IGCSE Physics · ⁨Physics IGCSE⁩

Matter exists in three states: solid 固体, liquid 液体 and gas 气体.

The changes of state are: melting (solid → liquid), boiling/evaporating (liquid → gas), condensation 凝结 (gas → liquid) and solidification 凝固 (liquid → solid). In condensation the gas particles slow down and come close together to form a liquid. In solidification the liquid particles lose energy and settle into fixed positions.

All matter is made of tiny particles 粒子 that are always moving. This is the kinetic particle model.

• In a solid the particles only vibrate 振动 about fixed positions.
• In a liquid they are still close but can slide past each other.
• In a gas they are far apart and move quickly in random directions.

Temperature and particle energy

When you heat a substance, its particles move faster, so they have more kinetic energy 动能. Temperature 温度 is a measure of the average kinetic energy of the particles.

The lowest possible temperature is absolute zero 绝对零度, $-273\,{}^{\circ}\text{C}$. At this point the particles have the least possible energy.

Gas pressure

Gas particles hit the walls of their container. Each hit is a tiny push. The pressure 压强 of the gas is the total force of these hits per unit area.

• Heating a gas (at constant volume) makes particles move faster and hit harder and more often, so the pressure rises.
• Squeezing a gas into a smaller volume (at constant temperature) means more hits per second on each unit of area, so the pressure rises.

For a fixed mass of gas at constant temperature:

So if the volume halves, the pressure doubles.

Worked example. A gas has a volume of $200\ \text{cm}^3$ at a pressure of $100\ \text{kPa}$. It is squeezed to $50\ \text{cm}^3$ at constant temperature. Find the new pressure.

Since $pV$ stays constant, $p_1 V_1 = p_2 V_2$:

Brownian motion

If you look at smoke in air under a microscope, you see tiny specks moving in a jerky, random way. This is Brownian motion 布朗运动. The specks are pushed by fast, invisible air particles hitting them. It is strong evidence for the kinetic particle model.

The kelvin scale

Scientists often use the kelvin 开尔文 (K) temperature scale, which starts at absolute zero. To convert:

So $0\,{}^{\circ}\text{C} = 273\ \text{K}$.

Worked example. Convert $25\,{}^{\circ}\text{C}$ to kelvin, and $200\ \text{K}$ to degrees Celsius.

When matter is heated, its particles move more and take up more space, so the material expands. Gases expand the most, then liquids, then solids. This order comes from the forces between the particles: in a solid the particles are held tightly in a lattice 晶格 (a fixed pattern), so they can only vibrate a little more; in a liquid they are held less strongly; in a gas the forces are very weak, so the particles spread out much more.

Everyday examples: gaps are left between railway lines; bridges sit on rollers; a tight metal lid loosens when heated.

The internal energy 内能 of an object is the total energy of all its particles. Heating an object raises its internal energy and usually its temperature.

The specific heat capacity 比热容 is the energy needed to raise the temperature of 1 kg of a material by 1 °C.

A material with a high specific heat capacity (like water) needs a lot of energy to warm up and cools down slowly.

Worked example. How much energy is needed to heat $2.0\ \text{kg}$ of water from $20\,{}^{\circ}\text{C}$ to $70\,{}^{\circ}\text{C}$? The specific heat capacity of water is $4200\ \text{J/(kg}\,{}^{\circ}\text{C)}$.

Rearranging $c = \frac{\Delta E}{m\,\Delta\theta}$ gives $\Delta E = mc\,\Delta\theta$, with $\Delta\theta = 70 - 20 = 50\,{}^{\circ}\text{C}$:

Measuring specific heat capacity. For a solid, use a metal block with two holes: one for an electric heater 加热器 and one for a thermometer 温度计. Measure the mass $m$ of the block. Switch on the heater for a measured time $t$ and record the current $I$ and the voltage $V$, so the energy supplied is $E = IVt$ (or read the energy from a joulemeter). Record the temperature rise $\Delta\theta$. Then $c = \dfrac{E}{m\,\Delta\theta}$. For a liquid, heat a measured mass of the liquid in a container and stir it, so the temperature is the same all through. Wrap the block or the container in insulating material 隔热材料, so less energy escapes to the air.

