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

1.3 CONDUCTORS AND INSULATORS Some substances readily allow passage of electricity through them, others do not. Those which allow electricity to pass through them easily are called conductors. They have electric charges (electrons) that are comparatively free to move inside the material. Metals, human and animal bodies and earth are conductors. Most of the non-metals like glass, porcelain, plastic, nylon, wood offer high resistance to the passage of electricity through them. They are called insulators. Most substances fall into one of the two classes stated above*. When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor. In contrast, if some charge is put on an insulator, it stays at the same place. You will learn why this happens in the next chapter. This property of the materials tells you why a nylon or plastic comb gets electrified on combing dry hair or on rubbing, but a metal article * There is a third category called semiconductors, which offer resistance to the movement of charges which is intermediate between the conductors and insulators. 3 Reprint 2025-26 Physics like spoon does not. The charges on metal leak through our body to the ground as both are conductors of electricity. However, if a metal rod with a wooden or plastic handle is rubbed without touching its metal part, it shows signs of charging. 1.4 BASIC PROPERTIES OF ELECTRIC CHARGE We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other. Here, we shall now describe some other properties of the electric charge. If the sizes of charged bodies are very small as compared to the distances between them, we treat them as point charges. All the charge content of the body is assumed to be concentrated at one point in space. 1.4.1 Additivity of charges FIGURE 1.2 Electroscopes: (a) We have not as yet given a quantitative definition of a The gold leaf electroscope, (b) charge; we shall follow it up in the next section. We shall Schematics of a simple tentatively assume that this can be done and proceed. If electroscope. a system contains two point charges q1 and q2, the total charge of the system is obtained simply by adding algebraically q1 and q2 , i.e., charges add up like real numbers or they are scalars like the mass of a body. If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn . Charge has magnitude but no direction, similar to mass. However, there is one difference between mass and charge. Mass of a body is always positive whereas a charge can be either positive or negative. Proper signs have to be used while adding the charges in a system. For example, the total charge of a system containing five charges +1, +2, –3, +4 and –5, in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the same unit. 1.4.2 Charge is conserved We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed. A picture of particles of electric charge enables us to understand the idea of conservation of charge. When we rub two bodies, what one body gains in charge the other body loses. Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found that the total charge of the isolated system is always conserved. Conservation of charge has been established experimentally. It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed 4 Reprint 2025-26 Electric Charges and Fields in a process. Sometimes nature creates charged particles: a neutron turns into a proton and an electron. The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation. 1.4.3 Quantisation of charge Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted by e. Thus charge q on a body is always given by q = ne where n is any integer, positive or negative. This basic unit of charge is the charge that an electron or proton carries. By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as –e and that on a proton as +e. The fact that electric charge is always an integral multiple of e is termed as quantisation of charge. There are a large number of situations in physics where certain physical quantities are quantised. The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday. It was experimentally demonstrated by Millikan in 1912. In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C. A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter. In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 1 of Class XI, Physics Textbook , Part I). In this system, the value of the basic unit of charge is e = 1.602192 × 10–19 C Thus, there are about 6 × 1018 electrons in a charge of –1C. In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 mC (micro coulomb) = 10–6 C or 1 mC (milli coulomb) = 10–3 C. If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples of e. Thus, if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (–e) = (n2 – n1) e. Since n1 and n2 are integers, their difference is also an integer. Thus the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e. The step size e is, however, very small because at the macroscopic level, we deal with charges of a few mC. At this scale the fact that charge of a body can increase or decrease in units of e is not visible. In this respect, the grainy nature of the charge is lost and it appears to be continuous. This situation can be compared with the geometrical concepts of points and lines. A dotted line viewed from a distance appears continuous to us but is not continuous in reality. As many points very close to 5 Reprint 2025-26 Physics each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution. At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e. Since e = 1.6 × 10–19 C, a charge of magnituOde, say 1 mC, contains something like 1013 times the electronic charge. At this scale, the fact that charge can increase or decrease only in units of e is not very different from saying that charge can take continuous values. Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored. However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored. It is the magnitude of scale involved that is very important. Example 1.1 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 C on the other body? Solution In one second 109 electrons move out of the body. Therefore the charge given out in one second is 1.6 × 10–19 × 109 C = 1.6 × 10–10 C. The time required to accumulate a charge of 1 C can then be estimated to be 1 C ÷ (1.6 × 10–10 C/s) = 6.25 × 109 s = 6.25 × 109 ÷ (365 × 24 × 3600) years = 198 years. Thus to collect a charge of one coulomb, from a body from which 109 electrons move out every second, we will 1.1 needunit forapproximatelymany practical200 years.purposes.One coulomb is, therefore, a very large It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material. A cubic piece of copper of side 1 cm contains about 2.5 × 1024 EXAMPLE electrons. Example 1.2 How much positive and negative charge is there in a cup of water? Solution Let us assume that the mass of one cup of water is 250 g. The molecular mass of water is 18g. Thus, one mole (= 6.02 × 1023 molecules) of water is 18 g. Therefore the number of 1.2 molecules in one cup of water is (250/18) × 6.02 × 1023. Each molecule of water contains two hydrogen atoms and one oxygen atom, i.e., 10 electrons and 10 protons. Hence the total positive and total negative charge has the same magnitude. It is equal to EXAMPLE (250/18) × 6.02 × 1023 × 10 × 1.6 × 10–19 C = 1.34 × 107 C. 1.5 COULOMB’S LAW Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges. Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly 6 proportional to the product of the magnitude of the two charges and Reprint 2025-26 Electric Charges and Fields acted along the line joining the two charges. Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by q q 2 1 F = k 2 (1.1) r How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres. When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges. However, the charges on the spheres were unknown, to begin with. How then could he discover a relation like Eq. (1.1)? Coulomb thought of the following simple way: Suppose the Charles Augustin de charge on a metallic sphere is q. If the sphere is put in contact Coulomb (1736 – 1806) Coulomb, a French with an identical uncharged sphere, the charge will spread over physicist, began his the two spheres. By symmetry, the charge on each sphere will career as a military be q/2*. Repeating this process, we can get charges q/2, q/4, engineer in the West etc. Coulomb varied the distance for a fixed pair of charges and Indies. In 1776, he measured the force for different separations. He then varied the returned to Paris andcharges in pairs, keeping the distance fixed for each pair. retired to a small estate CHARLES Comparing forces for different pairs of charges at different to do his scientific distances, Coulomb arrived at the relation, Eq. (1.1). research. He invented a Coulomb’s law, a simple mathematical statement, was torsion balance to initially experimentally arrived at in the manner described measure the quantity of a force and used it forabove. While the original experiments established it at a determination of forcesmacroscopic scale, it has also been established down to AUGUSTIN of electric attraction or subatomic level (r ~ 10–10 m). repulsion between small DE Coulomb discovered his law without knowing the explicit charged spheres. He magnitude of the charge. In fact, it is the other way round: thus arrived in 1785 at Coulomb’s law can now be employed to furnish a definition the inverse square law for a unit of charge. In the relation, Eq. (1.1), k is so far relation, now known as arbitrary. We can choose any positive value of k. The choice Coulomb’s law. The law of k determines the size of the unit of charge. In SI units, the had been anticipated by 2 Priestley and also by COULOMB Nm value of k is about 9 × 109 2 . The unit of charge that Cavendish earlier, C though Cavendishresults from this choice is called a coulomb which we defined never published his earlier in Section 1.4. Putting this value of k in Eq. (1.1), we results. Coulomb also (1736 see that for q1 = q2 = 1 C, r = 1 m found the inverse F = 9 × 109 N square law of force That is, 1 C is the charge that when placed at a distance between unlike and like magnetic poles. –1806)of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude * A torsion balance is a sensitive device to measure force. It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation. * Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q. 7 Reprint 2025-26 Physics 9 × 109 N. One coulomb is evidently too big a unit to be used. In practice, in electrostatics, one uses smaller units like 1 mC or 1 mC. The constant k in Eq. (1.1) is usually put as k = 1/4pe0 for later convenience, so that Coulomb’s law is written as 1 q1 q 2 F = 2 (1.2) 4 π ε0 r e0 is called the permittivity of free space . The value of e0 in SI units is =0 8.854 × 10–12 C2 N–1m–2 Since force is a vector, it is better to write Coulomb’s law in the vector notation. Let the position vectors of charges q1 and q2 be r1 and r2 respectively [see Fig.1.3(a)]. We denote force on q1 due to q2 by FIGURE 1.3 (a) Geometry and F12 and force on q2 due to q1 by F21. The two point (b) Forces between charges. charges q1 and q2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21: r21 = r2 – r1 In the same way, the vector leading from 2 to 1 is denoted by r12: r12 = r1 – r2 = – r21 The magnitude of the vectors r21 and r12 is denoted by r21 and r12, respectively (r12 = r21). The direction of a vector is specified by a unit vector along the vector. To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors: ɵ r21 ɵ r12 ɵ ɵ 12 = , r 21 − r 12 r 21 = , r r21 r12 Coulomb’s force law between two point charges q1 and q2 located at r1 and r2, respectively is then expressed as 1 q1 q 2 ɵ F21 = 2 r 21 (1.3) 4 π εo r21 Some remarks on Eq. (1.3) are relevant: · Equation (1.3) is valid for any sign of q1 and q2 whether positive or negative. If q1 and q2 are of the same sign (either both positive or both negative), F21 is along ˆr 21, which denotes repulsion, as it should be for like charges. If q1 and q2 are of opposite signs, F21 is along – ɵr 21(= ɵr 12), which denotes attraction, as expected for unlike charges. Thus, we do not have to write separate equations for the cases of like and unlike charges. Equation (1.3) takes care of both cases correctly [Fig. 1.3(b)]. Reprint 2025-26 Electric Charges and Fields · The force F12 on charge q1 due to charge q2, is obtained from Eq. (1.3), by simply interchanging 1 and 2, i.e., 1 q1 q 2 F12 = 2 rˆ12 = − F21 4 π ε0 r12 Thus, Coulomb’s law agrees with the Newton’s third law. · Coulomb’s law [Eq. (1.3)] gives the force between two charges q1 and q2 in vacuum. If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter. We shall consider electrostatics in matter in the next chapter. Example 1.3 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges and masses respectively. (a) Compare the strength of these forces by determining the ratio of their magnitudes (i) for an electron and a proton and (ii) for two protons. (b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are 1 Å (= 10-10 m) apart? (mp = 1.67 × 10–27 kg, me = 9.11 × 10–31 kg) Solution (a) (i) The electric force between an electron and a proton at a distance r apart is: 1 e 2 Fe = − 2 4 πε0 r where the negative sign indicates that the force is attractive. The corresponding gravitational force (always attractive) is: m p m e FG = −G 2 r where mp and me are the masses of a proton and an electron respectively. e 2 Fe = = 2.4 × 10 39 FG 4 πε0Gm p m e (ii) On similar lines, the ratio of the magnitudes of electric force to the gravitational force between two protons at a distance r apart is: Fe e 2 = = 1.3 × 1036 FG 4πε0Gm p m p However, it may be mentioned here that the signs of the two forces are different. For two protons, the gravitational force is attractive in nature and the Coulomb force is repulsive. The actual values of these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N, whereas, FG ~ 1.9 × 10–34 N. EXAMPLE The (dimensionless) ratio of the two forces shows that electrical forces are enormously stronger than the gravitational forces. 1.3 9 Reprint 2025-26 Physics (b) The electric force F exerted by a proton on an electron is same in magnitude to the force exerted by an electron on a proton; however, the masses of an electron and a proton are different. Thus, the magnitude of force is 1 e 2 |F| = 2 = 8.987 × 109 Nm2/C2 × (1.6 ×10–19C)2 / (10–10m)2 4 πε 0 r = 2.3 × 10–8 N Using Newton’s second law of motion, F = ma, the acceleration that an electron will undergo is a = 2.3×10–8 N / 9.11 ×10–31 kg = 2.5 × 1022 m/s2 Comparing this with the value of acceleration due to gravity, we 1.3 canthe motionconcludeof thatelectronthe effectand itofundergoesgravitationalveryfieldlargeis accelerationsnegligible on under the action of Coulomb force due to a proton. The value for acceleration of the proton is 2.3 × 10–8 N / 1.67 × 10–27 kg = 1.4 × 1019 m/s2 EXAMPLE Example 1.4 A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B held by an insulating 1.4 EXAMPLE 10 FIGURE 1.4 Reprint 2025-26 Electric Charges and Fields handle is brought close to A such that the distance between their centres is 10 cm, as shown in Fig. 1.4(a). The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen). Spheres A and B are touched by uncharged spheres C and D respectively, as shown in Fig. 1.4(b). C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in Fig. 1.4(c). What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes. Ignore the sizes of A and B in comparison to the separation between their centres. Solution Let the original charge on sphere A be q and that on B be q¢. At a distance r between their centres, the magnitude of the electrostatic force on each is given by 1 qq ′ F = 2 4 πε0 r neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q¢/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is 1 (q / 2 )( q ′ / 2 ) 1 (qq ′ ) = = F EXAMPLE F ′ = 4 πε0 (r / 2 )2 4 πε0 r 2 Thus the electrostatic force on A, due to B, remains unaltered. 1.4

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