The unit of conductivity \((K)\) is:

\(\Omega ^{-1}\, \ m^{-1}\)

The correct answer is option 2. \(\Omega ^{-1}\, \ m^{-1}\)

Conductivity (\(K\)) is a measure of a material's ability to conduct electric current. It is the reciprocal of resistivity (\(\rho\)), which quantifies how strongly a given material opposes the flow of electric current. The relationship between conductivity (\(K\)) and resistivity (\(\rho\)) is given by the equation:

\[ K = \frac{1}{\rho} \]

The unit of resistivity (\(\rho\)) is ohm-meter (\(\Omega \cdot m\)), and therefore, the unit of conductivity (\(K\)) is the reciprocal of ohm-meter, which is Siemens per meter (\(S/m\)).

Let us break it down:

1. Conductivity (\(K\)):
Conductivity is a measure of how well a material allows the flow of electric current. It is the inverse of resistivity and is represented by the symbol \(K\).

2. Resistivity (\(\rho\)):
Resistivity is the inherent property of a material that determines how strongly it resists the flow of electric current. It is represented by the symbol \(\rho\). The unit of resistivity is ohm-meter (\(\Omega \cdot m\)).

3. Relationship between Conductivity and Resistivity:
Conductivity (\(K\)) and resistivity (\(\rho\)) are inversely related by the equation \(K = \frac{1}{\rho}\).

4. Unit of Conductivity:
The unit of conductivity (\(K\)) is the reciprocal of the unit of resistivity (\(\rho\)). Therefore, the unit of conductivity is Siemens per meter (\(S/m\)), which is also equivalent to \(\Omega^{-1} \cdot m^{-1}\).

In summary, conductivity is a measure of how well a material conducts electric current, and its unit is Siemens per meter (\(S/m\)) or ohm-inverse per meter (\(\Omega^{-1} \cdot m^{-1}\)). This unit represents the ease with which electric current can flow through a material.