Melting 熔化 and boiling 沸腾 need energy, but the temperature stays the same while the state changes. This energy breaks the forces between particles. For water at normal air pressure, melting is at $0\,{}^{\circ}\text{C}$ and boiling at $100\,{}^{\circ}\text{C}$.

Evaporation 蒸发 is when a liquid changes to a gas at its surface, below the boiling point. The fastest particles escape from the surface. Because the fastest (most energetic) particles leave, the average energy of those left behind falls, so the liquid cools down 冷却. An object touching an evaporating liquid also cools, because the liquid takes energy from it: this is why sweat cools your skin.

Evaporation is faster when the temperature is higher, the surface area is larger, and there is more air movement over the surface.

Thermal energy moves from hotter places to colder places in three ways.

Conduction

Conduction 热传导 is the transfer of thermal energy through a material without the material moving. The particles in a solid sit in a lattice. Heated particles vibrate more and pass the energy to their neighbours. In metals, free electrons (delocalised electrons 自由电子) carry energy quickly, so metals are good thermal conductors 热导体.

Materials that conduct badly (like air, wood and plastic) are thermal insulators 热绝缘体.

Testing conductors. Fix a drawing pin with a little wax to one end of rods of copper, iron, glass and wood, and heat the other ends equally. The pin drops first from the copper rod, then from the iron rod, and last (or never) from the glass and the wood: metals are the best conductors. A metal rod held in a flame soon feels hot; a wooden rod does not.

Gases and most liquids are bad conductors because their particles are far apart and only pass on energy when they happen to collide. There is no lattice and there are no free electrons.

Convection

Convection 对流 happens in liquids and gases. When a fluid is heated it expands, becomes less dense, and rises. Cooler, denser fluid sinks to take its place. This circle of moving fluid is a convection current.

Showing convection. Drop a crystal of purple potassium permanganate 高锰酸钾 into a beaker of water and heat the beaker gently under the crystal. A coloured stream rises above the heat, spreads across the top and sinks at the sides. In air, a lit candle under one chimney of a smoke box pulls smoke down the other chimney and up above the flame.

Convection cannot happen in a solid because the particles cannot move from place to place.

Radiation

Thermal radiation 热辐射 is energy carried by infrared 红外线 waves. All objects emit it, and it needs no material to travel through — it can cross empty space (this is how energy reaches us from the Sun).

A surface that is dull 暗淡 and black is a good emitter 发射体 and a good absorber 吸收体 of infrared. A surface that is shiny and white is a poor emitter and a good reflector 反射体.

Comparing surfaces. Fill two cans, one dull black and one shiny silver, with hot water at the same temperature, and put a thermometer in each. The black can cools faster, so it is the better emitter. Then put the same two cans, full of cold water, at equal distances from a heater. The black can warms faster, so it is the better absorber. A Leslie cube 莱斯利立方体 (a metal box with one side of each finish) shows the same thing.

An object stays at a constant temperature when it emits energy at the same rate as it absorbs energy. The rate of emission is greater when the surface is hotter and larger. If an object absorbs energy faster than it emits it, its temperature rises; if it emits faster than it absorbs, its temperature falls.

The Earth's temperature. The Earth absorbs radiation from the Sun and emits infrared radiation back into space. Its average temperature stays constant only when the two rates are equal. Anything that changes this balance changes the temperature. Gases in the atmosphere 大气 such as carbon dioxide absorb some of the emitted infrared and send part of it back down, so more of these gases warms the Earth. Clouds and ice reflect incoming sunlight, so less ice means more energy is absorbed, and the Earth warms further.

