`color{blue} ✍️` Conductors contain mobile charge carriers. In metallic conductors, these charge carriers are electrons.

`color{blue} ✍️` In a metal, the outer (valence) electrons part away from their atoms and are free to move.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{blue} ✍️` In electrolytic conductors, the charge carriers are both positive and negative ions; but 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.

`color{blue} ✍️` We shall restrict our discussion to metallic solid conductors. Let us note important results regarding electrostatics of conductors.

`color{purple} ul{ 1." Inside a conductor, electrostatic field is zero Consider a conductor, neutral or charged".}`

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{purple} ul {"2. At the surface of a charged conductor, electrostatic field must be normal to the surface at every point."}`

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{purple}ul{ "3. The interior of a conductor can have no excess charge in the static situation "}`

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{purple}ul{ "4. Electrostatic potential is constant throughout the volume" }`
`color{purple}ul{"of the conductor and has the same value (as inside) on its surface"}`

`color{blue} ✍️` 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.

`color{blue} ✍️` That is, there is no potential difference between any two points inside or on the surface of the conductor. Hence, the result. If the conductor is charged electric field normal to the surface exists; this means potential will be different for the surface and a point just outside the surface.

`color{blue} ✍️` 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.

`color{purple} ul{"5. Electric field at the surface of a charged conductor"}`

`E = (sigma)/(epsilon_0)hatn` .............2.35

`color{blue} ✍️` where σ is the surface charge density and ˆn is a unit vector normal to the surface in the outward direction.

`color{blue} ✍️` 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.


`color{blue} ✍️` The pill box is partly inside and partly outside the surface of the conductor. It has a small area of cross section `δS` 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.

`color{blue} ✍️` This equals `± EδS` (positive for `σ > 0`, negative for `σ < 0`), since over the small area `δS, E` may be considered constant and `E` and `δS` are parallel or antiparallel. The charge enclosed by the pill box is `σδS`. By Gauss’s law

`EδS =(|σ| δ S)/(epsilon_0)`

`E= |σ|/(epsilon_0)` .......................2.36

`color{blue} ✍️` Including the fact that electric field is normal to the surface, we get the vector relation, Eq. (2.35), which is true for both signs of `σ`. For `σ > 0`, electric field is normal to the surface outward; for `σ < 0`, electric field is normal to the surface inward.

`color{purple}ul{"6. Electrostatic shielding"}`

`color{blue} ✍️` Consider a conductor with a cavity, with no charges inside the cavity.

`color{blue} ✍️` 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.

`color{blue} ✍️` We have proved a simple case of this result already: the electric field inside a charged spherical shell is zero.

`color{blue} ✍️` 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.

`color{blue} ✍️` A related result is that even if the conductor 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.


`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.


`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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 surface of the dielectric which produce a field that opposes the external field. Unlike in a conductor, however, the opposing field so induced does not exactly cancel the external field.

`color{blue} ✍️` 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.

`color{blue} ✍️` The molecules of a substance may be polar or non-polar. In a non-polar molecule, the centres of positive and negative charges coincide.


`color{blue} ✍️` The molecule then has no permanent (or intrinsic) dipole moment. Examples of non-polar molecules are oxygen `(O_2)` and hydrogen `(H_2)` 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 external field). Such molecules have a permanent dipole moment.

`color{blue} ✍️` An ionic molecule such as HCl or a molecule of water `(H_2O)` are examples of polar molecules.

`color{blue} ✍️` In an external electric field, the positive and negative charges of a nonpolar molecule are displaced in opposite directions.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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.)

`color{blue} ✍️` The induced dipole moments of different molecules add up giving a net dipole moment of the dielectric in the presence of the external field.

`color{blue} ✍️` A dielectric with polar molecules also develops a net dipole moment in an external field, but for a different reason.

`color{blue} ✍️` In the absence of any external field, the different permanent dipoles are oriented randomly due to thermal agitation; so the total dipole moment is zero.

`color{blue} ✍️` When an external field is applied, the individual dipole moments tend to align with the field. When summed over all the molecules, there is then a net dipole moment in the direction of the external field, i.e., the dielectric is polarised.


`color{blue} ✍️` 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.

`color{blue} ✍️` 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.

`color{blue} ✍️` 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,

`color{blue} {P =X_e E}` ....................... 2.37

where `χe` is a constant characteristic of the dielectric and is known as the electric susceptibility of the dielectric medium.

It is possible to relate χe to the molecular properties of the substance, but we shall not pursue that here.

`color{blue} ✍️` 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 `E_0` parallel to two of its faces.

`color{blue} ✍️` The field causes a uniform polarisation P of the dielectric. Thus every volume element `Δv` of the slab has a dipole moment P `Δv` in the direction of the field.

`color{blue} ✍️` 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).

`color{blue} ✍️` 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.


`color{blue} ✍️` 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.

`color{blue} ✍️` Thus the polarised dielectric is equivalent to two charged surfaces with induced surface charge densities, say `σp` and `–σp.` Clearly, the field produced by these surface charges opposes the external field.

`color{blue} ✍️` The total field in the dielectric is, thereby, reduced from the case when no dielectric is present. We should note that the surface charge density `±σp` arises from bound (not free charges) in the dielectric.