Electric ®elds in matter

@; E ˆ ÿ 1 r sin 

@

@ : …15:35†

Here there is no variation with and the components are Er ˆp cos 

2p0r³; Eˆ p sin 

4p0r³; E ˆ 0: …15:36†

The electric ®eld lines given by eqn (15.36) for small a=r are shown in Fig 15.24. This ®gure applies to any plane that includes the z-axis. Both the outward component Er of the electric ®eld and the component E

following circles of constant r and constant are proportional to 1=r³, falling off with distance faster than the ®eld due to a single charge. The terms in higher powers of a=r, which we have neglected, decrease faster still. Electrically neutral molecules may possess a dipole moment and, although their dipole ®elds may cause important interactions with other molecules, the higher terms are almost always negligible.

15.5 Electric ®elds in matter

Macroscopic electric ®elds

Inside a single atom the electric ®eld changes very rapidly with distance.

The atomic nucleus is extremely small, even on the atomic scale, and it carries a charge Ze, where Z is the atomic number of the atom (which determines the chemistry of the atom) and e is the electronic charge.

Close to the nucleus the positive charge on the nucleus is all that matters and the ®eld is directed away from the nucleus. Further out, the negative charge on the electrons tends to cancel out the effect of the nucleus, and outside the atom the ®eld is very small. On a microscopic scale these changes of ®eld within an atom are extremely important, and indeed in Chapter 11 the attraction given by Coulomb's law between an electron and a proton is used to work out the properties of the hydrogen atom.

In this section we are concerned with the electrical behaviour of pieces of matter made up of an emormous number of atoms. The ®elds within individual atoms are not of interest: we need to know how the average

®eld varies over volumes large enough to contain very many atoms. Such an average ®eld is called a macroscopic ®eld, to distinguish it from the microscopic ®eld, which varies rapidly within atoms.

Fig. 15.23 The arrows show the directions of the electric ®eld com-ponents Er, E, and E at the point P that has spherical polar coordinates …r, , †.

Fig. 15.24 The dipole ®eld due to a very small dipole with dipole moment p.

Before discussing the macroscopic ®eld in an assembly of many atoms, consider the average ®eld in a volume containing a single electrically neutral atom such as the inert gas argon. The atom is spherically symmetric, so that the ®eld within it is always pointing away from the centre of the atom. The ®eld has a high value near the centre, like the ®eld around a point charge, but it falls away even faster with distance because of the negative charge on the electrons. To work out the average ®eld, you have to remember that averaging a vector quantity is a bit different from averaging a scalar quantity. Directions as well as magnitudes must be taken into account. For a particular point with position vector r with respect to an origin at the centre of the argon atom, the ®eld points in the same direction as r. At the point diametrically opposite, which has position vector ÿr, the ®eld has the same magnitude but is in the opposite direction to the ®eld at r. The sum of the ®elds at r and ÿr is zero. The same applies to all possible points r, and the average ®eld in a volume including the atom is zero.

The example of an inert gas is a special case because the atoms are spherically symmetric. However, except for some special materials, in the absence of any electric ®eld applied from outside, the macroscopic electric

®eld in electrically neutral matter is zero. When charges are present, it is not necessary to calculate the macroscopic ®eld by adding up the contributions from every single particle carrying a charge e and then

®nd the averageÐmost of the contributions just cancel out. The average charge is determined within a volume small compared with everyday objects, but still large enough to contain many atoms. The electric ®eld caused by this average charge is then calculated.

Suppose that dV_jis a small volume located at a point having a position vector rjwith respect to the origin. Let the net amount of charge within dV_j be …r_j†dV_j: …r_j† is thus the charge density, that is, charge per unit volume, measured in coulombs per cubic metre. Now divide up the whole of the region containing charge into a lot of small volumes dVj. Each contributes to the macroscopic electric ®eld, which by substitution in eqn (15.11) is

E…r† ˆ 1 4p0

X

j

…rj†…r ÿ rj†dVj

jr ÿ rjj³ : …15:37†

The average charge density …r_j† varies smoothly with the position r_jand it is legitimate to replace the sum in eqn (15.37) with an integral, even though we started with volumes dVj that are large enough to contain many atoms. The macroscopic ®eld becomes

E…r† ˆ 1 4p₀

Z

volume

…r⁰†…r ÿ r⁰†dV⁰

jr ÿ r⁰j^{3 …15:38†}

zThe average ®eld due to electrically neutral atoms is zero

zMacroscopic ®elds are calculated from average charge densities

where the integral, labelled volume, is over all volumes that contain a net charge. From now on, when we refer to an electric ®eld E…r† without stating whether it is a microscopic or macroscopic ®eld, we shall mean the macroscopic ®eld that has been averaged over many atoms.