Everyday examples

• A kitchen pan has a metal base that conducts energy quickly to the food, and a plastic or wooden handle that is an insulator, so you can hold it.
• A radiator 散热器 heats a room by convection: the air next to it warms, rises, and cooler air flows in to take its place, so the whole room warms.
• Sitting near a fire you feel its radiation on your face at once; the air between you and the fire is a poor conductor.
• A vacuum flask 保温瓶 keeps a drink hot by stopping all three transfers: the vacuum between its double walls stops conduction and convection, and its shiny silver surfaces stop radiation.

• In the kinetic particle model, heating a gas makes its particles move faster and hit the walls harder and more often, so the pressure rises. The particles themselves do not get bigger.
• Evaporation happens only at the surface and at any temperature; boiling happens throughout the liquid at one fixed temperature. Evaporation cools the liquid left behind, because the fastest particles escape.
• During melting and boiling the temperature stays constant even though energy is still being supplied — that energy breaks the forces between particles.
• For specific heat capacity use $\Delta E = mc\,\Delta\theta$, where $\Delta\theta$ is the change in temperature, not the final temperature.
• Dull black surfaces are the best emitters and absorbers of infrared; shiny, light surfaces are the best reflectors. Convection needs a fluid to flow, so it cannot happen in a solid.

A wave 波 carries energy 能量 from place to place without carrying matter. For example, a water wave makes a floating cork bob up and down, but the cork does not travel along with the wave. You can make a wave on a rope by shaking one end up and down, or on a long spring by pushing one end back and forth. In a ripple tank 波纹水槽 (a shallow tray of water with a light above it) you can watch water waves on a screen below.

Describing a wave

• wavefront 波前: a line joining points on a wave that move together (for example, the top of one ripple)
• wavelength 波长 ($\lambda$): the distance between two neighbouring wavefronts (one full wave)
• frequency 频率 ($f$): the number of waves passing a point each second, measured in hertz (Hz)
• crest 波峰 (also called the peak 峰, the top) and trough 波谷 (bottom)
• amplitude 振幅: the largest distance a point moves from its rest position
• wave speed 波速 ($v$): how fast a wavefront travels

These are linked by the wave equation:

Worked example. A sound wave travels at $340\ \text{m/s}$ and has a frequency of $170\ \text{Hz}$. Find its wavelength.

Rearranging $v = f\lambda$ gives $\lambda = \dfrac{v}{f}$:

Two types of wave

• In a transverse wave 横波 the particles vibrate at right angles (90°) to the direction the wave travels (its propagation 传播). Light, water waves and seismic S-waves (the sideways shaking in an earthquake) are transverse.
• In a longitudinal wave 纵波 the particles vibrate along the same direction as the wave travels. Sound and seismic P-waves are longitudinal.

Wave behaviour

All waves can show three behaviours, which you can see in a ripple tank:

• reflection 反射: the wave bounces off a surface.
• refraction 折射: the wave changes speed (and usually direction) when it enters a different material or depth.
• diffraction 衍射: the wave spreads out after passing through a gap or around an edge. The spreading is greatest when the gap is about the same size as the wavelength.

Reflection

When light hits a mirror, it reflects. We measure angles from the normal 法线 (a line at 90° to the surface).

The law of reflection: the angle of incidence 入射角 equals the angle of reflection 反射角.

A plane mirror 平面镜 forms an image that is the same size as the object, the same distance behind the mirror, and virtual (it cannot be caught on a screen).

Refraction

When light passes from one material into another, it changes speed and bends. The angle in the second material is the angle of refraction 折射角.

Seeing refraction. Shine a narrow ray from a ray box 光线盒 into a rectangular glass block standing on paper. Mark the ray going in and the ray coming out, remove the block, and join the marks to draw the ray inside the glass. Draw the normal where the ray enters and measure the angle of incidence and the angle of refraction with a protractor. Repeat with a semicircular block or a prism: the ray always bends towards the normal on entering the glass and away from the normal on leaving.

