• Electrostatic Potential and Capacitance is characteristic of the present state of configuration, and not the way the state is achieved. Example 2.4 Four charges are arranged at the corners of a square ABCD of side d, as shown in Fig. 2.15.(a) Find the work required to put together this arrangement. (b) A charge q0 is brought to the centre E of the square, the four charges being held fixed at its corners. How much extra work is needed to do this? FIGURE 2.15 Solution (a)Since the work done depends on the final arrangement of the charges, and not on how they are put together, we calculate work needed for one way of putting the charges at A, B, C and D. Suppose, first the charge +q is brought to A, and then the charges –q, +q, and –q are brought to B, C and D, respectively. The total work needed can be calculated in steps: (i) Work needed to bring charge +q to A when no charge is present elsewhere: this is zero. (ii) Work needed to bring –q to B when +q is at A. This is given by (charge at B) × (electrostatic potential at B due to charge +q at A) ⎛ q ⎞ q 2 =− q × =−⎜⎟⎝ 4πε0d ⎠ 4πε0d (iii) Work needed to bring charge +q to C when +q is at A and –q is at B. This is given by (charge at C) × (potential at C due to charges at A and B)  +q −q ⎞ =+ q + ⎟⎝ 4πε0d 24πε0d⎠−q 2 ⎛ 1 ⎞ ⎜⎟= 1− 4πε0d ⎝ 2  (iv) Work needed to bring –q to D when +q at A,–q at B, and +q at C. This is given by (charge at D) × (potential at D due to charges at A, B and C)⎛+q −qq ⎞ =− q ++ ⎜⎝ 4πε0d 4πε0d 24πε0d⎟⎠ −q 2 ⎛ 1 ⎞ = 2 −⎜⎟4πε0d ⎝ 2 ⎠ EXAMPLE 2.4 Physics FIGURE 2.18 The electric field inside a cavity of any conductor is zero. All charges reside only on the outer surface of a conductor with cavity. (There are no charges placed in the cavity.) 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 outside electrical influence. Figure 2.19 gives a summary of the important electrostatic properties of a conductor. 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 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 charged and thus it will not attract small bits of paper. EXAMPLE 2.7 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. (c) Reason similar to (b). (d) Current passes only when there is difference in potential. 2.10 DIELECTRICSAND 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 chargeson the FIGURE 2.20 Difference in behavioursurface of the dielectric which produce a field that of a conductor and a dielectricopposes 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 polarexternal field). Such molecules have a and non-polar molecules.permanent dipole moment. An ionic molecule such as HCl or a molecule of water (HO) are examples of polar molecules. 712EXAMPLE 2.7 Electrostatic Potential and Capacitance every volume element Δv of the slab has a dipole moment PΔv in the direction of the field. The volume element Δv is macroscopically small but contains a very large number of molecular dipoles. Anywhere inside the dielectric, the volume element Δv 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 σ p and –σ p. Clearly, the field produced by these surface charges opposes the external field. The total field in the dielectric FIGURE 2.23 A uniformlyis, 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 ±σ parises from bound (not free charges) in the dielectric. density, but no volume charge density. 2.11 CAPACITORSAND CAPACITANCE A capacitor is a system of two conductors separated by an insulator (Fig. 2.24). The conductors have charges, say Q1 and Q2, and potentials V1 and V2. Usually, in practice, the two conductors have charges Q and – Q, with potential difference V = V1 – V2 between them. We shall consider only this kind of charge configuration of the capacitor. (Even a single conductor can be used as a capacitor by assuming the other at infinity.) The conductors may be so charged by connecting them to the two terminals of a battery. Q is called the charge of the capacitor, though this, in fact, is the charge on one of the conductors – the total charge of the capacitor is zero. The electric field in the region between the conductors is proportional to the charge Q. That is, if the charge on the capacitor is, say doubled, the electric field will also be doubled at every point. (This follows from the direct proportionality between field and charge implied by Coulomb’s law and the superposition principle.) Now, potential difference V is the work done per unit positive charge in taking a small test charge from the conductor 2 to 1 against the field. FIGURE 2.24 A system of two conductors Consequently, V is also proportional to Q, and separated by an insulator forms a capacitor. the ratio Q/V is a constant: QC = (2.38)V The constant C is called the capacitance of the capacitor. C is independent of Q or V, as stated above. The capacitance C depends only on the Physics 2.25 Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor. FIGURE 2.35 2.26 The plates of a parallel plate capacitor have an area of 90 cm2 each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply. (a) How much electrostatic energy is stored by the capacitor? (b) View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u. Hence arrive at a relation between u and the magnitude of electric field E between the plates. 2.27 A 4 µF capacitor is charged by a 200 V supply. It is then disconnected from the supply, and is connected to another uncharged 2 µF capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation? 2.28 Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the capacitor, and E is the magnitude of electric field between the plates. Explain the origin of the factor ½. 2.29 A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show FIGURE 2.36