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 \( \mathbf{N} \) electrons and the velocity of the \( i^{th} \) electron (\( i \) = 1, 2, 3, ... N ) at a given time is \( \mathbf{V}_i \), then
\[ \frac{1}{N} \sum_{t=1}^{N} \mathbf{v}_{t}=0 \](3.14)
A schematic picture of an electron moving from a point \( A \) to another point \( B \) through repeated collisions, and straight line travel between collisions (full lines). If an electric field is applied as shown, the electron ends up at point \( B^\prime \) (dotted lines). A slight drift in a direction opposite the electric field is visible.
Consider now the situation when an electric field is present. Electrons will be accelerated due to this field by
\[ \mathbf{a}=\frac{-e \mathbf{E}}{m} \](3.15)
where \( -e \) is the charge and \( m \) is the mass of an electron.Consider again the \( i^{th} \) electron at a given time \( t \). This electron would have had its last collision some time before \( t \), and let \( t_i \) be the time elapsed after its last collision. If \( v_i \) was its velocity immediately after the last collision, then its velocity \( V_i \) at time \( t \) is
\[ \mathbf{V}_{i}=\mathbf{v}_{i}+\frac{-e \mathbf{E}}{\boldsymbol{m}} \boldsymbol{t}_{i} \](3.16)
since starting with its last collision it was accelerated (Fig. 3.3) with an acceleration given by Eq. (3.15) for a time interval \( t_i \). The average velocity of the electrons at time \( t \) is the average of all the \( V_i \)’s. The average of \( v_i \)’s is zero [Eq. (3.14)] since immediately after any collision, the direction of the velocity of an electron is completely random. The collisions of the electrons do not occur at regular intervals but at random times. Let us denote by \( \tau \), the average time between successive collisions. Then at a given time, some of the electrons would have spent time more than \( \tau \) and some less than \( \tau \). In other words, the time \( t_i \) in Eq. (3.16) will be less than \( \tau \) for some and more than \( \tau \) for others as we go through the values of \( i \) = 1, 2 ..... N. The average value of \( t_i \) then is \( \tau \) (known as relaxation time). Thus, averaging Eq. (3.16) over the N-electrons at any given time \( t \) gives us for the average velocity \( \mathbf{v}_d \)
\[ \begin{array}{l} \mathbf{v}_{d} \equiv\left(\mathbf{V}_{i}\right)_{\text {average }}=\left(\boldsymbol{v}_{i}\right)_{\text {average }}-\frac{e \mathbf{E}}{m}\left(t_{i}\right)_{\text {average }} \\ =0-\frac{\boldsymbol{e} \mathrm{E}}{\boldsymbol{m}} \tau=-\frac{\boldsymbol{e} \mathrm{E}}{\boldsymbol{m}} \tau \end{array} \](3.17)
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 \( \mathbf{v}_d \) in Eq. (3.17) is called the drift velocity.
Current in a metallic conductor. The magnitude of current density in a metal is the magnitude of charge contained in a cylinder of unit area and length \( v_d \).
