Current electricity

An ampere-hour is current multiplied by time: 1\,\mathrm{Ah}=1\,\mathrm{A}\times3600\,\mathrm{s}=3600\,\mathrm{C}. Since \Delta Q=I\,\Delta t, the product of current and time is charge, so the unit is a unit of charge.

Current is measured in amperes (A), energy in joules (J) and resistance in ohms (\Omega), so options A, C and D are wrong.

A diode is a non-ohmic device. Its current does not rise in direct proportion to the voltage, so its V–I graph is curved (Figure 16.8 (d)). Non-ohmic devices break Ohm's law.

A copper wire and a carbon resistor are ohmic devices. The book lists both as examples of ohmic devices, so option D is wrong.

Electric current is measured with an ammeter, which is connected in series with the component (Section 16.8.1). A voltmeter measures potential difference (B), an ohmmeter measures resistance (C) and a meter rod measures length (D).

A voltmeter measures potential difference. It is connected in parallel with the component across which the potential difference is to be measured (Section 16.8.2).

Current is measured by an ammeter (B). Resistance and resistivity are calculated from measured values of V and I or from the material's dimensions, so they are not what a voltmeter reads.

The time must be in seconds: \Delta t=2\,\mathrm{min}=2\times60\,\mathrm{s}=120\,\mathrm{s}.

Option B comes from using 2 s instead of 120 s, and option A from leaving out the time altogether.

First find the charge that passes in 10 s, converting 1 mA to 1\times10^{-3}\,\mathrm{A}:

Each electron carries e=1.6\times10^{-19}\,\mathrm{C}, so the number of electrons is:

The other options are wrong by factors of 10, 100 and 1000, which come from misplacing the decimal point in the milliampere conversion.

Resistance depends on how many free charge carriers there are to carry the current. With more free electrons, the same voltage drives a larger current, so the resistance falls whatever the material. Option D is the reverse of this.

Temperature does not act the same way in every material. The resistance of a metal rises when it is heated, so option A is not always true. The resistance of a semiconductor or insulator falls when it is heated (Section 16.6.3), so option B is not always true either.

For a conductor, R=\rho\dfrac{L}{A}, so the resistance is inversely proportional to the cross-sectional area A (Section 16.6.2).

Length varies directly with resistance, not inversely (B is wrong). Temperature raises the resistance of a metal, so it is not an inverse relation (C is wrong), which makes D wrong too.

Conventional current is taken to flow from the positive to the negative terminal of the battery (Section 16.1.1). In a metal the charges that really move are electrons, which are negative, so they travel from the negative terminal to the positive terminal.

The two directions are therefore opposite, which makes B correct and D wrong.

In a semiconductor, rising temperature releases more charge carriers, so more current can flow and the resistance decreases (Section 16.6.3). Option A describes a metal, not a semiconductor.

The resistance does not stay constant (C) or become zero (D) when a semiconductor is heated.

The temperature coefficient is defined as:

The numerator and the denominator in this ratio both carry the unit of resistivity, so they cancel. What remains is one over the unit of temperature, \mathrm{K}^{-1} (or ^\circ\mathrm{C}^{-1}, Section 16.7.1). Option A is the unit of temperature itself, and options C and D keep an unwanted ohm.

The temperature coefficient of resistivity is negative for semiconductors (carbon, germanium and silicon in Table 16.2). Their resistivity therefore decreases steadily as the temperature rises.

Option A is the behaviour of metals, and options C and D do not describe semiconductors.

In series

In a series connection the sources are joined end to end. The positive terminal of one source is connected to the negative terminal of the next, so the emfs add up and the same current flows through every source.

In parallel

In a parallel connection all the positive terminals are joined to one wire and all the negative terminals to the other. The sources then share the load, and the current is divided between them. The emf of the group is the same as the emf of one source, provided that the four sources have equal emfs.

Both instruments are electrical measuring instruments used to find properties of a circuit. They measure different quantities, they are connected in different ways, and they are designed with opposite resistances so that they do not disturb the circuit.

