Question

Match List I with List II. 

Choose the correct answer from the options given below: 
 

• (a) $\rightarrow$ (iii), (b) $\rightarrow$ (iv), (c) $\rightarrow$ (ii), (d) $\rightarrow$ (i)
• (a) $\rightarrow$ (iv), (b) $\rightarrow$ (ii), (c) $\rightarrow$ (iii), (d) $\rightarrow$ (i)
• (a) $\rightarrow$ (iii), (b) $\rightarrow$ (ii), (c) $\rightarrow$ (iv), (d) $\rightarrow$ (i)
• (a) $\rightarrow$ (iv), (b) $\rightarrow$ (iii), (c) $\rightarrow$ (ii), (d) $\rightarrow$ (i)

Hint:

To solve dimensional analysis matching questions quickly, start with the quantities whose dimensions you remember or can derive easily (like Electric Field from $F=qE$). This can often eliminate some options immediately.

Answer 1

The Correct Option is C

Let's determine the dimensional formula for each physical quantity. We use M for mass, L for length, T for time, and A for electric current.
(a) Capacitance, C: The relation is $Q = CV$, so $C = Q/V$.
Dimension of charge $[Q] = [AT]$.
Dimension of voltage (Potential) $[V] = [\text{Work/Charge}] = [ML^2T^{-2}]/[AT] = [ML^2T^{-3}A^{-1}]$.
So, $[C] = \frac{[AT]}{[ML^2T^{-3}A^{-1}]} = [M^{-1}L^{-2}T^4A^2]$. This matches (iii).
(b) Permittivity of free space, $\epsilon_0$: From Coulomb's Law, $F = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}$.
So, $[\epsilon_0] = \frac{[q]^2}{[F][r]^2} = \frac{[AT]^2}{[MLT^{-2}][L]^2} = \frac{[A^2T^2]}{[ML^3T^{-2}]} = [M^{-1}L^{-3}T^4A^2]$. This matches (ii).
(c) Permeability of free space, $\mu_0$: From the formula for the speed of light, $c = \frac{1}{\sqrt{\epsilon_0\mu_0}}$.
So, $\mu_0 = \frac{1}{c^2 \epsilon_0}$.
$[\mu_0] = \frac{1}{[LT^{-1}]^2 [M^{-1}L^{-3}T^4A^2]} = \frac{1}{[L^2T^{-2}][M^{-1}L^{-3}T^4A^2]} = \frac{1}{[M^{-1}L^{-1}T^2A^2]} = [M^1L^1T^{-2}A^{-2}]$. This matches (iv).
(d) Electric field, E: From the relation $F = qE$, so $E = F/q$.
$[E] = \frac{[F]}{[q]} = \frac{[MLT^{-2}]}{[AT]} = [MLT^{-3}A^{-1}]$. This matches (i).
Combining the results: (a) $\rightarrow$ (iii), (b) $\rightarrow$ (ii), (c) $\rightarrow$ (iv), (d) $\rightarrow$ (i).
This corresponds to option (C).