2.10 DIELECTRICS AND POLARISATION Dielectrics are non-conducting substances. In contrast to conductors, they have no (or negligible number of ) charge carriers. Recall from Section

2.9 what happens when a conductor is placed in an external electric field. The free charge carriers move and charge distribution in the conductor adjusts itself in such a way that the electric field due to induced charges opposes the external field within the conductor. This happens until, in the static situation, the two fields cancel each other and the net electrostatic field in the conductor is zero. In a dielectric, this free movement of charges is not possible. It turns out that the external field induces dipole moment by stretching or re-orienting molecules of the dielectric. The collective effect of all the molecular dipole moments is net charges on the FIGURE 2.20 Difference in behavioursurface of the dielectric which produce a field that of a conductor and a dielectric opposes the external field. Unlike in a conductor, in an external electric field. however, the opposing field so induced does not exactly cancel the external field. It only reduces it. The extent of the effect depends on the nature of the dielectric. To understand the effect, we need to look at the charge distribution of a dielectric at the molecular level. The molecules of a substance may be polar or non-polar. In a non-polar molecule, the centres of positive and negative charges coincide. The molecule then has no permanent (or intrinsic) dipole moment. Examples of non-polar molecules are oxygen (O2) and hydrogen (H2) molecules which, because of their symmetry, have no dipole moment. On the other hand, a polar molecule is one in which the centres of positive and negative charges are separated (even when there is no FIGURE 2.21 Some examples of polar external field). Such molecules have a and non-polar molecules. permanent dipole moment. An ionic molecule such as HCl or a molecule of water (H2O) are examples of polar molecules. 65 Reprint 2025-26 Physics In an external electric field, the positive and negative charges of a non- polar molecule are displaced in opposite directions. The displacement stops when the external force on the constituent charges of the molecule is balanced by the restoring force (due to internal fields in the molecule). The non-polar molecule thus develops an induced dipole moment. The dielectric is said to be polarised by the external field. We consider only the simple situation when the induced dipole moment is in the direction of the field and is proportional to the field strength. (Substances for which this assumption is true are called linear isotropic dielectrics.) The induced dipole moments of different molecules add up giving a net dipole moment of the dielectric in the presence of the external field. A dielectric with polar molecules also develops a net dipole moment in an external field, but for a different reason. FIGURE 2.22 A dielectric develops a net dipole In the absence of any external field, the moment in an external electric field. (a) Non-polar different permanent dipoles are oriented molecules, (b) Polar molecules. randomly due to thermal agitation; so the total dipole moment is zero. When an external field is applied, the individual dipole moments tend to align with the field. When summed overall the molecules, there is then a net dipole moment in the direction of the external field, i.e., the dielectric is polarised. The extent of polarisation depends on the relative strength of two mutually opposite factors: the dipole potential energy in the external field tending to align the dipoles with the field and thermal energy tending to disrupt the alignment. There may be, in addition, the ‘induced dipole moment’ effect as for non-polar molecules, but generally the alignment effect is more important for polar molecules. Thus in either case, whether polar or non-polar, a dielectric develops a net dipole moment in the presence of an external field. The dipole moment per unit volume is called polarisation and is denoted by P. For linear isotropic dielectrics, P = ε0χe E (2.37) where ce is a constant characteristic of the dielectric and is known as the electric susceptibility of the dielectric medium. It is possible to relate ce to the molecular properties of the substance, but we shall not pursue that here. The question is: how does the polarised dielectric modify the original external field inside it? Let us consider, for simplicity, a rectangular dielectric slab placed in a uniform external field E0 parallel to two of its 66 faces. The field causes a uniform polarisation P of the dielectric. Thus Reprint 2025-26 Electrostatic Potential and Capacitance every volume element Dv of the slab has a dipole moment P Dv in the direction of the field. The volume element Dv is macroscopically small but contains a very large number of molecular dipoles. Anywhere inside the dielectric, the volume element Dv has no net charge (though it has net dipole moment). This is, because, the positive charge of one dipole sits close to the negative charge of the adjacent dipole. However, at the surfaces of the dielectric normal to the electric field, there is evidently a net charge density. As seen in Fig 2.23, the positive ends of the dipoles remain unneutralised at the right surface and the negative ends at the left surface. The unbalanced charges are the induced charges due to the external field. Thus, the polarised dielectric is equivalent to two charged surfaces with induced surface charge densities, say sp and –sp. Clearly, the field produced by these surface charges opposes the external field. The total field in the dielectric FIGURE 2.23 A uniformly is, thereby, reduced from the case when no dielectric is polarised dielectric amounts present. We should note that the surface charge density to induced surface charge ±sp arises from bound (not free charges) in the dielectric. density, but no volume charge density.