The refractive index 折射率 $n$ compares the speed of light in the two materials:

Worked example. A ray of light passes from air into glass. The angle of incidence is 60° and the angle of refraction is 35°. Find the refractive index of the glass. (Take $\sin i = 0.87$ and $\sin r = 0.57$.)

When light tries to leave glass or water and the angle is too large, it cannot get out and instead reflects completely. This is total internal reflection 全反射. It happens when the angle inside is bigger than the critical angle 临界角 $c$:

This effect is used in optical fibres 光纤, thin glass threads that carry light signals for telephones and the internet.

Lenses

A converging lens 凸透镜 (fat in the middle) bends parallel rays inwards to a point called the principal focus 焦点. The principal axis 主光轴 is the straight line through the centre of the lens at right angles to it; the principal focus lies on this axis. The distance from the centre of the lens to the principal focus is the focal length 焦距. A diverging lens 凹透镜 (thin in the middle) spreads parallel rays out, as if they came from a focus on the incoming side.

A converging lens can form a real image 实像 (rays really meet; can be shown on a screen) or, when the object is very close, a virtual image 虚像 (rays only seem to come from it). Used close to the eye, a converging lens is a magnifying glass 放大镜.

Describe an image with three pairs of words: enlarged 放大的 or diminished 缩小的 (or the same size), upright 正立 or inverted 倒立 (upside down), and real or virtual. An object far beyond the focus gives a diminished, inverted, real image (as in a camera). An object between the focus and the lens gives an enlarged, upright, virtual image (a magnifying glass).

Correcting sight. A person with short-sightedness 近视 sees near things clearly but distant things are blurred: the eye focuses the light in front of the retina 视网膜 (the back of the eye). A diverging lens in front of the eye spreads the rays out first, so they focus on the retina. A person with long-sightedness 远视 sees distant things clearly but near things are blurred: the eye focuses the light behind the retina. A converging lens brings the rays together a little before they enter the eye.

Dispersion

A glass prism 棱镜 splits white light into the colours of the spectrum 光谱. This splitting is called dispersion 色散.

The order of colours is red, orange, yellow, green, blue, indigo, violet. Red light has the longest wavelength and the lowest frequency; violet is the opposite. Light of a single frequency (one pure colour) is monochromatic 单色.

The electromagnetic spectrum is a family of waves that all travel at the same high speed in a vacuum 真空, $3.0 \times 10^8\ \text{m/s}$. In order from longest wavelength (lowest frequency) to shortest:

As you go from radio waves to gamma rays, the frequency rises, the wavelength falls, and the energy (and danger) rises. The speed of electromagnetic waves in air is almost the same as in a vacuum.

Satellites. Communication with artificial satellites 人造卫星 mainly uses microwaves, because they pass through the atmosphere. Some satellite phones use satellites in low orbit 低轨道, close to the Earth. Other satellite phones, and direct broadcast satellite television, use satellites in geostationary orbit 地球静止轨道, which stay above the same point on the Earth.

Digital and analogue signals

Radio waves and microwaves carry information as signals, and a signal is either analogue 模拟 or digital 数字. An analogue signal varies continuously and can take any value; a digital signal is a stream of just two values, on or off (1 or 0). Sound can be sent as either kind of signal: a microphone gives an analogue signal, which can be turned into a digital one.

Modern communication prefers digital signalling for two reasons the exam asks for:

• a higher rate of data transmission;
• a greater range: a weak digital signal can be regenerated back to a clean 1/0 exactly, because only two levels have to be told apart, so noise picked up along the way is removed. An analogue signal cannot be cleaned up this way, so its noise builds up.

Sound is made by a vibrating object. It is a longitudinal wave: the air is squeezed into a compression 压缩 (particles close together) and stretched into a rarefaction 稀疏 (particles far apart).

• Sound needs a medium 介质 (a material) to travel through, so it cannot travel through a vacuum.
• Humans can hear frequencies from about $20\ \text{Hz}$ to $20\,000\ \text{Hz}$.
• Sound travels much slower than light (about $340\ \text{m/s}$ in air), and faster in liquids and solids than in gases.
• A larger amplitude makes a louder sound (greater loudness 响度); a higher frequency makes a higher pitch 音调. So a dolphin's rapid clicks are high-pitched, and a hard drum-hit (big vibration) is loud.

