3 Electric current, resistance, energy and power

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1 3 3.1 Introduction Having looked at static charges, we will now look at moving charges in the form of electric current. We will examine how current passes through conductors and the nature of resistance to the flow of current. Resistors as elements of electrical circuits are considered in various configurations. The section concludes with an analysis of energy conversion and power in DC circuits. 3.2 Electric current Until now we have mostly considered the behaviour and effects of charges under electrostatic conditions, i.e. electric charges that are stationary. Now we will begin to discuss the physics of electric currents in other words, charges that are moving. Electric currents can take many forms. Most commonly we think of electrons moving through a metal conductor (a wire, for example) but a current could also consist of electrons moving through a vacuum or a semiconductor, or ions moving through a liquid, gas or plasma (these are just a few examples). Figure 3.1 Andre-Marie Ampere. ( ) It is important to note that, although all currents involve moving charges, not all moving charge constitutes an electric current! The free electrons in the metal of an isolated length of wire are in constant motion. Their thermal energy causes them to move very rapidly (with speeds of the order of 10 6 ms -1 ) 27 but the direction of movement is essentially random there is no net transport of charge and so no current flows. However, if the wire is connected to a battery, an electric field is set up along the length of the wire. The electrons will continue to be in thermal motion but there will also be a drift of electrons, moving in the opposite direction to the direction of the field, giving rise to a net transport of charge and, therefore, an electric current in the wire. 27 The thermal velocity of a particle at a given temperature (T) can be found by equating its kinetic energy (½mv 2 ) to the thermal energy, given by 3 / 2 kt, where k is the Boltzmann constant. World of the electron: part 1 35

2 Electric current is defined as the rate at which charge passes a point. So, if a total charge of Q coulombs flows past a given point at a uniform rate for t seconds, we can write I Q = (3.1) t where I is the current. The unit for current is the coulomb per second, or the ampere (A) commonly shortened to amp with 1 ampere being equal to 1 coulomb per second Conventional current direction When considering current flow, the convention is to describe the direction of the flow of current as the direction in which positive charge carriers would move, even if the actual charge carriers are negative (e.g. electrons) and move in the opposite direction. We can use this convention because, in most situations, the assumed motion of positive charges in one direction has the same effect as the actual motion of negative charge carriers in the opposite direction. The reason for this slightly counter-intuitive choice is that the convention of current direction was established before the discovery of the electron and its role as a charge carrier in metals. 3.4 Drift speed of electrons in a conductor As stated earlier, when a current flows through a conductor the electrons move randomly due to their thermal energy but, superimposed on this random motion, the electrons begin to drift in the opposite direction to the direction of the electric field that is giving rise to the current. We can derive a simple expression that relates the size of the current flowing through a conductor to the average drift speed (v) of the electrons. Consider the section of conductor shown in Figure 3.2, with length, L, and cross-sectional area of A. If we assume that the electrons have an average drift speed of v metres per second, then the time interval (t) in which all of the electrons in this section will pass through the end of the L section is simply given by t = v 28 In this form it may seem that the ampere, as a unit, was derived in terms of the coulomb. Actually, it is the other way round the ampere is one of the base SI units and the coulomb is formally defined as the quantity of charge which flows in one second past a point in a conductor which is carrying a steady current of one ampere. The definition of the ampere comes from the application of Ampere s force law. World of the electron: part 1 36

3 Figure 3.2 Current flow and electron drift in a conductor Now, we can work out the total quantity of charge that all these electrons carry, assuming that there are n free electrons per unit volume: The volume of the section of conductor is just L A So, the number of electrons in the section is given by n LA And the total quantity of charge on all these electrons is given by Q = e nla is the charge on a single electron. where e Now, since the current (I) is defined as the rate of transfer of charge, we can write I = Q enla enav t = L = v I v = (3.2) ena Example: What is the drift speed of the free electrons in a copper wire with cross-sectional area of 1 mm 2 when it has a steady current of 1 amp flowing through it? Assume that each copper atom contributes one conduction (free) electron to the current. Additional information: Avogadro s number (i.e. the number of atoms per mole of material) = mol -1 Molecular weight of copper = 64 gmol -1 (i.e. 1 mole of copper weighs 64 g) Density of copper = 8900 kgm -3 Elementary charge = C World of the electron: part 1 37

