Q.Describe the microscopic model of current and obtain the general form of Ohm's law.

Building I from first principles. Consider a conductor of cross-sectional area A carrying n free electrons per unit volume, all moving with the same drift velocity $v_d$. In a small time interval dt, every electron within a thin slice of thickness $dx=v_d\,dt$ crosses the cross-section; the number of electrons in that slice is $n\cdot A\,dx = n\,A\,v_d\,dt$, and since each carries charge e, the charge crossing in time dt is $dQ=neAv_d\,dt$. Dividing by dt gives the central microscopic-model result,

directly connecting the everyday, measurable current I to the microscopic quantities n (electron density), e (electron charge), A (cross-sectional area) and $v_d$ (drift velocity).

Current density. Dividing current by cross-sectional area defines the current density, $J=I/A$, SI unit $\text{A/m}^2$; substituting the result above gives $J=nev_d$. In general, current density is treated as a vector $\vec J=ne\vec v_d$, pointing along the direction of (conventional) current flow at a given point -- distinct from the total current I through a surface, which is the scalar quantity $I=\vec J\cdot\vec A$ built by dotting the current-density vector with the chosen surface's area vector.

Microscopic form of Ohm's law. Substituting the drift-velocity expression $\vec v_d=-(e\tau/m)\vec E$ into $\vec J=ne\vec v_d$ gives $\vec J=-(ne^2\tau/m)\vec E$; taking the direction of conventional current density along $\vec E$ (as is standard, since the negative sign is an artefact of tracking the electrons rather than the conventional positive-charge picture) gives the compact relation

where $\sigma=ne^2\tau/m$ is the material's conductivity. This relation is called the microscopic form of Ohm's law, and it is the true, fundamental starting point from which the everyday, macroscopic law $V=IR$ is later derived by integrating over a specific wire's geometry (length and area). It directly shows WHY a material conducts well or poorly: high electron density n, long mean free time $\tau$ (few collisions) and low electron mass m all push conductivity up, which is exactly the physical basis for why metals conduct far better than semiconductors.