A reflected sound is an echo 回声.

Measuring the speed of sound. Two students stand a known distance apart, for example 500 m, measured with a tape or a trundle wheel. One bangs two blocks of wood together. The other starts a stopwatch when they see the blocks hit and stops it when they hear the bang. Light arrives almost at once, so the time measured is the time the sound takes, and speed = distance ÷ time. Repeat and take an average to reduce the reaction-time error. The echo method needs one person: clap at a known distance from a large wall and time the reflected sound, remembering that the sound travels there and back.

Worked example. A student stands 85 m from a large wall and claps once. The echo returns 0.50 s later. Find the speed of sound.

The sound travels to the wall and back, a distance of $2 \times 85 = 170\ \text{m}$, in $0.50\ \text{s}$:

Ultrasound 超声波 is sound above $20\,000\ \text{Hz}$ – too high for humans to hear. Its uses rely on reflection: sonar measures water depth, medical scanning images a baby or soft tissue, and non-destructive testing finds cracks inside metal without cutting it open. Like an echo, an ultrasound pulse travels there and back, so halve the total distance to reach the object.

Worked example. A sonar pulse returns from the seabed $0.10\ \text{s}$ after it is sent; sound travels at $1500\ \text{m/s}$ in water. The pulse covers $s = v\times t = 1500\times0.10 = 150\ \text{m}$ there and back, so the sea is $150/2 = 75\ \text{m}$ deep.

• Learn $v = f\lambda$ and be ready to rearrange it. Measure all angles of incidence, reflection and refraction from the normal (the line at 90° to the surface), never from the surface itself.
• Transverse waves vibrate across the direction of travel (light, water); longitudinal waves vibrate along it (sound). Sound needs a medium, so it cannot cross a vacuum; light can.
• Total internal reflection happens only when light travels from a denser material to a less dense one and the angle inside is bigger than the critical angle.
• Know the electromagnetic spectrum in order — radio, microwave, infrared, visible, ultraviolet, X-ray, gamma. All travel at the same speed in a vacuum; frequency and energy rise from radio to gamma.
• Going into a denser material (air → glass) light slows down and bends towards the normal; leaving it (glass → air) light speeds up and bends away.

A magnet 磁体 can attract some metals. Every magnet has two ends, called poles 磁极: a north pole 北极 (N pole) and a south pole 南极 (S pole).

The rule for two poles is like the rule for charges:

• Poles that are the same (N and N, or S and S) repel 排斥 (push apart): this is repulsion 排斥力.
• Poles that are different (N and S) attract 吸引 (pull together): this is attraction 吸引力.

A magnet attracts a magnetic material 磁性材料, such as iron, steel, cobalt and nickel. Other materials (copper, plastic, wood) are non-magnetic and feel no force.

Induced magnetism

Put a piece of iron near or touching a magnet, and the iron becomes a magnet itself. This is induced magnetism 感应磁性. The end of the iron nearest the magnet gains the opposite pole, so the two then attract.

Temporary and permanent magnets

• Soft iron 软铁 is easy to magnetise 磁化 but loses its magnetism quickly. It is used for a temporary magnet 临时磁体.
• Steel 钢 is harder to magnetise but keeps its magnetism. It is used for a permanent magnet 永磁体.

So an unmagnetised 未磁化 piece of soft iron is the best core for an electromagnet, while steel is best for a bar magnet.

Magnetic fields

A magnetic field 磁场 is a region 区域 where a magnetic pole feels a force. We draw it with magnetic field lines 磁场线:

• The lines come out of the N pole and go into the S pole.
• The field direction at a point is the direction of the force on the N pole of a small test magnet placed there.
• Where the lines are close together the field is strong; where they are far apart the field is weak.

