UPB / ETTI O.DROSU Electrical Engineering 2 Lecture 3: Electrostatics: Electrostatic Potential; Charge Dipole; Visualization of Electric Fields; Potentials; Gauss’s Law and Applications; Conductors and Conduction Current 1

Objectives To continue our study of electrostatics with electrostatic potential; charge dipole; visualization of electric fields and potentials; Gauss’s law and applications; conductors and conduction current. 2

Electrostatic Potential of a Point Charge at the Origin Q P spherically symmetric 3

Electrostatic Potential Resulting from Multiple Point Charges Q2 P(R,q,f) Q1 O No longer spherically symmetric! 4

Electrostatic Potential Resulting from Continuous Charge Distributions  line charge  surface charge  volume charge 5

Charge Dipole An electric charge dipole consists of a pair of equal and opposite point charges separated by a small distance (i.e., much smaller than the distance at which we observe the resulting field). d +Q -Q 6

Dipole Moment Dipole moment p is a measure of the strength of the dipole and indicates its direction +Q -Q p is in the direction from the negative point charge to the positive point charge 7

Electrostatic Potential Due to Charge Dipole observation point d/2 +Q -Q z q P 8

Electrostatic Potential Due to Charge Dipole cylindrical symmetry 9

Electrostatic Potential Due to Charge Dipole q d/2 d/2 10

Electrostatic Potential Due to Charge Dipole in the Far-Field assume R>>d zeroth order approximation: not good enough! 11

Electrostatic Potential Due to Charge Dipole in the Far-Field first order approximation from geometry: q d/2 d/2 lines approximately parallel 12

Electrostatic Potential Due to Charge Dipole in the Far-Field Taylor series approximation: 13

Electrostatic Potential Due to Charge Dipole in the Far-Field In terms of the dipole moment: 14

Electric Field of Charge Dipole in the Far-Field 15

Visualization of Electric Fields An electric field (like any vector field) can be visualized using flux lines (also called streamlines or lines of force). A flux line is drawn such that it is everywhere tangent to the electric field. A quiver plot is a plot of the field lines constructed by making a grid of points. An arrow whose tail is connected to the point indicates the direction and magnitude of the field at that point. 16

Visualization of Electric Potentials The scalar electric potential can be visualized using equipotential surfaces. An equipotential surface is a surface over which V is a constant. Because the electric field is the negative of the gradient of the electric scalar potential, the electric field lines are everywhere normal to the equipotential surfaces and point in the direction of decreasing potential. 17

Visualization of Electric Fields Flux lines are suggestive of the flow of some fluid emanating from positive charges (source) and terminating at negative charges (sink). Although electric field lines do NOT represent fluid flow, it is useful to think of them as describing the flux of something that, like fluid flow, is conserved. 18

Faraday’s Experiment charged sphere (+Q) + insulator metal 19

Faraday’s Experiment Two concentric conducting spheres are separated by an insulating material. The inner sphere is charged to +Q. The outer sphere is initially uncharged. The outer sphere is grounded momentarily. The charge on the outer sphere is found to be -Q. 20

Faraday’s Experiment Faraday concluded there was a “displacement” from the charge on the inner sphere through the inner sphere through the insulator to the outer sphere. The electric displacement (or electric flux) is equal in magnitude to the charge that produces it, independent of the insulating material and the size of the spheres. 21

Electric Displacement (Electric Flux) +Q -Q 22

Electric (Displacement) Flux Density The density of electric displacement is the electric (displacement) flux density, D. In free space the relationship between flux density and electric field is 23

Electric (Displacement) Flux Density The electric (displacement) flux density for a point charge centered at the origin is 24

Gauss’s Law Gauss’s law states that “the net electric flux emanating from a close surface S is equal to the total charge contained within the volume V bounded by that surface.” 25

general, we can always write Gauss’s Law V S ds By convention, ds is taken to be outward from the volume V. Since volume charge density is the most general, we can always write Qencl in this way. 26

Applications of Gauss’s Law Gauss’s law is an integral equation for the unknown electric flux density resulting from a given charge distribution. known unknown 27

Applications of Gauss’s Law In general, solutions to integral equations must be obtained using numerical techniques. However, for certain symmetric charge distributions closed form solutions to Gauss’s law can be obtained. 28

Applications of Gauss’s Law Closed form solution to Gauss’s law relies on our ability to construct a suitable family of Gaussian surfaces. A Gaussian surface is a surface to which the electric flux density is normal and over which equal to a constant value. 29

Electric Flux Density of a Point Charge Using Gauss’s Law Consider a point charge at the origin: Q 30