4 We know that I = 1 A and A = m 2. In order to calculate v we just need to work out n (the number of free electrons per unit volume), which is equal to the number of copper atoms per unit volume. There are free electrons in kg of copper. The density of copper is 8900 kgm So, there are Substituting these figures into equation (3.2) gives I 1 v = = ena = ms 26 cm per hour! 28 = free electrons per m 3 of copper i.e. n = m Question: So why do the lights come on so quickly when you flick the switch? 3.5 Mechanism of conduction in metals In the previous section we showed that the average drift speed of electrons carrying a current through a metal conductor is of the order of centimetres per hour orders of magnitude less than the speeds of electrons in a vacuum under the influence of an electric field. Clearly, electrons travelling through a metal conductor encounter a considerable resistance to their movement. It is useful to think, on the microscopic level, about the mechanisms of electron transport in metals. At the microscopic level we can think of the metal as a crystal structure (or lattice) comprising an ordered 3D array of positive metal ions. The outer (valence) electrons from the metal atoms are free to move through the lattice, in the corridors between the rows of ions. Generally, the free electrons move randomly, their speed and direction continually changing as they collide with metal ions in the lattice 29. In the presence of an electric field, however, there will be a superimposed drift of these electrons along the corridors in the lattice. The electrons still undergo repeated collisions with the metal ions but, between collisions, the field s influence on the electrons tends to move them in the opposite direction to the field, and a current flows (see Figure 3.3a). The collisions contribute to the conductor s resistance. 29 The mean free path is defined as the average distance travelled between collisions. The mean free path of an electron in copper, at room temperature, is approximately 40 nm. This is roughly equivalent to the length of a row of 160 copper atoms World of the electron: part 1 38

5 Figure 3.3 Representation of the conduction pathways within the lattice of a metal conductor a) for an 'ideal' crystal lattice b) at elevated temperatures and c) with the addition of impurity atoms Of course thermal energy means that, rather than being stationary, the metal ions are constantly vibrating about their mean positions in the crystal lattice (Figure 3.3b). These vibrations effectively cause the ions to have a larger cross-section and so electrons are more likely to collide with them. The higher the temperature, the larger the vibrations are and the more resistance there is to the drift of electrons 30. We should also note that when a current is flowing the conduction electrons, between collisions, are gaining kinetic energy as they are accelerated by the field. Some of this energy is transferred to the metal ions, during collisions, in the form of heat. So, the temperature of 31 the conductor increases as it passes current. The increase in temperature, as described above, increases resistance to the passing of electrons which, in turn, generates more heat and so on (equilibrium is eventually reached, with the conductor losing as much heat to its surroundings as it gains from the passage of electrons). In addition to temperature, the resistance to the passage of electrons in metal conductors is affected by the presence of impurity (or alloying) atoms. If the impurity ion is a different size to the host ions then it will distort the electron pathways through the lattice and increase the 30 As the temperature increases, the mean free path becomes shorter 31 The conversion of electricity to heat in conductors is obviously an important effect. It s the basis of so-called resistive heaters but, in many cases, the heat represents a wasteful loss of energy for example, in the case of electric motor windings World of the electron: part 1 39