1.10 ELECTRIC DIPOLE An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a. The line connecting the two charges defines a direction in space. By convention, the direction from –q to q is said to be the direction of the dipole. The mid-point of locations of –q and q is called the centre of the dipole. The total charge of the electric dipole is obviously zero. This does not mean that the field of the electric dipole is zero. Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out. However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out. The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r 2 (the dependence on r of the field due to a single charge q). These qualitative ideas are borne out by the explicit calculation as follows: 1.10.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle. The results are simple for the following two cases: (i) when the point is on the dipole axis, and (ii) when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre. The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+q due to the charge q, by the parallelogram law of vectors. (i) For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in Fig. 1.17(a). Then q E − q = − 2 pˆ [1.13(a)] 4 πε0 (r + a ) where ˆp is the unit vector along the dipole axis (from –q to q). Also q E + q = pˆ [1.13(b)] 23 4 π ε0 (r − a )2 Reprint 2025-26 Physics The total field at P is q  1 1  pˆ − E = E + q + E − q =   (r + a )2 4 π ε0  (r − a )2  q 4 a r = ˆp (1.14) 4 π εo ( r 2 − a 2 )2 For r >> a 4 q a E = 3 pˆ (r >> a) (1.15) 4 πε0 r (ii) For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by q 1 E + q = 2 2 [1.16(a)] 4 πε0 r + a q 1 E – q = 2 2 [1.16(b)] 4 πε0 r + a FIGURE 1.17 Electric field of a dipole and are equal. at (a) a point on the axis, (b) a point The directions of E+q and E–q are as shown in on the equatorial plane of the dipole. Fig. 1.17(b). Clearly, the components normal to the dipole p is the dipole moment vector of axis cancel away. The components along the dipole axis magnitude p = q × 2a and add up. The total electric field is opposite to ˆp. We have directed from –q to q. E = – (E +q + E –q ) cosq ˆp 2 q a = − pˆ (1.17) 4 π εo (r 2 + a 2 )3 / 2 At large distances (r >> a), this reduces to 2 q a E = − pˆ (r >> a ) (1.18) 4 π εo r 3 From Eqs. (1.15) and (1.18), it is clear that the dipole field at large distances does not involve q and a separately; it depends on the product qa. This suggests the definition of dipole moment. The dipole moment vector p of an electric dipole is defined by p = q × 2a ˆp (1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q. In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis 2 p E = 3 (r >> a) (1.20) 4 πεor At a point on the equatorial plane p 3 (r >> a) (1.21) 24 E = −4 πεor Reprint 2025-26 Electric Charges and Fields Notice the important point that the dipole field at large distances falls off not as 1/r 2 but as1/r 3. Further, the magnitude and the direction of the dipole field depends not only on the distance r but also on the angle between the position vector r and the dipole moment p. We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite. Such a dipole is referred to as a point dipole. For a point dipole, Eqs. (1.20) and (1.21) are exact, true for any r. 1.10.2 Physical significance of dipoles In most molecules, the centres of positive charges and of negative charges* lie at the same place. Therefore, their dipole moment is zero. CO2 and CH4 are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules, H2O, is an example of this type. Various materials give rise to interesting properties and important applications in the presence or absence of electric field. Example 1.9 Two charges ±10 mC are placed 5.0 mm apart. Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in Fig. 1.18(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in Fig. 1.18(b). EXAMPLE FIGURE 1.18 1.9 * Centre of a collection of positive point charges is defined much the same way ∑ q i ri as the centre of mass: rcm = i . ∑ q i 25 i Reprint 2025-26 Physics Solution (a) Field at P due to charge +10 mC 10 −5 C 1 = − 12 2 −1 −2 × 2 −4 2 4 π (8.854 × 10 C N m ) (15 − 0.25) × 10 m = 4.13 × 106 N C–1 along BP Field at P due to charge –10 mC 10 –5 C 1 = −12 2 −1 −2 × 2 − 4 2 4 π (8.854 × 10 C N m ) (15 + 0.25) × 10 m = 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP. In this example, the ratio OP/OB is quite large (= 60). Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole. For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude 2 p E = 3 (r/a >> 1) 4 πε0 r where p = 2a q is the magnitude of the dipole moment. The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q). Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, 2 × 5 × 10 − 8 C m 1 E = −12 2 −1 −2 × 3 −6 3 = 2.6 × 105 N C–1 4 π (8.854 × 10 C N m ) (15) × 10 m along the dipole moment direction AB, which is close to the result obtained earlier. (b) Field at Q due to charge + 10 mC at B 10 −5 C 1 = −12 2 −1 − 2 × 2 2 −4 2 4 π (8.854 × 10 C N m ) [15 + (0.25) ] × 10 m = 3.99 × 106 N C–1 along BQ Field at Q due to charge –10 mC at A 10 −5 C 1 = − 12 2 −1 − 2 × 2 2 −4 2 4 π (8.854 × 10 C N m ) [15 + (0.25) ] × 10 m = 3.99 × 106 N C–1 along QA. Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA. Therefore, the resultant electric field at Q due to the two charges at A and B is 0.25 6 –1 = 2 × 2 2 × 3.99 × 10 N C along BA 1.9 15 + (0.25) = 1.33 × 105 N C–1 along BA. As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to EXAMPLE 26 the axis of the dipole: Reprint 2025-26 Electric Charges and Fields p E = 3 (r/a >> 1) 4 π 0ε r 5 × 10 −8 Cm 1 = −12 2 –1 –2 × 3 −6 3 4 π (8.854 × 10 C N m ) (15) × 10 m = 1.33 × 105 N C–1. The direction of electric field in this case is opposite to the direction EXAMPLE of the dipole moment vector. Again, the result agrees with that obtained before. 1.9