$\begin{array}{|c|}\hline Match\; List\; -\; I\; with\; List\; -\; II.\\ \hline\end{array}$

$\begin{array}{|c|}\hline \\ \hline\end{array}$
LIST - I LIST - II (A) [ M 0 L −2 T 0 A 1 ] (III) Current density (B) [ M −1 L −2 T 3 A 2 ] (I) Conductance (C) [ M L 3 T −3 A −2 ] (IV) Specific resistance (D) [ M −1 L −3 T +3 A +2 ] (II) Electrical conductivity

$\begin{array}{|c|}\hline \text{<br>}\\ \hline\end{array}$

$\begin{array}{|c|}\hline \text{Choose the correct answer from the options given below:}\\ \hline\end{array}$

The correct option is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).

To match List-I with List-II, we need to derive the dimensional formulas for each physical quantity listed in List-II.

1. Current density (III):
Current density is defined as the electric current per unit cross-sectional area.
$\text{Current density (J)}=\frac{\text{Current}}{\text{Area}}$
The dimensional formula for current is [A] and for area is [L²]. Therefore, the dimensional formula for current density is:
$\left[J\right]=\frac{\left[A\right]}{\left[L^2\right]}=\left[M^0{L}^{-2}{T}^0{A}^1\right]$
This matches with (A)-(III).

2. Conductance (I):
Conductance is the reciprocal of electrical resistance. First, let's find the dimensional formula of resistance (R) using Ohm's law:
$R=\frac{\text{Potential Difference (V)}}{\text{Current (I)}}$
Since potential difference V is work done per unit charge:
$V=\frac{\text{Work (W)}}{\text{Charge (q)}}=\frac{W}{I·t}$
This gives:
$R=\frac{W}{I^2·t}$
The dimensional formula for Work is [M L² T^-2], current is [A], and time is [T]:
$\left[R\right]=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[A^{2}T\right]}=\left[M{L}^2{T}^{-3}{A}^{-2}\right]$
Since Conductance (G) is the reciprocal of resistance:
$\left[G\right]=\frac{1}{\left[R\right]}=\left[M^{-1}{L}^{-2}{T}^3{A}^2\right]$
This matches with (B)-(I).

3. Specific resistance (IV):
Specific resistance (or resistivity, ρ) is related to resistance by the formula:
$R=\rho \frac{l}{A}\Rightarrow \rho =R\frac{A}{l}$
Substituting the dimensional formulas of resistance, area, and length:
$\left[\rho \right]=\left[M{L}^2{T}^{-3}{A}^{-2}\right]·\frac{\left[L^2\right]}{\left[L\right]}=\left[M{L}^3{T}^{-3}{A}^{-2}\right]$
This matches with (C)-(IV).

4. Electrical conductivity (II):
Electrical conductivity (σ) is the reciprocal of specific resistance (resistivity):
$\left[\sigma \right]=\frac{1}{\left[\rho \right]}=\left[M^{-1}{L}^{-3}{T}^3{A}^2\right]$
This matches with (D)-(II).