What each instrument measures

• An ammeter measures the electric current, in amperes (A). It reads the current flowing through the component it is connected to.
• A voltmeter measures the potential difference (p.d.), in volts (V). It reads the energy transferred per coulomb across the component it is connected across.

How each is connected

An ammeter is connected in series with the component, so that the whole current of the component passes through it. A voltmeter is connected in parallel with the component, so that it reads the p.d. across that component.

Design and effect on the circuit

An ammeter must not change the current it measures, so its resistance is made as low as possible (of the order of 1\,\Omega). A voltmeter must not change the p.d. it measures, so its resistance is made as high as possible (of the order of 10\,\mathrm{M}\Omega).

Why the connection matters

If an ammeter were connected in parallel, its very low resistance would divert most of the current away from the component, so the reading would be wrong and the circuit altered. If a voltmeter were connected in series, its very high resistance would cut the current almost to zero, so the component would hardly work at all.

In short, the ammeter tells you how much charge flows each second through a component, and the voltmeter tells you how much energy each coulomb gives up across it.

Silver is a slightly better conductor than copper. From Table 16.1, the resistivity of silver is 1.59\times10^{-8}\,\Omega\,\mathrm{m} and that of copper is 1.72\times10^{-8}\,\Omega\,\mathrm{m}, and the lower value means less resistance for the same wire. The difference is only about 8\%, so conductivity is not the deciding factor.

Copper is far cheaper than silver and is much more widely available, which keeps the cost of long lengths of household wiring low. It is also strong and ductile, so it can be drawn into long wires without breaking. Together these make copper the practical choice for wiring, even though silver conducts a little better.

Free electrons in a metal are in constant, random thermal motion, moving in every direction. Because the motion is random, the electrons on average cross any cross-section of the wire equally in both directions, so the net movement of charge is zero and there is no current (Section 16.1.2).

When a potential difference is applied, the battery sets up an electric field in the wire. The electrons then drift slowly in one direction, and the net flow of charge in that direction is the electric current. Without a potential difference there is no field, so the drift does not start.

In a metal, the actual current is a flow of electrons from the negative terminal of the battery through the external circuit to the positive terminal. The conventional current is taken in the opposite sense, from the positive terminal through the circuit back to the negative terminal (Section 16.1.1).

The two directions describe the same flow. Both give the same effects in the circuit, such as heating a filament, so the behaviour of the circuit does not depend on which direction is used. What matters is that each quantity is used consistently.

Conventional current is used in diagrams and calculations because it was adopted before the electron was discovered, and it is used in every circuit rule, such as Ohm's law and the arrangement of series and parallel components. Using it keeps the signs of the potential differences and emfs correct, so the calculations give the right answers.

The distinction matters because the source supplies the energy and the components use it. When the current in the circuit is worked out with Ohm's law, the potential difference across each component is what enters the calculation, while the emf describes the source that maintains it.

A metal wire has equal numbers of positive protons in its nuclei and negative electrons around them, so each atom is neutral. The free electrons that carry the current are only moved along the wire, not created or destroyed.

When current flows, electrons enter one end of the wire at the same rate as they leave the other end. The wire therefore never builds up a net charge, and the whole wire stays neutral while the current flows through it.

Yes, a filament bulb would still light. Reversing the battery reverses the potential difference across the bulb, and by Ohm's law the current reverses with it. The resistance of the filament is the same, and the heating effect of the current, which makes the filament glow, does not depend on the direction of the current.

The current would flow in the opposite direction through the circuit. Devices that only work one way round, such as a diode, would stop conducting in the reversed connection, so the circuit would not behave in the same way.

Ohm's law, V=IR, holds only for ohmic devices, such as metals at constant temperature. For these the V–I graph is a straight line through the origin (Figure 16.8 (a)).

• Non-ohmic materials. A filament bulb, a thermistor and a semiconductor diode do not obey Ohm's law. Their V–I graphs are curved (Figure 16.8 (b) to (d)).
• Temperature. The law assumes that the temperature is constant. When the current heats a conductor, its resistance changes, so the ratio V/I changes too.
• Range of voltage. A metal obeys the law only over a limited range of voltages. At high voltages the heating is large and the straight line no longer holds.
• Extreme conditions. High temperatures or very large voltages make the material depart from the straight-line graph, so the law can be used only for the range over which it is tested.

