Physics · Ch 2 — Current Electricity

Microscopic Model of Current

2.1.3

Microscopic Model of Current

Building the current formula from first principles: consider a conductor of cross-sectional area A with an electric field $\vec E$ applied along it, containing n free electrons per unit volume, all assumed to move with the same drift velocity $v_d$ . If an electron moves through a small distance dx in a small time interval dt, then $v_d = dx/dt$ , so $dx = v_d\,dt$ (2.7). The number of electrons contained in this thin slice of the conductor, of length dx and cross-sectional area A, is the slice's volume times the electron number density: $A\,dx\cdot n$ (2.8). Substituting $dx = v_d\,dt$ from (2.7) into (2.8), the number of electrons in the slice becomes $(A\,v_d\,dt)\,n$ , and since each electron carries charge e, the total charge in this volume element is

dQ = e\,(A\,v_d\,dt)\,n

Hence the current, $I = dQ/dt$ , works out to

I = ne\,A\,v_d \qquad (2.9)

This is the central microscopic-model result: current is directly proportional to the electron number density, the electron's charge, the conductor's cross-sectional area, and the drift velocity.

Current density (J) is defined as the current per unit cross-sectional area, $J = I/A$ , with SI unit $\text{A/m}^2$ . Substituting equation (2.9) gives $J = ne\,v_d$ (2.10) -- a relation that holds exactly when current flows perpendicular to A. In general, current density is treated as a vector quantity (its direction is the direction positive charge is actually flowing at a point), written $\vec J = ne\,\vec v_d$ . Substituting the drift-velocity expression $\vec v_d = -(e\tau/m)\vec E$ from equation (2.4) gives

\vec J = -\dfrac{ne^2\tau}{m}\vec E \qquad (2.11)

Since by convention the direction of conventional current density is taken along $\vec E$ (not opposite to it, as the raw electron-based derivation would suggest, because the flowing charge carriers are actually negative), this is written as

$\vec J = \sigma \vec E \qquad (2.12)$ …

Figure 2.5Microscopic model of current

What this figure shows. A conductor of cross-sectional area A is drawn with an electric field $\vec E$ applied along it and several electron symbols (with drift velocity $v_d$ ) shown inside a thin slice of thickness $dx$ marked off within the conductor; the slice's volume, multiplied by the number density n of free electrons, is used to count exactly how many electrons cross the area A in the small time interval dt -- the core geo …

Misc Example 2.3Drift velocity from current and electron density

Worked out. A copper wire of cross-sectional area $0.5\ \text{mm}^2$ carries a current of 0.2 A, and the free-electron density of copper is given as $8.4\times10^{28}\ \text{m}^{-3}$ ; the drift velocity is required. Rearranging $I = neAv_d$ gives $v_d = I/(neA)$ . Substituting $I = 0.2$ A, $n = 8.4\times10^{28}\ \text{m}^{-3}$ , $e = 1.6\times10^{-19}$ C and $A = 0.5\times10^{-6}\ \text{m}^2$ gives $v_d \approx 0.03\times10^{-3}\ \text{m/s}$ , i.e. only about $3\times10^{-5}$ m/s. This tiny value -- a few centimetres per hour -- is the standard illustration of just how slowly electrons actually drift even wh …

Misc Example 2.4Number of electrons flowing per second

Worked out. The question asks how many electrons flow per second through a conductor carrying a current of 32 A. Since $I = Q/t = ne/t$ for $t = 1$ s, rearranging gives $n = It/e$ . Substituting $I = 32$ A, $t = 1$ s and $e = 1.6\times10^{-19}$ C gives $n = 32/(1.6\times10^{-19}) = 20\times10^{19} = 2\times10^{20}$ electrons per second. This huge number of electrons crossing any cross-section every second is exactly why even a modest current corresponds to a very large charge flow, despite each individual electr …

Misc ~note-scalar-vectorWhy current density is a vector but current is a scalar

Worked out. This aside resolves an apparent puzzle: current density $\vec J$ is defined as a vector (it points in the direction of positive charge flow at a point), yet the total current I through a surface is a scalar. The resolution is that I is defined as the scalar (dot) product of the current density vector and the area vector through which the charges cross, $I = \vec J \cdot \vec A = JA\cos\theta$ , where $\theta$ is the angle between $\vec J$ and the outward normal to the surface. Because I is built from a dot product of two vectors, it comes out as a single number (which can even be positive or negative depending on which way the area's normal vector is chosen), even though the current density feeding into it is genuinely a ve …

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