.
• Capacitance ($C$): From $Q = CV$, where $Q$ is charge, $[C] = \frac{[Q]}{[V]} = \frac{AT}{ML^2T^{-3}A^{-1}} = M^{-1}L^{-2}T^4A^2$.
• Voltage ($V$): Dimension of voltage is $[V] = ML^2T^{-3}A^{-1}$.

We need to find the dimension of the expression $\frac{ℓ}{rCV}$.

Step 1: Write the dimensions of each quantity involved.

Step 2: Substitute these dimensions into the expression $\frac{ℓ}{rCV}$.

Step 3: Substitute the dimensional formulas.

Step 4: Simplify the denominator by multiplying the dimensions.

• Mass ($M$): $1 - 1 + 1 = 1$
• Length ($L$): $2 - 2 + 2 = 2$
• Time ($T$): $-3 + 4 - 3 = -2$
• Current ($A$): $-2 + 2 - 1 = -1$

Step 5: Now, the expression becomes:

Thus, the dimension of $\frac{ℓ}{rCV}$ is $[A^{-1}]$.

Students often make the following mistakes:

• Incorrect dimensional formulas: Confusing the dimensions of inductance, capacitance, or resistance. For example, treating inductance as $ML^2T^{-3}A^{-1}$ instead of $ML^2T^{-2}A^{-2}$.
• Sign errors in exponents: Miscounting the powers of $T$ or $A$ while multiplying dimensions, especially in the denominator.
• Overcomplicating the problem: Trying to relate the expression to time constants or other derived formulas instead of sticking to pure dimensional analysis.

Exam Tip: Always write down the dimensions of each quantity clearly before substituting. Double-check the arithmetic of exponents, especially when multiplying or dividing dimensions. This question tests dimensional consistency, so focus on the algebra of dimensions rather than physical interpretation.