3.5 DRIFT OF ELECTRONS AND THE ORIGIN OF RESISTIVITY As remarked before, an electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed but in random directions. If we consider all the electrons, their average velocity will be zero since their directions are random. Thus, if there are N electrons and the velocity of the ith electron (i = 1, 2, 3, ... N ) at a given time is vi, then 1 N v i = 0 (3.14) N =∑i 1 Consider now the situation when an electric field is present. Electrons will be accelerated due to this field by – e E a = (3.15) m where –e is the charge and m is the mass of an electron. Consider again the ith electron at a given time t. This electron would have had its last collision some time before t, and let ti be the time elapsed after its last collision. If vi was its velocity immediately after the last collision, then its velocity Vi at time t is −e E Vi = v i + t i (3.16) m FIGURE 3.3 A schematic picture of since starting with its last collision it was accelerated an electron moving from a point A to (Fig. 3.3) with an acceleration given by Eq. (3.15) for a another point B through repeated time interval ti. The average velocity of the electrons at collisions, and straight line travel time t is the average of all the Vi’s. The average of vi’s is between collisions (full lines). If an electric field is applied as shown, thezero [Eq. (3.14)] since immediately after any collision, electron ends up at point B¢ (dotted the direction of the velocity of an electron is completely lines). A slight drift in a direction random. The collisions of the electrons do not occur at opposite the electric field is visible. regular intervals but at random times. Let us denote by t, the average time between successive collisions. Then 85 at a given time, some of the electrons would have spent Reprint 2025-26 Physics time more than t and some less than t. In other words, the time ti in Eq. (3.16) will be less than t for some and more than t for others as we go through the values of i = 1, 2 ..... N. The average value of ti then is t (known as relaxation time). Thus, averaging Eq. (3.16) over the N-electrons at any given time t gives us for the average velocity vd e E v d ≡ ( Vi )average = ( v i )average − ( t i )average m e E e E = 0 – τ = − τ (3.17) m m 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 vd in Eq. (3.17) is called the drift velocity. 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 (Fig. 3.4). Then conductor. The magnitude of current because of the drift, in an infinitesimal amount of time density in a metal is the magnitude of Dt, all electrons to the left of the area at distances upto charge contained in a cylinder of unit |vd|Dt would have crossed the area. If n is the number area and length vd. of free electrons per unit volume in the metal, then there are n Dt |vd|A such electrons. Since each electron carries a charge –e, the total charge transported across this area A to the right in time Dt is –ne A|vd|Dt. 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 Dt is by definition [Eq. (3.2)] I Dt, where I is the magnitude of the current. Hence, I ∆=t + n e A v d ∆t (3.18) Substituting the value of |vd| from Eq. (3.17) e 2 A I ∆=t τn ∆t 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), ne 2 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 ne 2 j = τE (3.22) m Comparison with Eq. (3.13) shows that Eq. (3.22) is exactly the Ohm’s 86 law, if we identify the conductivity s as Reprint 2025-26 Current Electricity ne 2 σ = τ (3.23) m We thus see that a very simple picture of electrical conduction reproduces Ohm’s law. We have, of course, made assumptions that t and n are constants, independent of E. We shall, in the next section, discuss the limitations of Ohm’s law. Example 3.1 (a) Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area 1.0 × 10–7 m2 carrying a current of 1.5 A. Assume that each copper atom contributes roughly one conduction electron. The density of copper is 9.0 × 103 kg/m3, and its atomic mass is 63.5 u. (b) Compare the drift speed obtained above with, (i) thermal speeds of copper atoms at ordinary temperatures, (ii) speed of propagation of electric field along the conductor which causes the drift motion. Solution (a) The direction of drift velocity of conduction electrons is opposite to the electric field direction, i.e., electrons drift in the direction of increasing potential. The drift speed vd is given by Eq. (3.18) vd = (I/neA) Now, e = 1.6 × 10–19 C, A = 1.0 × 10–7m2, I = 1.5 A. The density of conduction electrons, n is equal to the number of atoms per cubic metre (assuming one conduction electron per Cu atom as is reasonable from its valence electron count of one). A cubic metre of copper has a mass of 9.0 × 103 kg. Since 6.0 × 1023 copper atoms have a mass of 63.5 g, 6.0 × 10 23 6 n = × 9.0 × 10 63.5 = 8.5 × 1028 m–3 which gives, 1.5 v d = 28 –19 –7 8.5 × 10 × 1.6 × 10 × 1.0 × 10 = 1.1 × 10–3 m s–1 = 1.1 mm s–1 (b) (i) At a temperature T, the thermal speed* of a copper atom of mass M is obtained from [<(1/2) Mv2 > = (3/2) kBT ] and is thus typically of the order of k B T/M , where kB is the Boltzmann constant. For copper at 300 K, this is about 2 × 102 m/s. This figure indicates the random vibrational speeds of copper atoms in a conductor. Note that the drift speed of electrons is much smaller, about 10–5 times the typical thermal speed at ordinary temperatures. (ii) An electric field travelling along the conductor has a speed of an electromagnetic wave, namely equal to 3.0 × 108 m s–1 EXAMPLE (You will learn about this in Chapter 8). The drift speed is, in comparison, extremely small; smaller by a factor of 10–11. 