As already mentioned, the macroscopic electric ®eld in electrically neutral matter is zero if there is no external electric ®eld. However, if an object is placed in an electric ®eld, this ®eld is modi®ed by the presence of the matter. To investigate how this comes about, we must consider electrical conductors and insulators separately.

Conductors in electric ®elds

Materials like copper and aluminium that are good electrical conductors are able to carry electric current because some of the electrons in the material are free to move. These electrons, which are called conduction electrons, are not ®xed to particular atoms, but are continually moving through the material. In the absence of an electric ®eld there is no net

¯ow of charge, because the electrons are moving at random in all directions. However, if a steady electric ®eld is applied, electrons, each carrying a charge ÿe, experience a force in the opposite direction to the

®eld. There is a net ¯ow in this direction, and the ¯ow may continue for an inde®nite time if the conductor is part of a complete electrical circuit.

On the other hand, if the conductor is isolated, the electrons cannot continue to move when they reach the boundaries of the conductor. In the slab of conductor shown in Fig 15.25, for example, the electric ®eld pointing to the right causes electrons to migrate from the right-hand side to the hand side. Negatively charged electrons accumulate on the left-hand surface, and the de®cit on the right-left-hand side causes a net positive charge to occur there.

The charges appearing on the surface of a conductor are called induced charges. The induced charges themselves generate an electric ®eld directed away from the positive charges towards the negative charges, tending to cancel out the external ®eld. Conduction electrons will continue to ¯ow, however small may be the resultant electric ®eld, and they ¯ow until the electric ®eld within the conductor is zero. The disposition of surface charges depends on the shape of the conductor and is, in general, very dif®cult to work out. Whatever the shape, the charges nevertheless arrange themselves so that the ®eld inside the conductor is exactly zero. This applies to any material that contains conduction electrons, and not just to very good conductors like copper. The semiconductors silicon and germanium, for example, have conduction electron densities billions of times smaller than copper at room temperature but, when placed in a steady external ®eld, they also have zero ®eld inside.

Fig. 15.25 Charges migrate to the surface of a conductor to ensure that the electric ®eld is zero inside the conductor.

zThe electrostatic ®eld inside a conductor is zero

Because the electric ®eld is zero throughout the conductor, its whole volume is at the same potential. In particular, its surface is an equi-potential surface. Since ®eld lines and equiequi-potentials are always perpen-dicular to one another, the external ®eld is normal to any conducting surface. Using Gauss's law we can relate the magnitude of the electric ®eld to the amount of charge on the conducting surface. In Fig 15.26 the closed surface S is shaped like a pillbox. The curved surface is parallel to the electric ®eld and there is no ¯ux through it. The ¯at surfaces of the pillbox, each of area dS, are parallel to the conducting surface, one inside and one outside the conductor. The electric ®eld and hence the ¯ux are zero on the inside. The total ¯ux out of S is E
dS and, if the charge inside the pillbox is dQ, Gauss's law gives

Z

SE
dS ˆ EdS ˆ dQ=0

or

₀E ˆ dQ=dS ˆ  …15:39†

where  is the surface charge density, which is measured in coulombs per square metre (C m^ÿ2). In the simple example of slab geometry illustrated in Fig 15.26 the surface charge density is given directly in terms of the external ®eld by eqn (15.39).

For other shapes of conductor the surface charge density must be distributed in such a way as to ensure that the external ®eld is normal to the conducting surface. This is illustrated schematically in Fig 15.27 which shows a conducting sphere in an electric ®eld that is uniform far from the sphere. Close to the sphere the surface charges modify the ®eld lines so that they curve towards the sphere and meet it normally.

