The Method of Images

2. Chapter 5 Current and Conductors The Method of Images • One important characteristic of the dipole field developed in Chapter 4 is the infinite plane at zero potential that exists midway between the two charges. • Such a plane may be represented by a thin infinite conducting plane. • The conductor is an equipotential surface at a potential V = 0. The electric field intensity, as for a plane, is normal to the surface.

3. Chapter 5 Current and Conductors The Method of Images • Thus, we can replace the dipole configuration (left) with the single charge and conducting plane (right), without affecting the fields in the upper half of the figure. • Now, we begin with a single charge above a conducting plane. • ► The same fields above the plane can be maintained by removing the plane and locating a negative charge at a symmetrical location below the plane. • This charge is called the image of the original charge, and it is the negative of that value.

4. Chapter 5 Current and Conductors The Method of Images • The same procedure can be done again and again. • Any charge configuration above an infinite ground plane may be replaced by an arrangement composed of the given charge configuration, its image, and no conducting plane.

5. Chapter 5 Current and Conductors The Method of Images • Example • Find the surface charge density at P(2,5,0) on the conducting plane z = 0 if there is a line charge of 30 nC/m located at x = 0, z = 3, as shown below. • We remove the plane and install an image line charge • The field at P may now be obtained by superposition of the known fields of the line charges

6. Chapter 5 Current and Conductors The Method of Images x = 0, z = 3 P(2,5,0) x = 0, z = –3 • Normal to the plane

7. Chapter 5 Current and Conductors Semiconductors • In an intrinsic semiconductor material, such as pure germanium or silicon, two types of current carriers are present: electrons and holes. • The electrons are those from the top of the filled valence band which have received sufficient energy to cross the small forbidden band into conduction band. • The forbidden-band energy gap in typical semiconductors is of the order of 1 eV. • The vacancies left by the electrons represent unfilled energy states in the valence band. They may also move from atom to atom in the crystal. • The vacancy is called a hole, and the properties of semiconductor are described by treating the hole as a positive charge of e, a mobility μh, and an effective mass comparable to that of the electron.

8. Chapter 5 Current and Conductors Semiconductors • The conductivity of a semiconductor is described as: • As temperature increases, the mobilities decrease, but the charge densities increase very rapidly. • As a result, the conductivity of silicon increases by a factor of 100 as the temperature increases from about 275 K to 330 K.

9. Chapter 5 Current and Conductors Semiconductors • The conductivity of the intrinsic semiconductor increases with temperature, while that of a metallic conductor decreases with temperature. • The intrinsic semiconductors also satisfy the point form of Ohm's law: the conductivity is reasonably constant with current density and with the direction of the current density.

10. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • A dielectric material in an electric field can be viewed as a free-space arrangement of microscopic electric dipoles, a pair of positive and negative charges whose centers do not quite coincide. • These charges are notfree charges, not contributing to the conduction process. They are called bound charges, can only shift positions slightly in response to external fields. • All dielectric materials have the ability to store electric energy. This storage takes place by means of a shift (displacement) in the relative positions of the bound charges against the normal molecular and atomic forces.

11. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • The mechanism of this charge displacement differs in various dielectric materials. • Polar molecules have a permanent displacement existing between the centers of “gravity” of the positive and negative charges, each pair of charges acts as a dipole. • Dipoles are normally oriented randomly, and the action of the external field is to align these molecules in the same direction. • Nonpolar molecules does not have dipole arrangement until after a field is applied. • The negative and positive charges shift in opposite directions against their mutual attraction and produce a dipole which is aligned with the electric field.

12. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • Either type of dipole may be described by its dipole moment p: • If there are n dipoles per unit volume, then there are nΔv dipoles in a volume Δv. The total dipole moment is: • We now define the polarization P as the dipole moment per unit volume: • The immediate goal is to show that the bound-volume charge density acts like the free-volume charge density in producing an external field ► We shall obtain a result similar to Gauss’s law.

13. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • Take a dielectric containing nonpolar molecules. No molecules has p, and P = 0 throughout the material. • Somewhere in the interior of the dielectric we select an incremental surface element ΔS, and apply an electric field E. • The electric field produces a moment p = Qd in each molecule, such that p and d make an angle θ with ΔS. • Due to E, any positive charges initially lying below the surface ΔS and within ½dcosθ must have crossed ΔS going upward. • Any negative charges initially lying above the surface ΔS and within ½dcosθ must have crossed ΔS going downward.

14. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • For n molecules/m3, the net total charge (positive and negative) which crosses the elemental surface in upward direction is: • The notation Qb means the bound charge. In terms of the polarization, we have: • If we interpret ΔS as an element of a closed surface, then the direction of ΔS is outward. • The net increase in the bound charge within the closed surface is:

15. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • Seeing some similarity to Gauss’s law, we may now generalize the definition of electric flux density so that it applies to media other than free space. • We write Gauss’s law in terms of ε0E and QT, the total enclosed charge (bound charge plus free charge): • Combining the last three equations: • We may now define D in more general terms: • There is an added term to D when a material is polarized

16. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • For equations with volume charge densities, we now have: • With the help of the divergence theorem, we may transform the equations into equivalent divergence relationships:

17. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • To utilize the new concepts, it is necessary to know the relationship between E and P. • This relationship will be a function of the type of material. We will limit the discussion to isotropic materials for which E and P are linearly related. • In an isotropic material, the vectors E and P are always parallel, regardless of the orientation of the field. • The linear relationship between P and E can be described as: • We now define: χe : electric susceptibility, a measure of how easily a dielectric polarizes in response to an electric field εr : relative permittivity

18. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • In summary, we now have a relationship between D and E which depends on the dielectric material present:

19. Chapter 6 Dielectrics and Capacitance The Nature of Dielectric Materials • Example • We locate a slab of Teflon in the region 0 ≤ x ≤ a, and assume free space where x < 0 and x > a. Outside the Teflon there is a uniform field Eout = E0ax V/m. Find the values for D, E, and P everywhere. • No dielectric materials outside 0 ≤ x ≤ a • No relations yet established over the boundary • This will be discussed in the next section

20. Chapter 5 Current and Conductors Homework 7 • D5.6. • D5.7. • D6.1. (7th Edition) or D5.8 (6th Edition) • Deadline: 17.03.11, at 07:30 am.