6 resistance to current flow (Figure 3.3c). The mass of the impurity atom is also important because it affects how much it vibrates when an electron hits it and so influences the transfer of heat energy into the lattice. The charge on the ion (if it is different to that of the host ions) can also affect resistance by creating electrical irregularities in the electron pathways. 3.6 Resistance and resistivity For the case of a metal conductor we have discussed some of the factors that might make it either a good conductor (i.e. one with a low resistance to the passage of electrons) or a bad conductor (i.e. one with a high resistance). We should now quantify exactly what we mean by resistance. Resistance, R, is defined as the ratio of the potential difference (V) across a conductor to the current (I) that passes through it as a result, thus V R = or V = IR (3.3) I The unit of resistance is, therefore, the volt per ampere, which is called the ohm (Ω) 1 ohm = 1 Ω = 1 volt per ampere Examples: a) There is a current of 0.20 A in a wire when the potential difference between its ends is 5.0 V. What is its resistance? b) The opposite faces of a sheet of polythene are covered with metal foil. When the potential difference between the two layers of foil is 12 V, the current through the polythene is A. What is the resistance of the polythene? and c) What potential difference must be applied to a 10MΩ resistor to drive a current of 5.0 µa through it? These answers to all these problems come from simple applications of equation (3.3). a) V 5.0 R = = I 0.20 R = 25.0 Ω V I R b) R = = = Ω c) V = IR = V = 50 V 6 6 World of the electron: part 1 40

7 From our discussion of the mechanisms of conduction (section 3.5) it should be obvious that the resistance of a conductor (e.g. a length of wire) depends on the dimensions of that conductor. If we double the cross-sectional area of the conductor (A in Figure 3.2), while keeping everything else constant, then twice as much charge can pass and so the current is doubled. Alternatively, if we keep everything the same but double the length (L) of the conductor, then the charge has travel twice as far and encounters twice the resistance so the current will be halved (because I = V/R). In other words, the resistance of a conductor is proportional to the length, and inversely proportional to the cross-sectional area, of the conductor 32. We can write R L ρ A = (3.4) where the constant of proportionality, ρ, is known as the resistivity of the material from which the conductor is made. Since L is measured in metres and A in square metres, it can be seen by rearranging equation (3.4) that the unit of resistivity is the Ωm (ohm-metre). Material Resistivity (Ωm) Typical metals (conductors) Silver Copper Aluminium Iron Tungsten Typical semiconductor Silicon Typical insulators Alumina Rubber approx Teflon Table 3.1 Resistivities of various materials at room temperature. 1 this is for pure silicon the resistivity of silicon can be as low as 10-4, depending on impurities. It is useful to remember this important distinction: resistance is a property of an object, resistivity is a property of a material. In choosing materials for making a conductor (such as domestic wiring, overhead power cables, bulb filaments etc) resistivity is clearly an important consideration. Generally, the lower the resistivity, the less energy is wasted as heat when a current flows. However, other factors (such as cost, weight, melting point) will also be important. When talking about material properties of conductors, the term conductivity is often used. Conductivity (σ) is simply the reciprocal of resistivity a material with a high resistivity has a low conductivity, and vice versa: The unit of conductivity is Ω -1 m σ = ρ 32 We assume that the cross-section is uniform and the material of the conductor is isotropic and homogeneous meaning that the resistance doesn t depend on the direction of the current or which bit of the conductor the charge is passing through. World of the electron: part 1 41

8 3.7 Ohm s law Ohm s law (first published by Georg Ohm in 1827) can be stated in the following way: the current through a device that obeys Ohm s law is always directly proportional to the potential difference applied across the device 33. In equation form it is most appropriately written V I = constant where the constant will be equal to the resistance of the device. It may seem that this is just a restatement of equation (3.3). Indeed, the equation R = V/I is often referred to as Ohm s law, but this is incorrect. Equation (3.3) defines resistance but only for one particular pair of values of V and I. In many cases the resistance of a device is dependent upon the applied potential difference and so, if the potential difference is changed, dividing by the resulting current will still give a value of resistance, but this value will have changed. A conducting device obeys Ohm s law when the resistance of the device is independent of the magnitude and polarity of the applied potential difference. The simplest way of determining whether something obeys Ohm s law is to plot its I-V characteristics; in other words to draw a graph of I versus V for the device. If Ohm s law applies (V/I is constant) then clearly the graph will be a straight line and the resistance can be found by taking the reciprocal of the gradient. If the graph is not a straight line, then it means that the resistance is not constant but depends on the applied voltage and Ohm s law does not apply. Figure 3.4 illustrates typical I-V plots for various devices in various situations. Figure 3.4 Typical I-V characteristics for a number of devices a) a metal resistor at two different temperatures b) wire conductors with different dimensions c) bulb filament 33 In this context we take device to mean anything from a length of wire or metal resistor to something more complicated like a semiconductor diode World of the electron: part 1 42

