Current Density And Conductivity

The microscopic or local form of Ohm's law relates the current density vector to the electric field through the conductivity, written as $\vec{J}=\sigma \vec{E}$, where $\sigma$ is the conductivity, a formulation given in NCERT Class 12, Chapter 3 (Current Electricity). This is the field-level statement equivalent to the familiar $V=IR$, and it holds point by point inside the material. Since conductivity is the reciprocal of resistivity, $\sigma =1/\rho$, one may equivalently write $\vec{E}=\rho \vec{J}$. The option stating current density equals resistivity times field is wrong because resistivity and conductivity are reciprocals, so substituting $\rho$ for $\sigma$ would invert the true relationship. The option that field equals conductivity times current density is physically incorrect because it swaps the roles of cause and effect and gives the wrong dimensions. The option dividing conductivity by field is wrong because Ohm's law is a linear proportionality between $J$ and $E$, not an inverse relation, so $J$ cannot fall as the field rises. A consistency check confirms the form: a higher conductivity should produce more current density for the same applied field, and the relation $\vec{J}=\sigma \vec{E}$ indeed makes $J$ increase in direct proportion to $\sigma$, matching both intuition and the macroscopic law it generalises.