3.1 * See Eq. (12.23) of Chapter 12 from Class XI book. 87 Reprint 2025-26 Physics Example 3.2 (a) In Example 3.1, the electron drift speed is estimated to be only a few mm s–1 for currents in the range of a few amperes? How then is current established almost the instant a circuit is closed? (b) The electron drift arises due to the force experienced by electrons in the electric field inside the conductor. But force should cause acceleration. Why then do the electrons acquire a steady average drift speed? (c) If the electron drift speed is so small, and the electron’s charge is small, how can we still obtain large amounts of current in a conductor? (d) When electrons drift in a metal from lower to higher potential, does it mean that all the ‘free’ electrons of the metal are moving in the same direction? (e) Are the paths of electrons straight lines between successive collisions (with the positive ions of the metal) in the (i) absence of electric field, (ii) presence of electric field? Solution (a) Electric field is established throughout the circuit, almost instantly (with the speed of light) causing at every point a local electron drift. Establishment of a current does not have to wait for electrons from one end of the conductor travelling to the other end. However, it does take a little while for the current to reach its steady value. (b) Each ‘free’ electron does accelerate, increasing its drift speed until it collides with a positive ion of the metal. It loses its drift speed after collision but starts to accelerate and increases its drift speed again only to suffer a collision again and so on. On the average, therefore, electrons acquire only a drift speed. 3.2 (c) Simple,~1029 m–3.because the electron number density is enormous, (d) By no means. The drift velocity is superposed over the large random velocities of electrons. (e) In the absence of electric field, the paths are straight lines; in the EXAMPLE presence of electric field, the paths are, in general, curved. 3.5.1 Mobility As we have seen, conductivity arises from mobile charge carriers. In metals, these mobile charge carriers are electrons; in an ionised gas, they are electrons and positive charged ions; in an electrolyte, these can be both positive and negative ions. An important quantity is the mobility m defined as the magnitude of the drift velocity per unit electric field: | vd | µ= (3.24) E The SI unit of mobility is m2/Vs and is 104 of the mobility in practical units (cm2/Vs). Mobility is positive. From Eq. (3.17), we have e τ E 88 vd = m Reprint 2025-26 Current Electricity Hence, v d eτ µ= = (3.25) E m where t is the average collision time for electrons.

2.9 ELECTROSTATICS OF CONDUCTORS Conductors and insulators were described briefly in Chapter 1. Conductors contain mobile charge carriers. In metallic conductors, these charge carriers are electrons. In a metal, the outer (valence) electrons part away from their atoms and are free to move. These electrons are free within the metal but not free to leave the metal. The free electrons form a kind of ‘gas’; they collide with each other and with the ions, and move randomly in different directions. In an external electric field, they drift against the direction of the field. The positive ions made up of the nuclei and the bound electrons remain held in their fixed positions. In electrolytic 61conductors, the charge carriers are both positive and negative ions; but Reprint 2025-26 Physics the situation in this case is more involved – the movement of the charge carriers is affected both by the external electric field as also by the so-called chemical forces (see Chapter 3). We shall restrict our discussion to metallic solid conductors. Let us note important results regarding electrostatics of conductors. 1. Inside a conductor, electrostatic field is zero Consider a conductor, neutral or charged. There may also be an external electrostatic field. In the static situation, when there is no current inside or on the surface of the conductor, the electric field is zero everywhere inside the conductor. This fact can be taken as the defining property of a conductor. A conductor has free electrons. As long as electric field is not zero, the free charge carriers would experience force and drift. In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside. Electrostatic field is zero inside a conductor. 2. At the surface of a charged conductor, electrostatic field must be normal to the surface at every point If E were not normal to the surface, it would have some non-zero component along the surface. Free charges on the surface of the conductor would then experience force and move. In the static situation, therefore, E should have no tangential component. Thus electrostatic field at the surface of a charged conductor must be normal to the surface at every point. (For a conductor without any surface charge density, field is zero even at the surface.) See result 5. 3. The interior of a conductor can have no excess charge in the static situation A neutral conductor has equal amounts of positive and negative charges in every small volume or surface element. When the conductor is charged, the excess charge can reside only on the surface in the static situation. This follows from the Gauss’s law. Consider any arbitrary volume element v inside a conductor. On the closed surface S bounding the volume element v, electrostatic field is zero. Thus the total electric flux through S is zero. Hence, by Gauss’s law, there is no net charge enclosed by S. But the surface S can be made as small as you like, i.e., the volume v can be made vanishingly small. This means there is no net charge at any point inside the conductor, and any excess charge must reside at the surface. 4. Electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface This follows from results 1 and 2 above. Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface. That is, there is no potential difference between any two points inside or on 62 the surface of the conductor. Hence, the result. If the conductor is charged, Reprint 2025-26 Electrostatic Potential and Capacitance electric field normal to the surface exists; this means potential will be different for the surface and a point just outside the surface. In a system of conductors of arbitrary size, shape and charge configuration, each conductor is characterised by a constant value of potential, but this constant may differ from one conductor to the other. 5. Electric field at the surface of a charged conductor σ E = nˆ (2.35) ε0 where s is the surface charge density and ˆn is a unit vector normal to the surface in the outward direction. To derive the result, choose a pill box (a short cylinder) as the Gaussian surface about any point P on the surface, as shown in Fig. 2.17. The pill box is partly inside and partly outside the surface of the conductor. It has a small area of cross section d S and negligible height. Just inside the surface, the electrostatic field is zero; just outside, the field is normal to the surface with magnitude E. Thus, the contribution to the total flux through the pill box comes only from the outside (circular) cross-section of the pill box. This equals ± EdS (positive for s > 0, negative for s < 0), since over the small area dS, E may be considered constant and E and dS are parallel or antiparallel. The charge enclosed by the pill box is sdS. By Gauss’s law σδS EdS = ε0 σ E = (2.36) ε0 Including the fact that electric field is normal to the FIGURE 2.17 The Gaussian surface surface, we get the vector relation, Eq. (2.35), which (a pill box) chosen to derive Eq. (2.35) is true for both signs of s. For s > 0, electric field is for electric field at the surface of a normal to the surface outward; for s < 0, electric field charged conductor. is normal to the surface inward. 6. Electrostatic shielding Consider a conductor with a cavity, with no charges inside the cavity. A remarkable result is that the electric field inside the cavity is zero, whatever be the size and shape of the cavity and whatever be the charge on the conductor and the external fields in which it might be placed. We have proved a simple case of this result already: the electric field inside a charged spherical shell is zero. The proof of the result for the shell makes use of the spherical symmetry of the shell (see Chapter 1). But the vanishing of electric field in the (charge-free) cavity of a conductor is, as mentioned above, a very general result. A related result is that even if the conductor 63 Reprint 2025-26 Physics is charged or charges are induced on a neutral conductor by an external field, all charges reside only on the outer surface of a conductor with cavity. The proofs of the results noted in Fig. 2.18 are omitted here, but we note their important implication. Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded from outside electric influence: the field inside the cavity is always zero. This is known as electrostatic shielding. The effect can be made use of in protecting sensitive instruments from FIGURE 2.18 The electric field inside a outside electrical influence. Figure 2.19 gives a cavity of any conductor is zero. All summary of the important electrostatic properties charges reside only on the outer surface of a conductor.of a conductor with cavity. (There are no charges placed in the cavity.) FIGURE 2.19 Some important electrostatic properties of a conductor. Example 2.7 (a) A comb run through one’s dry hair attracts small bits of paper. Why? What happens if the hair is wet or if it is a rainy day? (Remember, a paper does not conduct electricity.) (b) Ordinary rubber is an insulator. But special rubber tyres of aircraft are made slightly conducting. Why is this necessary? (c) Vehicles carrying inflammable materials usually have metallic ropes touching the ground during motion. Why? (d) A bird perches on a bare high power line, and nothing happens to the bird. A man standing on the ground touches the same line and gets a fatal shock. Why? Solution (a) This is because the comb gets charged by friction. The molecules 2.7 in the paper gets polarised by the charged comb, resulting in a net force of attraction. If the hair is wet, or if it is rainy day, friction between hair and the comb reduces. The comb does not get EXAMPLE charged and thus it will not attract small bits of paper. 64 Reprint 2025-26 Electrostatic Potential and Capacitance (b) To enable them to conduct charge (produced by friction) to the ground; as too much of static electricity accumulated may result in spark and result in fire. EXAMPLE (c) Reason similar to (b). (d) Current passes only when there is difference in potential. 2.7

2.4 A spherical conductor of radius 12 cm has a charge of 1.6 × 10–7C distributed uniformly on its surface. What is the electric field (a) inside the sphere (b) just outside the sphere (c) at a point 18 cm from the centre of the sphere?