9 It is immediately obvious that the devices represented in Figure 3.4 (a) and (b) obey Ohm s law (at least over this range of potential difference), whereas the device in part (c) does not. Figure 3.4 (a) shows the behaviour of a simple metal resistor at two different temperatures. At the higher temperature, as we would expect (see section 3.5), the resistance of the device is higher (remember, resistance is given by 1/gradient). But, over the range of applied potential difference shown here, Ohm s law is obeyed at both the high and low temperature. Metals, under moderate conditions, generally obey Ohm s law. However, if a relatively large current flows through an appreciable resistance, as we have mentioned before, then heat will be generated and this can cause the resistance to increase. This is the situation illustrated in Figure 3.4 (c), which shows the I-V characteristic of a light-bulb filament. The higher the potential difference, the more current flows and the hotter the filament gets and as it gets hotter, the resistance increases (shown by the gradually decreasing gradient). Figure 3.4 (b) shows two different ohmic plots and illustrates how the resistance of a wire conductor depends on its dimensions. The resistance of the shorter and/or thicker conductor is less than that of the longer and/or thinner one. 3.8 Resistors in series and in parallel Having discussed resistance in general terms we will now look at resistors as discrete electronic components that, along with capacitors and inductors, form the basic building blocks of almost all electrical circuits. As with capacitors (see section 2.6) resistors can be connected together either in series or in parallel. When designing or analysing circuits, it is therefore necessary to be able to work out the total equivalent resistance of both these configurations. First let s look at resistors in series. Figure 3.5 shows three resistors, with resistances of R 1, R 2 and R 3, connected in series. Figure 3.5 Resistors in series World of the electron: part 1 43

10 The same current (I) flows through all three resistors (a charge moving around the circuit can only follow one route a route that takes it through all three resistors, one after the other). It is also apparent that the applied potential difference, V, across all three resistors is equal to sum of the potential differences across the three individual resistors 34. In equation form And from equation (3.3) we can write V = V + V + V V = IR where R is the resistance that is equivalent to the three resistors in series. For the individual resistors, R 1, R 2 and R 3, we can write Combining these expressions gives V = IR, V = IR, V = IR IR = IR + IR + IR And, dividing by I (which is the same throughout) gives us the following expression for the total equivalent resistance of three resistors in series R= R + R + R (3.5) This equation can be extended to apply to any number of resistors. Now let s look at the situation when resistors are connected in parallel, as shown in Figure 3.6. In such cases it is apparent, looking at the circuit diagram, that there is the same potential difference, V, across each resistor. The total current, I, is divided between the three resistors and, because charge is conserved, we can say From equation (3.3) I = I + I + I V V V V I =, I =, I = and I = R R R R where R is the equivalent resistance of the parallel combination. 34 In microscopic terms, charge passing through the three resistors will have lost an amount of energy in the form of heat equivalent to the total drop in potential (V) across the resistors World of the electron: part 1 44

11 Figure 3.6 Resistors in parallel Combining these expressions gives V V V V = + + R R R R and dividing through by V gives the following expression for determining the equivalent resistance of three resistors in parallel (although this equation too can be extended for any number of resistors) = + + (3.6) R R R R Example: Calculate a) the total resistance and b) the current I for the circuit shown below: World of the electron: part 1 45

