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 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • Consider the interface between two dielectrics having permittivities ε1 and ε2, as shown below. • We first examine the tangential components around the small closed path on the left, with Δw<< and Δh<<< :

21. Chapter 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • The tangential electric flux density is discontinuous, • The boundary conditions on the normal components are found by applying Gauss’s law to the small cylinder shown at the right of the previous figure (net tangential flux is zero). • ρS cannot be a bound surface charge density because the polarization already counted in by using dielectric constant different from unity • ρS cannot be a free surface charge density, for no free charge available in the perfect dielectrics we are considering • ρS exists only in special cases where it is deliberately placed there

22. Chapter 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • Except for this special case, we may assume ρS is zero on the interface: • The normal component of electric flux density is continuous. • It follows that:

23. Chapter 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • Combining the normal and the tangential components of D, • After one division,

24. Chapter 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • The direction of E on each side of the boundary is identical with the direction of D, because D = εE.

25. Chapter 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • The relationship between D1 and D2 may be derived as: • The relationship between E1 and E2 may be derived as:

26. Chapter 6 Dielectrics and Capacitance Boundary Conditions for Perfect Dielectric Materials • Example • Complete the previous example by finding the fields within the Teflon. • E only has normal component

27. Chapter 6 Dielectrics and Capacitance Boundary Conditions Between a Conductor and a Dielectric • The boundary conditions existing at the interface between a conductor and a dielectric are much simpler than those previously discussed. • First, we know that D and E are both zero inside the conductor. • Second, the tangential E and D components must both be zero to satisfy: • Finally, the application of Gauss’s law shows once more that both D and E are normal to the conductor surface and that DN = ρS and EN = ρS/ε. • The boundary conditions for conductor–free space are valid also for conductor–dielectric boundary, with ε0 replaced by ε.

28. Chapter 6 Dielectrics and Capacitance Boundary Conditions Between a Conductor and a Dielectric • We will now spend a moment to examine one phenomena: “Any charge that is introduced internally within a conducting material will arrive at the surface as a surface charge.” • Given Ohm’s law and the continuity equation (free charges only): • We have:

29. Chapter 6 Dielectrics and Capacitance Boundary Conditions Between a Conductor and a Dielectric • If we assume that the medium is homogenous, so that σ and ε are not functions of position, we will have: • Using Maxwell’s first equation, we obtain; • Making the rough assumption that σ is not a function of ρv, it leads to an easy solution that at least permits us to compare different conductors. • The solution of the above equation is: • ρ0 is the charge density at t = 0 • Exponential decay with time constant of ε/σ

30. Chapter 6 Dielectrics and Capacitance Boundary Conditions Between a Conductor and a Dielectric • Good conductors have low time constant. This means that the charge density within a good conductors will decay rapidly. • We may then safely consider the charge density to be zero within a good conductor. • In reality, no dielectric material is without some few free electrons (the charge density is thus not completely zero), but the charge introduced internally in any of them will eventually reach the surface. ρv ρ0 ρ0/e ε/σ t

31. Chapter 6 Dielectrics and Capacitance Homework 7 • D5.6. • D5.7. • D6.1. • D6.2. (Bonus Question, + 20 points if correctly made) • Deadline: 12.06.2012, at 08:00 am.