You can show the pattern by sprinkling iron filings 铁屑 on paper over a magnet, or by moving a small plotting compass 指南针 around it (the needle points along the field).

Uses of magnets

• Permanent magnets: fridge door catches, compass needles, loudspeakers.
• Electromagnets 电磁铁 (magnetic only while a current flows): cranes that lift scrap iron, electric bells, relays. An electromagnet can be switched on and off and made stronger, which a permanent magnet cannot.

Magnetic forces happen because the fields of the two magnets push and pull on each other.

There are two kinds of electric charge 电荷: positive charge 正电荷 and negative charge 负电荷. The rule is like the rule for poles — same charges repel, different charges attract.

Charging by friction

When you rub two different materials together, electrons 电子 (tiny negative particles) move from one to the other. This is charging by friction 摩擦起电.

• The material that gains electrons becomes negative.
• The material that loses electrons becomes positive.

Only the negative electrons move; the positive charge stays in place. For example, a plastic rod rubbed with a cloth becomes negative because electrons move from the cloth onto the rod, leaving the cloth positive.

Detecting a charge. A charged rod picks up small pieces of paper, and it bends a thin stream of water from a tap. Hang one charged rod from a thread and bring another charged rod near it: if the two have the same kind of charge, the hanging rod swings away. Repulsion is the only sure test for a charge, because attraction also happens with uncharged objects.

Conductors and insulators

• A conductor 导体 (copper and other metals) lets charge flow through it easily, because it has free electrons that can move.
• An insulator 绝缘体 (plastic, rubber, glass) does not let charge flow, because its electrons cannot move freely.

To test a material, join it in a circuit with a lamp and a battery: the lamp lights only for a conductor.

Electric fields and ions

Charge is measured in coulombs 库仑 (C). An electric field 电场 is a region where an electric charge feels a force. Its direction at a point is the direction of the force on a positive charge. Simple field patterns (the arrows show the direction):

• around a point charge 点电荷: straight lines, pointing away from a positive charge and towards a negative charge;
• around a charged ball (sphere): the same, spreading out evenly all around;
• between two parallel plates with opposite charges: straight, evenly spaced lines from the positive plate to the negative plate.

An atom is normally neutral. If it loses electrons it becomes a positive ion 离子; if it gains electrons it becomes a negative ion.

An electric current 电流 is a flow of electric charge. It is measured in amperes 安培 (A) with an ammeter 电流表, joined in series in the circuit.

An analogue 模拟 ammeter has a needle that moves over a scale; a digital 数字 ammeter shows the reading as a number. Both come with different ranges 量程. Choose the range just above the biggest current you expect, so the reading is accurate: a 0–1 A range for a current of about 0.4 A, or a 0–100 mA range for a few milliamps.

Current is the charge that passes a point each second:

Here $I$ is current (A), $Q$ is charge (C) and $t$ is time (s). Rearranged, $Q = It$.

Worked example. A current of $3.0\ \text{A}$ flows through a lamp for $20\ \text{s}$. How much charge passes through it?

In a metal the current is a flow of free electrons 自由电子. There are two "directions" to know:

• Conventional current 常规电流 is taken to flow from + to − round the outside of the cell.
• The free electrons really drift the other way, from − to +.

A direct current 直流电 (d.c.) always flows one way, as from a battery. An alternating current 交流电 (a.c.) keeps swapping direction many times each second, as from the mains supply 市电.

A cell gives energy 能量 to the charges that move through it. Two quantities describe this, and both are measured in volts 伏特 (V).

• The electromotive force 电动势 (e.m.f.) of a source is the electrical work it does to drive unit charge all the way round a complete circuit 电路.
• The potential difference 电势差 (p.d.) across a component is the work done by unit charge as it passes through that component.

A voltmeter 电压表 measures e.m.f. or p.d. It is joined in parallel (across the component). Like ammeters, voltmeters are analogue or digital and have different ranges: pick the range just above the largest voltage you expect.

