1. We need the dimension of

$\frac{\ell}{rCV}$

where:

• $\ell$ = inductance
• $r$ = resistance
• $C$ = capacitance
• $V$ = voltage

2. Write SI dimensions of each quantity.

(i) Inductance $\ell$

Using

$V = L\,\frac{dI}{dt}$

so,

$[L] = \frac{[V][T]}{[I]}$

Now,

$[V] = [M L^2 T^{-3} A^{-1}]$

Hence,

$[\ell] = [M L^2 T^{-3} A^{-1}]\,[T]\,[A^{-1}] = [M L^2 T^{-2} A^{-2}]$

(ii) Resistance $r$

Using Ohm's law $V = IR$,

$[r] = \frac{[V]}{[A]} = [M L^2 T^{-3} A^{-2}]$

(iii) Capacitance $C$

Using $C = Q/V$ and $Q = AT$,

$[C] = \frac{[A T]}{[M L^2 T^{-3} A^{-1}]} = [M^{-1} L^{-2} T^4 A^2]$

(iv) Voltage $V$

$[V] = [M L^2 T^{-3} A^{-1}]$

3. Now find dimension of the denominator $rCV$:

$[rCV] = [M L^2 T^{-3} A^{-2}]\,[M^{-1} L^{-2} T^4 A^2]\,[M L^2 T^{-3} A^{-1}]$

Multiply step-by-step:

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

So,

$[rCV] = [M L^2 T^{-2} A^{-1}]$

4. Therefore,

$\left[\frac{\ell}{rCV}\right] = \frac{[M L^2 T^{-2} A^{-2}]}{[M L^2 T^{-2} A^{-1}]} = [A^{-1}]$

5. Match with options:

• A: $[A^{-1}]$ ✅
• B: $[LTA]$ ❌
• C: $[LA^{-2}]$ ❌
• D: $[LT^2]$ ❌

Therefore, the correct option is A.