In practice, Ohm's law is a useful model for metals in ordinary conditions, and it must be checked before it is used for other materials or extreme conditions.

The resistance of a wire depends on its length and cross-sectional area, as R=\rho\dfrac{L}{A}. A longer wire has a larger resistance, because the electrons collide more often along a longer path. A wire with a smaller cross-sectional area has a larger resistance, because there is less room for the current to pass through.

Bending a wire does not change its resistance on its own. The wire keeps the same length, cross-section and material, so the value of R is unchanged. Only if the bend stretches or squeezes the wire, so that L or A changes, is the resistance changed.

The temperature coefficient of resistivity is positive for copper, so its resistivity increases as the temperature rises. The relation is:

With \alpha=0.004\,^\circ\mathrm{C}^{-1}, a rise of 25\,^\circ\mathrm{C} raises the resistivity by 0.004\times25=0.10, or 10\%. The resistance of a copper wire rises by the same fraction, so for a fixed voltage the current falls.

In electrical systems, copper wires in a warm environment carry less current for the same voltage and lose more energy as heat in the wire. The designer must allow for this when a system operates over a wide range of temperatures. (The book's own value for copper is \alpha=3.9\times10^{-3}\,\mathrm{K}^{-1}, which the rounded value 0.004 approximates.)

Fluid resistance is the opposition that a pipe offers to the flow of a fluid through it. It is defined in the same way as electrical resistance, which the book describes as the opposition to the flow of charges.

By analogy with the electrical case, the fluid resistance of a pipe rises with its length and falls with its cross-sectional area, as R\propto\dfrac{L}{A}. A long, narrow pipe therefore opposes the flow of water more than a short, wide pipe does.

Electric current

Electric current is the time rate of flow of electric charge through a cross-sectional area of a conductor. If a charge \Delta Q flows through a cross-section in time \Delta t, the current is:

The SI unit of current is the ampere (A). One ampere is the current when one coulomb of charge flows through a cross-section in one second, so 1\,\mathrm{A}=1\,\mathrm{C\,s^{-1}}. Smaller units are the milliampere, 1\,\mathrm{mA}=10^{-3}\,\mathrm{A}, and the microampere, 1\,\mu\mathrm{A}=10^{-6}\,\mathrm{A}.

Charge flow

Current can be produced by positive charges moving in one direction, by negative charges moving in the opposite direction, or by both at once. In a metal the charges that move are electrons, which are negative. In a gas or an electrolyte, positive and negative charges both move.

Conventional and actual current

The actual current is the flow of the charges that really move. In a metal this is a flow of electrons from the negative terminal to the positive terminal of the battery. The conventional current is the flow of positive charge, taken from the positive terminal through the circuit to the negative terminal. The two currents are equal in size and opposite in direction (Section 16.1.1).

When the distinction matters

The distinction does not change the effects of the current, such as heating or the light from a bulb. It matters when the physical motion of the charges is being described, for example when the electrons in a metal are considered. For diagrams, for Ohm's law and for every calculation, the conventional direction is used, so that the signs of the currents and potential differences are consistent.

Conduction in metals

In a metal the valence electrons are not tightly bound to the nuclei, so they move freely through the metal like gas molecules in a container. With no battery connected, their net motion is zero and there is no current. When a battery is connected, the electric field makes the electrons drift in one direction, and the net drift of electrons is the current. The electrons keep colliding with the ions of the metal, which slows them down (Section 16.1.2).

Effect of temperature

When the temperature rises, the positive ions vibrate with greater amplitude. The electrons then collide with the ions more often, so their motion is hindered and the current for a given electric field is reduced (Section 16.7.1). The resistivity of a metal therefore rises with temperature, and the conduction efficiency falls.

