What is Charge density; current density? — Physics

Definition 10.1 University Physics — Year 2 · Chapter 10 — Charges, Currents and Conduction

At the mesoscopic scale, matter carries a volume charge density $\rho(M, t)$ ( $\mathrm{C}/\mathrm{m}^{3}$ ): the charge $\rho\,\dd\tau$ in $\dd\tau$ . A surface may carry a surface density $\sigma$ ( $\mathrm{C}/\mathrm{m}^{2}$ ), a wire a linear density $\lambda$ ( $\mathrm{C}/\mathrm{m}$ ). If the carriers of species $i$ (charge $q_i$ , number density $n_i$ ) move at the mean velocity $\vect v_i$ , the current density is

\vect j = \sum_in_iq_i\,\vect v_i \qquad (\mathrm{A}/\mathrm{m}^{2}) ,

and the intensity through an oriented surface $S$ is the flux $I = \iint_S\vect j\cdot\vect n\,\dd S$ : the charge crossing $S$ per unit time. For a single species, $\vect j = \rho_m\vect v$ with $\rho_m = nq$ the density of mobile charge.

Left: a tube of current — in a stationary regime the same intensity crosses every section. Right: an electron in a metal zigzags between collisions at about 106 m/s; the field adds a slow drift, opposite to E, of a fraction of a millimetre per second. — Left: a tube of current — in a stationary regime the same intensity crosses every section. Right: an electron in a metal zigzags between collisions at about $10^6$ m/s; the field adds a slow drift, opposite to $\vect E$ , of a fraction of a millimetre per second.

Examples

Example 10.2 (How slowly electrons drift)

Copper has one conduction electron per atom: $n = \rho_{\text{Cu}}N_A/M = 8900 \times 6.02 \times 10^{23}/0.0635 = 8.5 \times 10^{28}\,\mathrm{m}^{-3}$ . A current of $10\,\mathrm{A}$ in a $1.5\,\mathrm{mm}^{2}$ wire is $j = 6.7 \times 10^{6}\,\mathrm{A}/\mathrm{m}^{2}$ , and the drift velocity is $v = j/ne = 6.7 \times 10^6/(8.5 \times 10^{28} \times 1.6 \times 10^{-19}) = 0.5\,\mathrm{mm}/\mathrm{s}$ — two hours per metre. The lamp lights at once because the field that pushes the electrons is set up along the whole wire within nanoseconds (it travels at nearly the speed of light, as the cable of Chapter 8 showed): all the electrons start together, like water in a full hose.

Example 10.3 (Orders of magnitude)

Lightning, $30\,\mathrm{kA}$ in a channel of $1\,\mathrm{cm}$ radius: $j \approx 1 \times 10^{8}\,\mathrm{A}/\mathrm{m}^{2}$ . A household wire: $10^6$ – $10^7$ . An electron beam in a cathode-ray tube, $1\,\mathrm{mA}$ over $1\,\mathrm{mm}^{2}$ : $10^3$ . A nerve fibre, $1\,\mathrm{nA}$ through $10\,\text{µ}\mathrm{m}^{2}$ : $10^2$ . The beam of a particle accelerator, $1\,\mathrm{A}$ in a $0.1\,\mathrm{mm}^{2}$ spot: $10^7$ — in vacuum, with no collisions at all.

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Examples

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