12 It can help to redraw the circuit in a different layout the circuit below is exactly equivalent electrically but makes the problem clearer. a) Evaluating the parallel combination first using equation (3.6) = + = R R = 6 Ω Therefore, from equation (3.5), the total resistance of the circuit becomes b) From equation (3.3), we have R = = 12 Ω I V = = R I = 2 A Problem 6: What is the smallest number of resistors needed to provide a resistance of a) 5 Ω, given a supply of 3 Ω resistors b) 7 Ω, given a supply of 4 Ω resistors In each case draw a diagram to show how you would connect them. [Answers: a) four b) five] World of the electron: part 1 46

13 3.9 Practical resistors Practical resistors come in a huge variety of shapes, sizes and materials and with resistance values ranging from ohms to megohms (1 MΩ = 10 6 Ω). They can be discrete devices for connecting into circuits or micron-scale components of complex integrated circuits or systems. The resistive elements are typically made from carbon, metal or conducting ceramics (metal oxides) and may be in the form of thick or thin films, wire sections or wire coils. Figure 3.7 Various resistors Wirewound resistors can have multiple tapping (connection) points or an adjustable (sliding) tapping point to provide variable resistance from a single device Batteries, EMF and internal resistance An electric cell is a device that uses chemical energy to raise charge through a potential difference. Equation (2.3) (W = qv) tells us that a 12 V cell will give 12 joules of energy to each coulomb passing through it. The resulting electrical potential energy can then be utilised when the cell is connected into a circuit. A battery is simply a number of electric cells connected together in one single unit. When a cell is not being used, no current is being drawn and no charge is moving through the interior of the cell (see Figure 3.8a). The potential difference between the terminals (the terminal potential difference) in such a case is known as the electromotive force or e.m.f. of the cell. Electromotive force is denoted by E, or sometimes E. The e.m.f. is the amount of energy per coulomb converted into electrical energy by the cell. However, if a cell is being used to maintain a current, for example through the resistance R shown in Figure 3.8b, then it is found that the potential difference, V, across the terminals of the cell, and so across the resistor, is less than the e.m.f. This is because the cell itself has its own internal resistance (denoted r) Figure 3.8 Circuit diagrams representing electromotive force (E) and internal World of the electron: part 1 47

14 through which the current must be driven. Part of the cell s e.m.f. is used to drive the current through this internal resistance, which leaves a reduced potential difference to drive the current through the external circuit. In other words, energy is consumed by the cell as well as by what it is connected to. It is convenient to think of, and represent, the internal resistance as an additional resistor connected in series with the cell and the external circuit, as shown in Figure 3.8c. In such a case, the total resistance is (R + r) and so, from equation (3.3) E = I( R+ r) = IR + Ir but V = IR, which represents the potential difference across the external resistance, and so we have E = V + Ir (3.7) V = E Ir In these expressions Ir is the inaccessible voltage that is lost across the resistance of the cell as heat. Notice that the useful voltage, V, of the cell decreases as the current drawn from the cell increases. Also, it is clear that when I = 0 the terminal potential difference is equal to the e.m.f., as previously stated. It has already been said that a battery is just a number of cells connected together. The cells can be connected either in series or in parallel and the way in which they are connected will obviously determine the overall e.m.f. and internal resistance of the battery. If the cells are connected in series, as shown in Figure 3.9a, then the total e.m.f. is simply found by adding the individual components. Similarly, as you d expect for resistors in series, the total internal resistance is just the sum of the individual internal resistances. So and E = E 1 + E 2 + E E n r = r 1 + r 2 + r r n So, for example, 12 V car batteries are typically made up of six 2.0 V lead-acid cells connected in series and the internal resistance, r battery, is equal to 6 r cell. Figure 3.9 Batteries consisting of identical cells connected a) in series and b) in parallel World of the electron: part 1 48