Magnetic fields

This unit does not treat the effect of magnetic fields on moving charges. That topic belongs to the unit on Electromagnetism (Unit 19), so the answer for this unit is limited to the electric field and the temperature. The electric field supplies the drift of the electrons, while the temperature decides how often they collide and so how efficiently the metal conducts.

Direct and alternating current

In direct current (DC) the electrons flow in one fixed direction. The current may be steady or may vary in size, as shown in Figure 16.4 (a). In alternating current (AC) the electrons go forward for half the time and backward for the other half, so the direction of the current changes continuously (Section 16.2).

Behaviour in circuit elements

At any instant, a resistor obeys Ohm's law for both kinds of current, so V=IR holds whether the current is DC or AC. The current through a resistor follows the polarity of the voltage. A diode, which is non-ohmic, conducts only in one direction, so it does not behave the same way for AC, where the current reverses.

Pros and cons

In summary, DC is simple and steady, and it suits batteries and electronics. AC is easy to generate and to change in voltage, so it is preferred for transmission, but at high voltage it is more hazardous to people.

Definition

The emf of a source is the energy supplied by the source per unit positive charge when the charge flows through the complete circuit. If W is the work done by the battery on a charge q, then:

The SI unit of emf is the same as that of electric potential, \mathrm{J\,C^{-1}}=\mathrm{V}. A source of emf converts non-electrical energy into electrical energy. Examples are batteries (chemical to electrical), solar panels (light to electrical), thermocouples (heat to electrical) and generators (mechanical to electrical).

Role in maintaining current

Inside the battery, positive charge moves from the low potential terminal to the high potential terminal. Work must be done on the charge to drive it uphill in potential, and the battery does this using its stored energy. Outside the battery, the charge gives up its energy in the devices in the circuit.

Without an emf source, the charges would lose their energy and stop moving once the potential difference had been used up. The emf is therefore the cause of the steady current in a closed circuit.

Definitions

The emf of a battery is the energy it gives to each coulomb of charge when the charge goes round the closed circuit, \varepsilon=W/q. The potential difference across a component is the energy each coulomb gives up in that component, V=\Delta U/\Delta q (Equation 16.2).

Real-world example: a torch

Two cells of 1.5\,\mathrm{V} each, connected in series, have a total emf of 2\times1.5\,\mathrm{V}=3.0\,\mathrm{V} (the same arrangement as Example 16.2). This is the energy that the cells supply to each coulomb of charge. When the bulb is lit, the potential difference across the bulb is 3.0\,\mathrm{V} and the current through it is 0.15\,\mathrm{A}, so the bulb uses 3.0\,\mathrm{J} of energy from each coulomb that passes through it.

The battery's emf and the bulb's potential difference have the same value here because the wires have negligible resistance, so the whole emf appears across the bulb. The two quantities are different in concept: the emf is what the battery provides, and the potential difference is what the bulb receives.

Why the distinction matters

The emf is a property of the source. The potential difference is a property of a component, and it changes with the component. Ohm's law and all circuit calculations use the potential difference across each component, while the emf describes the source that maintains it.

Ohm's law and its statement

Ohm's law states that the current through a conductor is directly proportional to the potential difference across it, as long as the temperature and the physical state of the conductor are kept constant (Section 16.5). It is written V=IR, where R is a constant for the conductor.

Limitations

• Ohm's law is applicable only to materials whose V–I graph is a straight line through the origin.
• Only metallic conductors show this behaviour, and only over a limited range of voltages. The temperature must also stay constant.
• Many important devices, such as a filament bulb, a thermistor and a semiconductor diode, are non-ohmic. Their graphs are curved, so V/I is not constant (Figure 16.8).

The importance of I-V graphs

A plot of current against voltage shows directly whether a device is ohmic. A straight line through the origin means that R=V/I is the same at every voltage. A curve means that the resistance changes with voltage or current, and the shape of the curve shows how it changes. The graph below sketches the three shapes described in the book, on the same axes.