Exercise 15.5 The electric ®eld at the surface of a conductor is 10⁴V cm^ÿ1. What is the surface charge density on the conductor, and what average area has a charge equal to one electronic charge?

Answer The coulomb is a very large unit, and charge is frequently expressed in microcoulombs (1 mC  10^ÿ6 C). The surface charge in this exercise is 8:85 mC m^ÿ2. This is equivalent to one electronic charge on an area 1:8  10^ÿ14 m² or 1:8  10⁴nm², an area large enough to accommodate about one million atoms.

The induced charges on the surface of a conductor are located in a very thin layer. The amount of induced charge is given by the surface charge density  and a surface integral must be added to eqn (15.38) to account

Fig. 15.26 Gauss's theorem relates the induced surface charge to the electric ®eld outside the conductor.

Fig. 15.27 When a conducting sphere is placed in an external electric ®eld E, the ®eld lines bend to meet the conducting surface normally.

for the contribution of the induced charges to the ®eld. Including the surface charges, the general expression for the macroscopic electric ®eld is

E…r† ˆ 1

where the labels volume and surface indicate that the volume integral is over all volumes containing a volume charge density and the surface integral is over all surfaces on which there is a surface charge density.

Similarly, the potential is

In an insulator, all the electrons are ®xed to particular atoms. Over long time periods, practically no migration of charge occurs when an insulating material is placed in an electric ®eld. We can understand how the material responds to the presence of a steady ®eld by considering just one atom.

Imagine that a neutral atom is supported so that it does not fall under gravity, but is free to move horizontally. If a horizontal electric ®eld is switched on, there is no net force on the atom since its charge is zero.

However, the nucleus and the electrons experience forces in opposite directions and they tend to move apart, without shifting the centre of mass of the atom. As the centre of the distribution of negatively charged electrons moves away from the positively charged nucleus, the mutual attraction of nucleus and electrons creates a restoring force that balances the force caused by the external ®eld.

Under all conditions that are met in the laboratory, the restoring force is proportional to the distance x between the nucleus and the centre of the electron distribution. Calling the constant of proportionality k, the restoring force is kx. Figure 15.28 shows the forces acting on the nucleus, but greatly exaggerates the relative movement of the electrons and the nucleus: on the scale of the ®gure, the shift would not be visible for realistic electric ®elds. For an atom with atomic number Z and nuclear charge Ze, the force due to the external ®eld E is ZeE. This force is balanced by the restoring force when kx ˆ ZeE, that is when x ˆ ZeE=k.

When the nucleus and the centre of electronic charge do not coincide, the atom is said to be polarized. As for the point charges discussed in Section 15.4, the vector in the x-direction and with magnitude equal to zInduced charges on the

surfaces of conductors contribute to the electric ®eld

Fig. 15.28 The force on the nucleus of the atom due to the electric ®eld E is balanced by a restoring force caused by the mutual attraction of the nucleus and the electrons.

the product of the distance x and the charge Ze is called the dipole moment of the atom and is measured in coulomb metres (C m). The dipole moment is denoted by the vector p: the vectors x, p, and E all point in the same direction and

p ˆ Zex ˆ…Ze†²

k E ˆ 0E …15:42†

where the constant  is called the polarizability of the atom.

How does polarization affect the macroscopic electric ®eld in an insulator? Let us ®rst consider a slab of uniform insulating material placed in an electric ®eld normal to the faces of the slab. Within the slab the macroscopic electric ®eld must be in the same direction as the ®eld ouside the slab, and we shall assume for the moment that it is constant throughout the slab, having a value Eint, say. Each atom of the insulator acquires a dipole momen ₀E_intand, according to eqn (15.42), the centre of the electron distribution is displaced a distance 0Eint=Ze from the nucleus.

The nucleus is much more massive than the electrons, and the centre of mass almost coincides with the nucleus. We may picture the polarization as if only the electrons move: this simpli®es the argument without altering the results. Figure 15.29 represents a section through a slab of insulator placed in an electric ®eld perpendicular to the sides of the slab.

The dashed lines show the boundaries of imaginary closed boxes with faces of area dS perpendicular to the ®eld: we shall apply Gauss's law to these boxes.

