Electricity

An electric current 电流 is a flow of charge carriers 载流子. In a metal the carriers are negative conduction electrons 电子; in an electrolyte 电解质 they are positive and negative ions 离子; in a semiconductor 半导体 they may be electrons or "holes" 空穴. The conventional current 常规电流 direction is the way positive charge would flow — opposite to the real flow of electrons in a wire.

Charge

Charge is quantised 量子化: the smallest free unit of charge is the elementary charge 基本电荷

Every free charge in this syllabus is a whole-number multiple of $e$. The unit of charge is the coulomb 库仑, $\text{C}$.

Current as the rate of flow of charge

If charge $Q$ passes a point in time $t$, the current is

Unit of current: ampere, $\text{A}$ ($= \text{C s}^{-1}$). For a changing current, the charge that has flowed in a time is the area under an $I$–$t$ graph.

Worked example. A current of $0.50\ \text{A}$ flows for $2.0$ minutes. Find the charge that passes, and how many electrons this represents. ($e = 1.60 \times 10^{-19}\ \text{C}$.)

Drift velocity equation

For a uniform conductor of cross-section area $A$, with $n$ charge carriers per unit volume (the number density 数密度), each carrying charge $q$, moving with average drift velocity 漂移速度 $v$:

Charge carriers drifting inside a conductor: positive carriers drift with $I$, electrons against it

Worked example. A copper wire of cross-sectional area $1.0 \times 10^{-6}\ \text{m}^{2}$ carries a current of $5.0\ \text{A}$. Copper has $n = 8.5 \times 10^{28}$ free electrons per $\text{m}^{3}$. Find the drift velocity. ($e = 1.60 \times 10^{-19}\ \text{C}$.)

Rearranging $I = Anvq$ gives $v = \dfrac{I}{Anq}$:

The electrons drift very slowly — less than a millimetre per second.

Use this to compare currents:

• a thinner wire (smaller $A$) at the same $I$ needs a faster drift $v$.
• a semiconductor has far fewer free carriers than a metal (smaller $n$), so for the same $I$ the drift velocity is much larger.
• in series 串联 components, $I$ is the same everywhere, so if $A$ stays the same but the material changes, $nv$ changes the other way.

The potential difference 电势差 (p.d.) across a component is the energy 能量 transferred per unit charge as that charge passes through it:

Unit: volt 伏特, $\text{V}$ ($= \text{J C}^{-1}$).

If $1\ \text{J}$ of electrical energy changes into other forms (thermal, light, kinetic, …) when $1\ \text{C}$ of charge passes through a component, the p.d. across it is $1\ \text{V}$.

The electromotive force 电动势 (e.m.f.) of a source is the energy given per unit charge by the source. The formula is the same as for p.d.; the difference is direction: e.m.f. is energy given to the charge by the source; p.d. is energy given up by the charge to the component.

High-voltage power lines carry electrical energy across the country.

Combining $V = W/Q$ and $I = Q/t$:

Using Ohm's law $V = IR$:

Pick the form with the quantities you know. Examples:

• two heaters of equal resistance — the one with the larger current gives more power 功率 ($P = I^{2}R$).
• two resistors in parallel 并联 across the same voltage 电压 — the one with smaller $R$ gives more power ($P = V^{2}/R$).
• a kettle marked "$2.4\ \text{kW}, 240\ \text{V}$" draws $I = P/V = 10\ \text{A}$ and has resistance $R = V^{2}/P = 24\ \Omega$.

Energy transferred in time $t$ is $E = P t$.

The resistance 电阻 $R$ of a component is

Unit: ohm 欧姆, $\Omega$ ($= \text{V A}^{-1}$). Resistance depends on the conditions (such as temperature) when it is measured.

Real fixed resistors — the coloured bands code the resistance in ohms

Ohm's law

A conductor obeys Ohm's law 欧姆定律 when the current through it is proportional to the p.d. across it, as long as the conditions (especially temperature) stay constant. For such a conductor $R$ is constant and the $I$–$V$ graph is a straight line through the origin.

Ohm's law is an experimental result, not a definition. The definition $R = V/I$ works for any component; only ohmic ones have constant $R$.

$I$–$V$ characteristics

You should be able to sketch these:

• metal wire at constant temperature — a straight line through the origin (constant $R$). Reversing the p.d. drives the current the other way, giving a straight line in both directions.
• filament lamp 灯丝灯泡 — through the origin, steep at first, then flatter as $V$ (and $I$) grow. Reason: more current heats the filament, so its resistance rises and the gradient $1/R$ falls.
• semiconductor diode 二极管 — almost no current for negative $V$ or small positive $V$. Above a "switch-on" voltage (about $0.7\ \text{V}$ for silicon), the current rises sharply.

$I$–$V$ characteristic of an ohmic conductor (metal wire at constant temperature)

$I$–$V$ characteristic of a filament lamp

$I$–$V$ characteristic of a semiconductor diode

Resistivity

For a uniform conductor of length $L$ and cross-section area $A$,

$\rho$ is the resistivity 电阻率, a property of the material, with unit $\Omega\ \text{m}$. Doubling the length doubles $R$; doubling the area halves it; halving the diameter quarters the area and so makes $R$ four times bigger.

Worked example. A copper wire of length $2.0\ \text{m}$ and cross-sectional area $1.7 \times 10^{-7}\ \text{m}^{2}$ has resistivity $1.7 \times 10^{-8}\ \Omega\ \text{m}$. Find its resistance.

Typical values: copper at room temperature $\rho \sim 1.7 \times 10^{-8}\ \Omega\ \text{m}$; an insulator 绝缘体 $\rho \sim 10^{15}\ \Omega\ \text{m}$ or more.

A longer conductor has more resistance — doubling $L$ doubles $R$

A wider conductor has less resistance — doubling $A$ halves $R$

The resistivity of a metal rises with temperature (more lattice vibration 晶格振动 scatters 散射 the electrons), which is why the filament lamp's $I$–$V$ line curves.

A light-dependent resistor 光敏电阻 (LDR) is a semiconductor whose resistance falls as the light intensity rises. In bright light $R$ may be a few hundred $\Omega$; in the dark it can be in the megaohms. LDRs are used in light-sensing circuits (street lamps, camera light meters). Here the light intensity 光强 controls the resistance.

Resistance of an LDR decreases as light intensity increases

In this syllabus a thermistor 热敏电阻 has a negative temperature coefficient 负温度系数: its resistance falls as its temperature rises. This is useful for sensing temperature — put it in a potential divider 分压器 and the output voltage changes with temperature.

Resistance of a thermistor falls as temperature rises

This is the opposite of a metal: in a semiconductor, more thermal energy frees more charge carriers, and this matters more than the extra scattering.

• Use $I = Q/t$, $V = W/Q$ (energy per unit charge) and $P = VI = I^2 R = V^2/R$.
• Ohm's law ($V = IR$) applies only to an ohmic conductor at constant temperature — a filament lamp is non-ohmic.
• Sketch and interpret the $I$-$V$ characteristics of a resistor, filament lamp and diode.
• An LDR's resistance falls with light; a thermistor's falls as temperature rises.