Experiments to explore non-linear materials

Relationship between resistance and resistivity

The resistance of a uniform wire depends on its length and cross-sectional area. Keeping the temperature constant, the resistance is directly proportional to the length and inversely proportional to the cross-sectional area (Section 16.7):

The resistivity \rho is a property of the material only. It is the resistance of a wire of length 1\,\mathrm{m} and cross-sectional area 1\,\mathrm{m}^2, so the resistance of a given wire is its resistivity multiplied by a factor that depends on its shape.

Experiment design

Factors affecting accuracy

• Temperature. The current heats the wire, so its resistance rises during the reading. Keep the current small and let the wire cool between readings.
• Zero error. If the needles do not rest on zero, every reading is shifted. Correct the zero before starting.
• Least count. The smallest division limits the precision of each reading, so use the most sensitive range that does not overload the instrument.
• Contact and connections. Poor joints add a resistance that is not part of the wire, so the connections must be clean and tight.

With these steps and precautions, the average of several readings gives a reliable value of the resistance of the wire.

Factors affecting resistance

Practical applications

• Wiring. Connecting wires must have very low resistance, so that little energy is wasted as heat. Copper, with a low resistivity, is the usual choice, and the wire is made thick (large A) and kept short (small L).
• Heating elements. A heater needs a large resistance so that the current produces a lot of heat. A material of high resistivity is chosen, and the wire is made long and thin.
• Filaments. A tungsten filament is a conductor with a fairly high resistivity. When it is cold its resistance is low, so a large current flows at switch-on, which heats the filament to a high temperature.

Ways to optimise resistance

To reduce resistance, use a material with a low resistivity, a short length and a large cross-sectional area. To increase it, do the reverse. The temperature of the material must also be considered, because a warm metal has a higher resistance than the same metal when cold.

Metals

For a metal, the resistance increases as the temperature rises, because the ions vibrate more and collide with the electrons more often (Section 16.6.3). The relation between the resistivity and the temperature is:

The graph below shows this straight-line increase for copper, starting from \rho_o=1.7\times10^{-8}\,\Omega\,\mathrm{m} at 0\,^\circ\mathrm{C}, with \alpha=3.9\times10^{-3}\,\mathrm{K}^{-1}.

Semiconductors and insulators

For insulators and semiconductors the resistance decreases as the temperature rises, because the thermal energy releases more charge carriers (Section 16.6.3). The temperature coefficient of these materials is negative (Table 16.2).

Implications for components

• Hot environments. A copper or tungsten wire has a higher resistance when it is hot, so for a fixed voltage it carries less current. A circuit designed in a cold place may therefore work differently in a hot one.
• Cold start. A tungsten filament is cold and has a low resistance when first switched on, so a large current flows and the filament heats up quickly. This is why a bulb often fails when it is switched on.
• Sensors. A thermistor is made of a semiconductor, and its resistance falls as it gets hot. It can therefore be used to measure temperature, because its resistance shows how hot its surroundings are.

Correct placement

An ammeter is placed in series with the component, so that all the current passes through it (Section 16.8.1). A voltmeter is placed in parallel with the component, so that it reads the potential difference across it (Section 16.8.2).

Effect of misconnection

• Ammeter in parallel. The ammeter has a very low resistance, of about 1\,\Omega. Connected in parallel with a component, it takes most of the current and gives a reading that is not the current through the component. The large current can also damage the meter.
• Voltmeter in series. The voltmeter has a very high resistance, of about 10\,\mathrm{M}\Omega. Connected in series, it limits the current to a tiny value, so the circuit hardly works and the reading is not the potential difference across the component.
• Wrong range. Choosing a range that is too small can overload the meter. Choosing one that is too large gives a reading with little precision.

Strategies to avoid errors

1. Draw the circuit first and check that the ammeter is in series and the voltmeter is in parallel.
2. Connect the positive terminal of each meter towards the positive side of the circuit, and the negative terminal towards the negative side.
3. Set the needles to zero before taking readings, and note the full-scale value and least count.
4. Start with the highest range and reduce it until the needle is in the upper half of the scale.
5. Disconnect the supply before changing the connections, and check the circuit once more before closing the switch.