When the atoms are polarized, electrons move through both surfaces of the box (b), which is completely inside the insulator. Negative charge has moved out of the left-hand side of the box, but just as much has moved in at the right-hand side. The net charge inside the box is zero, as it was before the atoms were polarized. Gauss's law tells us that the net ¯ux of E out of the box is zero. This requires the ¯ux entering the left-hand side to equal the ¯ux leaving the right-hand side, and the assumption that the

®eld is uniform within the slab is justi®ed.

Now look at the box (c), which straddles the right-hand surface of the insulator. Negative charge has moved out of the left-hand side of the box but there are no atoms at the right-hand side and the box has acquired a net positive charge. If the number of atoms per unit volume is N, the charge density of electrons is ÿNZe. All the electrons in the slab have moved the same distance x, and the charge moving out of the area dS on the left of the box is ÿNZexdS ˆ ÿNpdS. The box now encloses a net charge ‡NpdS, and the surface charge density caused by polarization is

pˆ Np. The opposite face of the slab acquires a surface charge density (ÿNp† as electrons move into box (a). The slab as a whole is electrically neutral, as it must be since it is composed entirely of neutral atoms.

Fig. 15.29 The movement of charge in the slab of insulator builds up charge on the surface but leaves the interior electrically neutral.

If the surface of the insulator is at an angle  to the electric ®eld, as in Fig 15.30, the surface charge density is reduced. If the area of the insulator surface inside the box is dS, the projected area normal to the ®eld is dS cos . The net charge inside the box is now _pdS ˆ NpdS cos . At a surface where the electric ®eld enters the insulator, pis given by the same expression except for a minus sign. Remembering that the vector dS is the outward normal to the surface, we ®nd that both the sign of the surface charge density _pand its angular dependence can be expressed concisely by using the vector notation

pdS ˆ Np
dS ˆ P
dS …15:43†

where the vector P ˆ Np is the dipole moment per unit volume of the insulator. The vector P is called the polarization density. The polarization density, like the dipole moment of a single atom, is in the direction from negative to positive polarization surface charge.

The polarization density is useful because it is related to the electric

®eld inside the polarized material. In slab geometry this relation is easily found from Gauss's law. The closed surface S in Fig 15.31 has surfaces of area dS normal to the ®eld. The polarization charge within S is

pdS ˆ PdS. According to Gauss's law the net ¯ux out of S is therefore PdS=₀. The ®eld entering S from the left is E_int and the ®eld leaving on the right is Eext, and the net ¯ux is …Eextÿ Eint†dS ˆ PdS=0. Hence

Eextˆ Eint‡ P=0 …15:44†

or, since P ˆ Np ˆ N₀E_int,

E_extˆ E_int…1 ‡ N† ˆ E_int…1 ‡ _E†: …15:45†

The dimensionless constant Eˆ N is called the electric suscept-ibility of the insulating material. The presence of the polarization charges has reduced the ®eld inside the insulator by the factor (1 ‡ _E). This is represented by drawing a reduced density of lines inside: in Fig 15.31 some lines of the external ®eld end on negative polarization charges and start on positive polarization charges.

Worked Example 15.6 At 20^C and one atmosphere pressure helium gas contains 2:7  10²⁵atoms m^ÿ3, and the electric susceptibility of the gas is 6:5  10^ÿ5. Calculate the separation of the centres of the positive and negative charges in a helium atom when it is placed in an electric ®eld of 10⁶V m^ÿ1.

Answer From eqns (15.42) and (15.45) the separation is x ˆ0E

Ze ˆE0E ZeN : Fig. 15.30 If the ®eld inside the

insulator is not perpendicular to its surface, the charges move the same distance, but the surface charge is now spread out over a bigger area.

Fig. 15.31 The induced surface charge is related to the electric ®eld inside and outside the insulator.

Substituting the values given, the separation is 6:5  10^ÿ5 8:85  10^ÿ12 10⁶

2  1:6  10^ÿ19 2:7  10²⁵ ˆ 6:7  10^ÿ17m ˆ 6:7  10^ÿ8nm, a shift of about one-millionth of the radius of the helium atom.

2  1:6  10^ÿ19 2:7  10²⁵ ˆ 6:7  10^ÿ17m ˆ 6:7  10^ÿ8nm, a shift of about one-millionth of the radius of the helium atom.