Because of the drift, there will be net transport of charges across any area perpendicular to \(\mathbf{E}\). Consider a planar area \(\mathbf{A}\), located inside the conductor such that the normal to the area is parallel to \(\mathbf{E}\) (Fig. 3.4). Then because of the drift, in an infinitesimal amount of time \(\Delta t\), all electrons to the left of the area at distances upto \(|\mathbf{v}_d|\Delta t\) would have crossed the area. If \(n\) is the number of free electrons per unit volume in the metal, then there are \(n \Delta t |\mathbf{v}_d|\mathbf{A}\) such electrons. Since each electron carries a charge \(–e\), the total charge transported across this area \(\mathbf{A}\) to the right in time \(\Delta t\) is \(–ne \mathbf{A}|\mathbf{v}_d|\Delta t\). \(\mathbf{E}\) is directed towards the left and hence the total charge transported along \(\mathbf{E}\) across the area is negative of this. The amount of charge crossing the area \(\mathbf{A}\) in time \(\Delta t\) is by definition [Eq. (3.2)] \(\mathbf{I} \Delta t\), where \(\mathbf{I}\) is the magnitude of the current. Hence,
\[ \text { I } \Delta \boldsymbol{t}=+\boldsymbol{n} \boldsymbol{e} \boldsymbol{A}\left|\mathbf{v}_d\right| \Delta \boldsymbol{t} \](3.18)
Substituting the value of \( |\mathbf{v}_d| \) from Eq. (3.17)
\[ I \Delta t=\frac{e^{2} A}{m} \tau n \Delta t|\mathbf{E}| \](3.19)
By definition \( \mathbf{I} \) is related to the magnitude \( |\mathbf{j}| \) of the current density by
\[ I=|j|\mathbf{A} \](3.20)
Hence, from Eqs.(3.19) and (3.20),
\[ |\mathbf{j}|=\frac{\boldsymbol{n} \boldsymbol{e}^{2}}{\boldsymbol{m}} \tau|\mathbf{E}| \](3.21)
The vector \( \mathbf{j} \) is parallel to \( \mathbf{j} \) and hence we can write Eq. (3.21) in the vector form
\[ \mathbf{j}=\frac{n e^{2}}{m} \tau \mathbf{E} \](3.22)
Comparison with Eq. (3.13) shows that Eq. (3.22) is exactly the Ohm’s law, if we identify the conductivity \( \sigma \) as
\[ \sigma=\frac{n e^{2}}{m} \tau \](3.23)
We thus see that a very simple picture of electrical conduction reproduces Ohm’s law. We have, of course, made assumptions that \( \tau \) and \( n \) are constants, independent of \( E \). We shall, in the next section, discuss the limitations of Ohm’s law.
Example 3.1
- Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area \( 1.0*10^{-7}m^2 \) carrying a current of \( 1.5 A \). Assume that each copper atom contributes roughly one conduction electron. The density of copper is \( 9.0*10^3 kg/m^3 \), and its atomic mass is \( 63.5 u \).
-
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.
-
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 \( v_d \) is given by Eq. (3.18) \( v_d = (I/neA)\)
Now, \( e=1.6*10^{-19}C\), \(A=1.0*10^{-7}m^2\), \(I=1.5A\). 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 \( 9.0*10^3 kg \). Since \( 6.0*10^{23} \) copper atoms have a mass of \( 63.5g \), \[ \begin{aligned} n &=\frac{6.0 \times 10^{23}}{63.5} \times 9.0 \times 10^{6} \\ &=8.5 \times 10^{28} \mathrm{~m}^{-3} \end{aligned} \]
which gives, \[ \begin{aligned} v_{d} &=\frac{1.5}{8.5 * 10^{28} * 1.6 * 10^{-19} * 1.0 * 10^{-7}} \\ &=1.1 * 10^{-3} \mathrm{~m} \mathrm{~s}^{-1}=1.1 \mathrm{~mm} \mathrm{~s}^{-1} \end{aligned} \]
-
(i) At a temperature \( T \), the thermal speed* of a copper atom of mass \( M \) is obtained from \( \left[<(1 / 2) M v^{2}>=(3 / 2) k_{\mathrm{B}} T\right] \) and is thus typically of the order of \( \sqrt{k_{B} T / M} \), where \( k_B \) is the Boltzmann constant. For copper at \( 300K \), this is about \( 2*10^2 m/s \). 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 \( 3.0*10^8 m s^{-1} \) (You will learn about this in Chapter 8). The drift speed is, in comparison, extremely small; smaller by a factor of \( 10^{-11} \).
*See Eq. (13.23) of Chapter 13 from Class XI book.
Example 3.2
- 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?
- 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?
- 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?
- 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?
- 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?
- 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.
- 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.
- Simple, because the electron number density is enormous, \( \sim 10^{29} m^{-3}\).
- By no means. The drift velocity is superposed over the large random velocities of electrons.
- In the absence of electric field, the paths are straight lines; in the 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 μ defined as the magnitude of the drift velocity per unit electric field:
\[ \mu=\frac{\left|\mathbf{v}_{d}\right|}{E} \](3.24)
The SI unit of mobility is \( m^2/Vs \) and is \( 10^4 \) of the mobility in practical units (\( cm^2/Vs \)). Mobility is positive. From Eq. (3.17), we have \[ v_{d}=\frac{e \tau E}{m} \]
Hence,
\[ \mu=\frac{v_{d}}{E}=\frac{e \tau}{m} \] (3.25)
where \( \tau \) is the average collision time for electrons.