Current Electricity number of electrons travelling in any direction will be equal to the number of electrons travelling in the opposite direction. So, there will be no net electric current. Let us now see what happens to such a piece of conductor if an electric field is applied. To focus our thoughts, imagine the conductor in the shape of a cylinder of radius R (Fig. 3.1). Suppose we now take two thin circular discs of a dielectric of the same radius and put positive charge +Q distributed over one disc and similarly –Q at the other disc. We attach the two discs on the two flat surfaces of the cylinder. An electric field will be created and is directed from the positive towards the FIGURE 3.1 Charges +Q and –Q put at the ends of a metallic cylinder. The electrons will drift because of the electric field created to neutralise the charges. The current thus will stop after a while unless the charges +Q and –Q are continuously replenished. negative charge. The electrons will be accelerated due to this field towards +Q. They will thus move to neutralise the charges. The electrons, as long as they are moving, will constitute an electric current. Hence in the situation considered, there will be a current for a very short while and no current thereafter. We can also imagine a mechanism where the ends of the cylinder are supplied with fresh charges to make up for any charges neutralised by electrons moving inside the conductor. In that case, there will be a steady electric field in the body of the conductor. This will result in a continuous current rather than a current for a short period of time. Mechanisms, which maintain a steady electric field are cells or batteries that we shall study later in this chapter. In the next sections, we shall study the steady current that results from a steady electric field in conductors. 3.4 OHM’S LAW A basic law regarding flow of currents was discovered by G.S. Ohm in 1828, long before the physical mechanism responsible for flow of currents was discovered. Imagine a conductor through which a current I is flowing and let V be the potential difference between the ends of the conductor. Then Ohm’s law states that V ∝ I or, V = R I (3.3) where the constant of proportionality R is called the resistance of the conductor. The SI units of resistance is ohm, and is denoted by the symbol Ω. The resistance R not only depends on the material of the conductor but also on the dimensions of the conductor. The dependence of R on the dimensions of the conductor can easily be determined as follows. Consider a conductor satisfying Eq. (3.3) to be in the form of a slab of length l and cross sectional area A [Fig. 3.2(a)]. Imagine placing two such identical slabs side by side [Fig. 3.2(b)], so that the length of the combination is 2l. The current flowing through the combination is the same as that flowing through either of the slabs. If V is the potential difference across the ends of the first slab, then V is also the potential difference across the ends of the second slab since the second slab is FIGURE 3.2 Illustrating the relation R = ρl/A for a rectangular slab of length l and area of cross-section A. Physics time more than τ and some less than τ. In other words, the time t in i Eq. (3.16) will be less than τ for some and more than τ for others as we go through the values of i = 1, 2 ..... N. The average value of ti then is τ (known as relaxation time). Thus, averaging Eq. (3.16) over the N-electrons at any given time t gives us for the average velocity v d e E v ≡(V ) =(v ) −(t ) dii i average average average m e E e E = 0– τ=− τ (3.17) mm This last result is surprising. It tells us that the electrons move with an average velocity which is independent of time, although electrons are accelerated. This is the phenomenon of drift and the velocity v in Eq. (3.17) is called the drift velocity. d Because of the drift, there will be net transport of charges across any area perpendicular to E. Consider a planar area A, located inside the conductor such that FIGURE 3.4 Current in a metallic the normal to the area is parallel to E conductor. The magnitude of current (Fig. 3.4). Then because of the drift, in an infinitesimal density in a metal is the magnitude of amount of time Δt, all electrons to the left of the area at charge contained in a cylinder of unit distances upto |vd|Δt would have crossed the area. If area and length v. dn is the number of free electrons per unit volume in the metal, then there are n Δt |v d|A such electrons. Since each electron carries a charge –e, the total charge transported across this area A to the right in time Δt is –ne A|vd|Δt. E is directed towards the left and hence the total charge transported along E across the area is negative of this. The amount of charge crossing the area A in time Δt is by definition [Eq. (3.2)] I Δt, where I is the magnitude of the current. Hence, I Δt =+ neA v Δt (3.18) d Substituting the value of |v | from Eq. (3.17) d e 2 A I Δt =τ n Δt E (3.19) m By definition I is related to the magnitude |j| of the current density by I = |j|A (3.20) Hence, from Eqs.