Current, Drift Velocity and Resistivity: Quick Revision

A brief note on Current, Drift Velocity and Resistivity. Read it before a test. Try each question at the end before you open its answer.

Current and the drift

• Current: $I = Q/t$, in A. The number of electrons is $N = Q/e$, with $e = 1.6 \times 10^{-19}$ C.
• Without a field: the free electrons move at random and the motions cancel.
• With a field: a slow drift, $v_d = \dfrac{eE}{m}\tau$, against the field.
• Mobility: $\mu = \dfrac{v_d}{E} = \dfrac{e\tau}{m}$, in $\text{m}^2/(\text{V s})$.
• Current: $I = nAev_d$. Current density: $J = \dfrac{I}{A} = nev_d$.
• Drift speeds are fractions of a millimetre each second. The field spreads at nearly the speed of light.

Resistivity and conductivity

• Ohm's law at the level of the material: $J = \sigma E$, with $\sigma = \dfrac{ne^2\tau}{m} = ne\mu$.
• Resistivity: $\rho = \dfrac{1}{\sigma} = \dfrac{m}{ne^2\tau}$, in $\Omega$ m.
• A wire: $R = \rho\dfrac{L}{A}$, with $A = \dfrac{\pi d^2}{4}$ for a round wire.
• Materials: metals about $10^{-8}\ \Omega$ m, semiconductors in between, insulators $10^{10}\ \Omega$ m and more.
• A wire cut into $n$ pieces joined side by side has $R/n^2$.

Temperature and the colour code

• Metals: $R_T = R_0[1 + \alpha(T - T_0)]$, with $\alpha$ positive. Resistance rises with temperature.
• Semiconductors: $\alpha$ is negative. Alloys such as manganin: $\alpha$ is almost zero.
• Superconductors: zero resistance below a critical temperature.
• Not ohmic: a filament lamp and a diode do not give a straight line of $I$ against $V$.
• Colour code: digit, digit, power of ten, tolerance. Gold $\pm 5\ \%$, silver $\pm 10\ \%$, none $\pm 20\ \%$.

A worked example

A wire of cross section $4.0\ \text{mm}^2$ carries $12$ A. Find the current density, and the drift speed if $n = 8.5 \times 10^{28}\ \text{m}^{-3}$.

1. Current density. $J = \dfrac{12\ \text{A}}{4.0 \times 10^{-6}\ \text{m}^2} = 3.0 \times 10^6\ \text{A/m}^2$.
2. Drift speed. $v_d = \dfrac{J}{ne} = \dfrac{3.0 \times 10^6}{8.5 \times 10^{28} \times 1.6 \times 10^{-19}} = 2.2 \times 10^{-4}\ \text{m/s}$.

Practise with new numbers

Check yourself

1. A wire is made twice as long and its diameter is made half, with the same metal. By what factor does its resistance change?

2. A coil of copper has $50\ \Omega$ at $20\ ^\circ$C. Estimate its resistance at $120\ ^\circ$C, with $\alpha = 3.9 \times 10^{-3}$ per $^\circ$C.

3. Why does the resistance of a thermistor fall when it warms?