Both are work done per unit charge:

where $W$ is the energy transferred (J) and $Q$ is the charge (C). For example, if a cell does $120\ \text{J}$ of work moving $60\ \text{C}$ of charge round a circuit, its e.m.f. is $120 / 60 = 2\ \text{V}$.

Resistance 电阻 measures how hard it is for current to flow. A bigger resistance gives a smaller current. It is measured in ohms 欧姆 (Ω).

Measuring resistance. Connect the component in series with a cell, a switch, an ammeter and a variable resistor 可变电阻器, and connect a voltmeter across the component. Close the switch, read $I$ and $V$, then change the variable resistor to get several more pairs of readings. Either calculate $R = V/I$ for each pair and take the average, or plot $V$ against $I$: for a metal wire at constant temperature the graph is a straight line through the origin, and its gradient is $R$.

Worked example. A $12\ \text{V}$ battery drives a current of $0.50\ \text{A}$ through a lamp. Find the lamp's resistance.

For a metal wire at constant temperature:

• a longer wire has a bigger resistance — resistance is directly proportional 成正比 to length, $R \propto l$;
• a thicker wire has a smaller resistance — resistance is inversely proportional 成反比 to cross-sectional area 横截面积, $R \propto \dfrac{1}{A}$.

So doubling the length doubles the resistance; doubling the area halves it. (Area depends on the diameter squared, so a small change in diameter changes the resistance a lot.)

Current–voltage graphs show how three components behave:

A circuit transfers energy from a source (a cell or the mains) to the components, and then into the surroundings (often as heat, light or sound).

Electrical power 电功率 is the energy transferred each second, measured in watts 瓦特 (W):

The electrical energy 电能 transferred in a time $t$, measured in joules 焦耳 (J), is:

Worked example. A $12\ \text{V}$ motor draws a current of $2.0\ \text{A}$. Find its power, and the energy it transfers in $30\ \text{s}$.

The kilowatt-hour

Home electricity bills use a bigger energy unit, the kilowatt-hour 千瓦时 (kWh). One kilowatt-hour is the energy used by a $1\ \text{kW}$ appliance 电器 in $1$ hour.

For example, a $2\ \text{kW}$ heater 加热器 used for $3$ hours uses $2 \times 3 = 6\ \text{kWh}$. If one kWh costs $20$ cents, the cost is $6 \times 20 = 120$ cents.

Circuit components

You should know these components and how they behave:

Series and parallel

In a series 串联 circuit the parts form one single loop:

• the current is the same at every point;
• the supply p.d. is shared between the components;
• cells in series add their e.m.f.s (when they point the same way);
• resistors in series add up: $R_{\text{total}} = R_1 + R_2 + \dots$

In a parallel 并联 circuit the parts are on separate branches:

• each branch gets the full supply p.d.;
• the current splits, so the current from the source is bigger than the current in any one branch;
• the total resistance 总电阻 is less than the smallest single resistance.

Lamps in a home are wired in parallel: each gets the full voltage, and if one fails the others stay on.

For calculations (with the rules above):

• at a junction 节点, the total current flowing in equals the total current flowing out;
• in series, the supply p.d. equals the sum of the p.d.s across the components;
• in parallel, the p.d. across each branch is the same;
• two resistors in parallel combine as

Worked example. A $6\ \Omega$ resistor and a $3\ \Omega$ resistor are connected in parallel. Find their combined resistance.

Notice the combined resistance ($2\ \Omega$) is smaller than the smaller of the two resistors.

Potential dividers

Two resistors in series share the supply voltage. This is a potential divider 分压器. For the same current, a bigger resistance has a bigger p.d. across it, so the bigger resistor takes the bigger share:

A variable potential divider (or a variable resistor) lets you change the output voltage smoothly, for example from $0\ \text{V}$ up to the full supply. If you use a thermistor or an LDR as one of the resistors, the output voltage changes with temperature or light — handy for switching a heater or a lamp on and off automatically.