(3.19) and (3.20), 2 ne j =τ E (3.21) m The vector j is parallel to E and hence we can write Eq. (3.21) in the vector form 2 ne j=τ E (3.22) m Comparison with Eq. (3.13) shows that Eq. (3.22) is exactly the Ohm’s98 law, if we identify the conductivity σ as Current Electricity Resistors in the higher range are made mostly from carbon. Carbon resistors are compact, inexpensive and thus find extensive use in electronic circuits. Carbon resistors are small in size and hence their values are given using a colour code. TABLE 3.2 RESISTOR COLOUR CODES Colour Number Multiplier Tolerance (%) Black 0 1 Brown 1 101 Red 2 102 Orange 3 103 Yellow 4 104 Green 5 105 Blue 6 106 Violet 7 107 Gray 8 108 White 9 109 Gold 10–1 5 Silver 10–2 10 No colour 20 The resistors have a set of co-axial coloured rings on them whose significance are listed in Table 3.2. The first two bands from the end indicate the first two significant figures of the resistance in ohms. The third band indicates the decimal multiplier (as listed in Table 3.2). The last band stands for tolerance or possible variation in percentage about the indicated values. Sometimes, this last band is absent and that indicates a tolerance of 20% (Fig. 3.8). For example, if the four colours are orange, blue, yellow and gold, the resistance value is 36 × 104 Ω, with a tolerence value of 5%. 3.8 TEMPERATURE DEPENDENCE OF RESISTIVITY The resistivity of a material is found to be dependent on the temperature. Different materials do not exhibit the FIGURE 3.8 Colour coded resistors same dependence on temperatures. Over a limited range (a) (22 × 102 Ω) ± 10%, of temperatures, that is not too large, the resistivity of a (b) (47 × 10 Ω) ± 5%. metallic conductor is approximately given by, ρ T = ρ 0 [1 + α (T–T 0)] (3.26) where ρT is the resistivity at a temperature T and ρ0 is the same at a reference temperature T0. α is called the temperature co-efficient of resistivity, and from Eq. (3.26), the dimension of α is (Temperature)–1. 103 Physics and V(B) respectively. Since current is flowing from A to B, V(A) > V (B) and the potential difference across AB is V = V(A) – V(B) > 0. In a time interval Δt, an amount of charge ΔQ = I Δt travels from A to B. The potential energy of the charge at A, by definition, was Q V(A) and similarly at B, it is Q V(B). Thus, change in its potential energy ΔU is pot ΔU pot = Final potential energy – Initial potential energy = ΔQ[(V (B) – V (A)] = –ΔQV = –I VΔt < 0 (3.28) If charges moved without collisions through the conductor, their kinetic energy would also change so that the total energy is unchanged. Conservation of total energy would then imply that, ΔK = –ΔUpot (3.29) that is, ΔK = I VΔt > 0 (3.30) Thus, in case charges were moving freely through the conductor under the action of electric field, their kinetic energy would increase as they move. We have, however, seen earlier that on the average, charge carriers do not move with acceleration but with a steady drift velocity. This is because of the collisions with ions and atoms during transit. During collisions, the energy gained by the charges thus is shared with the atoms. The atoms vibrate more vigorously, i.e., the conductor heats up. Thus, in an actual conductor, an amount of energy dissipated as heat in the conductor during the time interval Δt is, ΔW = I VΔt (3.31) The energy dissipated per unit time is the power dissipated P = ΔW/Δt and we have, P = I V (3.32) Using Ohm’s law V = IR, we get P = I 2 R = V 2/R (3.33) as the power loss (“ohmic loss”) in a conductor of resistance R carrying a current I. It is this power which heats up, for example, the coil of an electric bulb to incandescence, radiating out heat and light. Where does the power come from? As we have reasoned before, we need an external source to keep a steady current through the conductor. It is clearly this source which must supply this power. In the simple circuit shown with a cell (Fig.3.12), it is the chemical energy of the cell which supplies this power for as long as it can. FIGURE 3.12 Heat is produced in the The expressions for power, Eqs. (3.32) and (3.33), resistor R which is connected across show the dependence of the power dissipated in a the terminals of a cell. The energy resistor R on the current through it and the voltage dissipated in the resistor R comes from across it. the chemical energy of the electrolyte. Equation (3.33) has an important application to power transmission. Electrical power is transmitted from power stations to homes and factories, which Physics 3.11 CELLS, EMF, INTERNAL RESISTANCE We have already mentioned that a simple device to maintain a steady current in an electric circuit is the electrolytic cell. Basically a cell has two electrodes, called the positive (P) and the negative (N), as shown in Fig. 3.18. They are immersed in an electrolytic solution. Dipped in the solution, the electrodes exchange charges with the electrolyte. The