15 If, as in Figure 3.9b, the battery consists of identical cells connected in parallel then the combined e.m.f. is the same as that of the individual cells 35. The combined internal resistance can be worked out from equation (3.6) = + + r r r r but, since r 1 = r 2 = r 3 = r cell 1 3 = r r cell r = r 3 cell and, more generally, for a battery made up of n cells r = battery r n cell Problem 7: A battery is connected across a variable resistor. With the resistance set to 21 Ω, the current through the resistor is 0.48 A. When the resistor is at 36 Ω, the current is 0.30 A. What are the e.m.f. (E) and the internal resistance (r) of the battery? [Answer: E = 12 V and r = 4 Ω] 3.11 Energy and power in DC circuits Consider a simple circuit in which a steady current (I) flows through a load, such as a resistor or an electric motor. As the current flows, energy is dissipated in the load. In the case of a resistor the energy is lost in the form of heat. In the case of a motor, the electrical energy is converted to mechanical energy although some energy will be lost as heat due to the resistance of the motor windings. In all cases the energy dissipated is equal to the electrical potential energy lost by the charge as it moves through the potential difference that exists across the input and output terminals of the load. 35 Although the e.m.f. of a battery consisting of cells connected in parallel is the same as that of the individual cells, the capacity of the battery is increased correspondingly. Capacity is measured in ampere-hours (Ah), with 1 ampere-hour being the amount of charge transferred by a steady current of 1 amp in 1 hour (1 Ah = 3600 C). World of the electron: part 1 49

16 It follows from our definition of potential difference (section 2.2) and from equation (2.3) that the energy dissipated is given by W = qv where W is the energy dissipated in some time, t q is the charge which flows in time, t V is the potential difference across the load Since the current, I, is steady we can say q = It and then substituting this in the above expression gives W = ItV Power, P, is defined as the rate of conversion of energy and, since I and V are constant, we can express power simply as W/t (joules per second). So, now we can write P = IV (3.8) The unit of power is the watt, W, where 1 W is equal to 1 joule per second 36. Equation (3.8) applies to any load it tells us the rate at which electrical energy is converted in a device, regardless of what form the energy is being converted into. For example, it applies equally to resistors, where the energy is converted to heat, and electric motors, where the energy is converted into a combination of mechanical energy and heat. In the case of resistors (or other devices where all the energy is being dissipated as heat) we can use equation (3.3), our definition of resistance, to rewrite equation (3.8) and eliminate I or V, as required V R = I 2 2 V P= IR= R It must, however, be remembered that these expressions only apply to devices that are purely resistive in nature. 36 You may sometimes see energy values expressed in kilowatt hours (kwh). One kilowatt hour is the energy dissipated, or consumed, by a power of 1000 W in 1 hour i.e. 1 kwh = 1000 W 3600 seconds = J. World of the electron: part 1 50

17 Example: An electric motor running on 30 V draws a current of 2 A. The motor windings have a resistance of 4 Ω. What is the rate at which the motor is converting energy into mechanical energy? The total power of the motor can be found from equation (3.8) P = IV = 2 30 = 60 W However, this 60 W of energy is split between the mechanical output of the motor and the heat generated by the resistance of the motor windings. The windings can be treated as resistors with R = 4 Ω, so we can use the expression 37 P = I 2 R = = 16 W This tells us that 16 W of power is lost in heating the motor windings. Therefore, the power left for the production of mechanical energy is given by = 44 W. So, 44 W is the rate of production of mechanical energy by the motor. 37 When finding the power lost in the windings, we use the expression for power in which voltage (V) has been eliminated. Don t make the mistake of using P = V 2 /R, with V = 30 V. The figure of 30 V is the potential difference between the input and output terminals of the motor as a whole. It is not the same as the potential difference just across the windings (if we want, we can work this voltage out from V = IR = 2 4 = 8 Ω). World of the electron: part 1 51