Electric Flux Density of a Point Charge Using Gauss’s Law (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces spherical symmetry spheres of radius r where 31

Electric Flux Density of a Point Charge Using Gauss’s Law (3) Evaluate the total charge within the volume enclosed by each Gaussian surface 32

Electric Flux Density of a Point Charge Using Gauss’s Law Q Gaussian surface R 33

Electric Flux Density of a Point Charge Using Gauss’s Law (4) For each Gaussian surface, evaluate the integral surface area of Gaussian surface. magnitude of D on Gaussian surface. 34

Electric Flux Density of a Point Charge Using Gauss’s Law (5) Solve for D on each Gaussian surface 35

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Consider a spherical shell of uniform charge density: a b 36

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces spheres of radius r where 37

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Here, we shall need to treat separately 3 sub-families of Gaussian surfaces: 1) a b 2) 3) 38

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Gaussian surfaces for which Gaussian surfaces for which Gaussian surfaces for which 39

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (3) Evaluate the total charge within the volume enclosed by each Gaussian surface 40

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law For 41

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law For 42

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (4) For each Gaussian surface, evaluate the integral surface area of Gaussian surface. magnitude of D on Gaussian surface. 43

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (5) Solve for D on each Gaussian surface 44

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law 45

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Notice that for r > b Total charge contained in spherical shell 46

Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R D r (C/m) 47

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law Consider a infinite line charge carrying charge per unit length of qel: z 48

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces cylinders of radius r where 49

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (3) Evaluate the total charge within the volume enclosed by each Gaussian surface cylinder is infinitely long! 50

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (4) For each Gaussian surface, evaluate the integral surface area of Gaussian surface. magnitude of D on Gaussian surface. 51

Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (5) Solve for D on each Gaussian surface 52

Gauss’s Law in Integral Form V S 53

Recall the Divergence Theorem Also called Gauss’s theorem or Green’s theorem. Holds for any volume and corresponding closed surface. V S 54

Applying Divergence Theorem to Gauss’s Law  Because the above must hold for any volume V, we must have Differential form of Gauss’s Law 55

Fields in Materials Materials contain charged particles that respond to applied electric and magnetic fields. Materials are classified according to the nature of their response to the applied fields. 56

Classification of Materials Conductors Semiconductors Dielectrics Magnetic materials 57

Conductors A conductor is a material in which electrons in the outermost shell of the electron migrate easily from atom to atom. Metallic materials are in general good conductors. 58

Conduction Current In an otherwise empty universe, a constant electric field would cause an electron to move with constant acceleration. -e e = 1.602  10-19 C magnitude of electron charge 59

Conduction Current In a conductor, electrons are constantly colliding with each other and with the fixed nuclei, and losing momentum. The net macroscopic effect is that the electrons move with a (constant) drift velocity vd which is proportional to the electric field. Electron mobility 60

Conductor in an Electrostatic Field To have an electrostatic field, all charges must have reached their equilibrium positions (i.e., they are stationary). Under such static conditions, there must be zero electric field within the conductor. (Otherwise charges would continue to flow.) 61

Conductor in an Electrostatic Field If the electric field in which the conductor is immersed suddenly changes, charge flows temporarily until equilibrium is once again reached with the electric field inside the conductor becoming zero. In a metallic conductor, the establishment of equilibrium takes place in about 10-19 s - an extraordinarily short amount of time indeed. 62

Conductor in an Electrostatic Field There are two important consequences to the fact that the electrostatic field inside a metallic conductor is zero: The conductor is an equipotential body. The charge on a conductor must reside entirely on its surface. A corollary of the above is that the electric field just outside the conductor must be normal to its surface. 63

Conductor in an Electrostatic Field + - 64

Macroscopic versus Microscopic Fields In our study of electromagnetics, we use Maxwell’s equations which are written in terms of macroscopic quantities. The lower limit of the classical domain is about 10-8 m = 100 angstroms. For smaller dimensions, quantum mechanics is needed. 65

Boundary Conditions on the Electric Field at the Surface of a Metallic Conductor - - - - - E = 0 + + + + + 66

Induced Charges on Conductors The BCs given above imply that if a conductor is placed in an externally applied electric field, then the field distribution is distorted so that the electric field lines are normal to the conductor surface a surface charge is induced on the conductor to support the electric field 67

Applied and Induced Electric Fields The applied electric field (Eapp) is the field that exists in the absence of the metallic conductor (obstacle). The induced electric field (Eind) is the field that arises from the induced surface charges. The total field is the sum of the applied and induced electric fields. 68