positive electrode has a potential difference V(V > 0) between ++ itself and the electrolyte solution immediately adjacent to it marked A in the figure. Similarly, the negative electrode develops a negative potential – (V ) (V ≥ 0) relative to the electrolyte adjacent to it, –– marked as B in the figure. When there is no current, the electrolyte has the same potential throughout, so that the potential difference between P and N is V + – (–V –) = V + + V – . This difference is called the electromotive force (emf) of the cell and is denoted by ε. Thus ε = V++V– > 0 (3.55) Note that ε is, actually, a potential difference and not a force. The name emf, however, is used because of historical reasons, and was given at a time when the phenomenon was not understood properly. To understand the significance of ε, consider a resistor R connected across the cell (Fig. 3.18). A current I flows across R from C to D. As explained before, a steady current is maintained because current flows from N to P through the electrolyte. Clearly, across the electrolyte the same current flows through the electrolyte FIGURE 3.18 (a) Sketch of but from N to P, whereas through R, it flows from P to N. an electrolyte cell with The electrolyte through which a current flows has a finite positive terminal P and resistance r, called the internal resistance. Consider first the negative terminal N. The gap between the electrodes situation when R is infinite so that I = V/R = 0, where V is the is exaggerated for clarity. A potential difference between P and N. Now, and B are points in the V = Potential difference between P and Aelectrolyte typically close to + Potential difference between A and B P and N. (b) the symbol for + Potential difference between B and N a cell, + referring to P and = ε (3.56) – referring to the N Thus, emf ε is the potential difference between the positive and electrode. Electrical negative electrodes in an open circuit, i.e., when no current is connections to the cell are flowing through the cell. made at P and N. If however R is finite, I is not zero. In that case the potential difference between P and N is V = V + V – Ir +– = ε – I r (3.57) Note the negative sign in the expression (I r ) for the potential difference between A and B. This is because the current I flows from B to A in the electrolyte. In practical calculations, internal resistances of cells in the circuit may be neglected when the current I is such that ε >> I r. The actual values of the internal resistances of cells vary from cell to cell. The internal resistance of dry cells, however, is much higher than the common electrolytic cells. Physics 3.16 POTENTIOMETER This is a versatile instrument. It is basically a long piece of uniform wire, sometimes a few meters in length across which a standard cell is connected. In actual design, the wire is sometimes cut in several pieces placed side by side and connected at the ends by thick metal strip. (Fig. 3.28). In the figure, the wires run from A to C. The small vertical portions are the thick metal strips connecting the various sections of the wire. A current I flows through the wire which can be varied by a variable resistance (rheostat, R) in the circuit. Since the wire is uniform, the potential difference between A and any point at a distance l from A is ε(l )=φ l (3.89) whereφ is the potential drop per unit length. Figure 3.28 (a) shows an application of the potentiometer to compare the emf of two cells of emf ε1 and ε2 . The points marked 1, 2, 3 form a two way key. Consider first a position of the key where 1 and 3 are connected so that the galvanometer is connected to ε1. The jockey is moved along the wire till at a point N1, at a distance l 1 from A, there is no deflection in the galvanometer. We can apply Kirchhoff’s loop rule to the closed loop AN1G31A and get, φl1 + 0 – ε1 = 0 (3.90) Similarly, if another emf ε is balanced against l (AN) 2 22 φl 2 + 0 – ε 2 = 0 (3.91) From the last two equations ε l 1 = 1 (3.92) ε2 l2 This simple mechanism thus allows one to compare the emf’s of any two sources. In practice one of the cells is chosen as a standard cell whose emf is known to a high degree of accuracy. The emf of the other cell is then easily calculated from Eq. (3.92). We can also use a potentiometer to measure internal resistance of a cell [Fig. 3.28 (b)]. For this the cell (emf ε ) whose internal resistance (r) is to be determined is connected across a resistance box through a key K2, as FIGURE 3.28 A potentiometer. G is shown in the figure. With key K2 open, balance is a galvanometer and R a variable obtained at length l 1 (AN1). Then, resistance (rheostat). 1, 2, 3 are ε = φ l [3.93(a)] terminals of a two way key 1 (a) circuit for comparing emfs of two When key K2 is closed, the cell sends a current (I ) cells; (b) circuit for determining through the resistance box (R). If V is the terminal internal resistance of a cell. potential difference of the cell and balance is obtained at length l 2 (AN2), 122 V = φ l 2 [3.93(b)]