JEE PYQ: Units & Measurements - Question ID 983f67639b7b (JEE Main 2025)

Core Formula & Concept:

In electromagnetism, two fundamental constants define the properties of free space:

• Vacuum permeability, $\mu_0$, measures the resistance of free space to the formation of magnetic fields. Its SI unit is henry per meter ($\text{H/m}$), and its dimensional formula is $[M L T^{-2} I^{-2}]$.
• Vacuum permittivity, $\epsilon_0$, measures the resistance of free space to the formation of electric fields. Its SI unit is farad per meter ($\text{F/m}$), and its dimensional formula is $[M^{-1} L^{-3} T^{4} I^{2}]$.

The expression $\sqrt{\frac{\mu_0}{\epsilon_0}}$ appears in the context of electromagnetic wave propagation in vacuum, where it represents the intrinsic impedance of free space. The question asks for the dimension of this expression, which we will derive step-by-step.

Step-by-Step Derivation:

We begin by writing the dimensional formulas for $\mu_0$ and $\epsilon_0$:

$[\mu_0] = M L T^{-2} I^{-2}$ $[\epsilon_0] = M^{-1} L^{-3} T^{4} I^{2}$

Now, compute the dimension of the ratio $\frac{\mu_0}{\epsilon_0}$:

$\left[\frac{\mu_0}{\epsilon_0}\right] = \frac{[\mu_0]}{[\epsilon_0]} = \frac{M L T^{-2} I^{-2}}{M^{-1} L^{-3} T^{4} I^{2}} = M^{2} L^{4} T^{-6} I^{-4}$

Next, take the square root of this ratio:

$\left[\sqrt{\frac{\mu_0}{\epsilon_0}}\right] = \left(M^{2} L^{4} T^{-6} I^{-4}\right)^{1/2} = M^{1} L^{2} T^{-3} I^{-2}$

Now, let’s compare this with the dimensional formulas of the given options:

• Voltage (Option A): $[V] = M L^{2} T^{-3} I^{-1}$
• Inductance (Option B): $[L] = M L^{2} T^{-2} I^{-2}$
• Resistance (Option C): $[R] = M L^{2} T^{-3} I^{-2}$
• Capacitance (Option D): $[C] = M^{-1} L^{-2} T^{4} I^{2}$

Comparing $M^{1} L^{2} T^{-3} I^{-2}$ with the above, we see an exact match with the dimension of Resistance.

Common Traps & Exam Tip:

Students often confuse the dimensions of $\mu_0$ and $\epsilon_0$, especially their exponents on mass, length, and time. A frequent mistake is misapplying the exponents during division or square root operations, leading to incorrect dimensional results.

Another common error is confusing the dimension of $\sqrt{\mu_0 \epsilon_0}$ (which has the dimension of inverse velocity) with $\sqrt{\frac{\mu_0}{\epsilon_0}}$. The former relates to wave speed, while the latter relates to impedance.

Exam Tip: Always write down the full dimensional formula before performing algebraic operations. Double-check each exponent during multiplication, division, and exponentiation. Remember that $\sqrt{\frac{\mu_0}{\epsilon_0}}$ has the dimension of resistance, which is a key result in electromagnetic theory.

Thus, the correct answer is Option C: Resistance.