1 EEE 498/598 Overview of Electrical Engineering Lecture 3: Electrostatics: Electrostatic Potential; Charge Dipole; Visualization of Electric Fields; Potentials; Gauss’s Law and Applications; Conductors and Conduction Current

Lecture 3 2 Lecture 3 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. 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.

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

Lecture 3 4 Electrostatic Potential Resulting from Multiple Point Charges Q1Q1 P(R,  ) O Q2Q2 No longer spherically symmetric!

Lecture 3 5 Electrostatic Potential Resulting from Continuous Charge Distributions  line charge  surface charge  volume charge

Lecture 3 6 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). 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

Lecture 3 7 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

Lecture 3 8 Electrostatic Potential Due to Charge Dipole observation point d/2 +Q -Q z d/2  P

Lecture 3 9 Electrostatic Potential Due to Charge Dipole (Cont’d) cylindrical symmetry

Lecture 3 10 Electrostatic Potential Due to Charge Dipole (Cont’d) d/2  P

Lecture 3 11 Electrostatic Potential Due to Charge Dipole in the Far-Field assume R>>d zeroth order approximation: not good enough!

Lecture 3 12 Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d) first order approximation from geometry: d/2  lines approximately parallel

Lecture 3 13 Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d) Taylor series approximation:

Lecture 3 14 Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d)

Lecture 3 15 Electrostatic Potential Due to Charge Dipole in the Far-Field (Cont’d) In terms of the dipole moment:

Lecture 3 16 Electric Field of Charge Dipole in the Far-Field

Lecture 3 17 Visualization of Electric Fields An electric field (like any vector field) can be visualized using flux lines (also called streamlines or lines of force ). 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 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. 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.

Lecture 3 18 Visualization of Electric Potentials The scalar electric potential can be visualized using equipotential surfaces. The scalar electric potential can be visualized using equipotential surfaces. An equipotential surface is a surface over which V is a constant. 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. 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.

Lecture 3 19 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 ). 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. 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.

Lecture 3 20 Faraday’s Experiment charged sphere (+Q) insulator metal

Lecture 3 21 Faraday’s Experiment (Cont’d) Two concentric conducting spheres are separated by an insulating material. Two concentric conducting spheres are separated by an insulating material. The inner sphere is charged to + Q. The outer sphere is initially uncharged. The inner sphere is charged to + Q. The outer sphere is initially uncharged. The outer sphere is grounded momentarily. The outer sphere is grounded momentarily. The charge on the outer sphere is found to be - Q. The charge on the outer sphere is found to be - Q.

Lecture 3 22 Faraday’s Experiment (Cont’d) Faraday concluded there was a “ displacement ” from the charge on the inner sphere through the inner sphere through the insulator to the outer sphere. 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. 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.

Lecture 3 23 Electric Displacement (Electric Flux) +Q -Q

Lecture 3 24 Electric (Displacement) Flux Density The density of electric displacement is the electric (displacement) flux density, D. The density of electric displacement is the electric (displacement) flux density, D. In free space the relationship between flux density and electric field is In free space the relationship between flux density and electric field is

Lecture 3 25 Electric (Displacement) Flux Density (Cont’d) The electric (displacement) flux density for a point charge centered at the origin is The electric (displacement) flux density for a point charge centered at the origin is

Lecture 3 26 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.” 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.”

Lecture 3 27 Gauss’s Law (Cont’d) V S dsds By convention, ds is taken to be outward from the volume V. Since volume charge density is the most general, we can always write Q encl in this way.

Lecture 3 28 Applications of Gauss’s Law Gauss’s law is an integral equation for the unknown electric flux density resulting from a given charge distribution. Gauss’s law is an integral equation for the unknown electric flux density resulting from a given charge distribution. known unknown

Lecture 3 29 Applications of Gauss’s Law (Cont’d) In general, solutions to integral equations must be obtained using numerical techniques. 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. However, for certain symmetric charge distributions closed form solutions to Gauss’s law can be obtained.

Lecture 3 30 Applications of Gauss’s Law (Cont’d) Closed form solution to Gauss’s law relies on our ability to construct a suitable family of Gaussian surfaces. 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. A Gaussian surface is a surface to which the electric flux density is normal and over which equal to a constant value.

Lecture 3 31 Electric Flux Density of a Point Charge Using Gauss’s Law Consider a point charge at the origin: Q

Lecture 3 32 Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d) (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces spheres of radius r where spherical symmetry

Lecture 3 33 Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d) (3) Evaluate the total charge within the volume enclosed by each Gaussian surface

Lecture 3 34 Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d) Q R Gaussian surface

Lecture 3 35 Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d) (4) For each Gaussian surface, evaluate the integral magnitude of D on Gaussian surface. surface area of Gaussian surface.

Lecture 3 36 Electric Flux Density of a Point Charge Using Gauss’s Law (Cont’d) (5) Solve for D on each Gaussian surface

Lecture 3 37 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Consider a spherical shell of uniform charge density: a b

Lecture 3 38 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces spheres of radius r where

Lecture 3 39 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) Here, we shall need to treat separately 3 sub- families of Gaussian surfaces: Here, we shall need to treat separately 3 sub- families of Gaussian surfaces: 1) 2) 3) a b

Lecture 3 40 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) Gaussian surfaces for which Gaussian surfaces for which Gaussian surfaces for which

Lecture 3 41 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) (3) Evaluate the total charge within the volume enclosed by each Gaussian surface

Lecture 3 42 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) For For

Lecture 3 43 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) For For

Lecture 3 44 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) (4) For each Gaussian surface, evaluate the integral magnitude of D on Gaussian surface. surface area of Gaussian surface.

Lecture 3 45 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) (5) Solve for D on each Gaussian surface

Lecture 3 46 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d)

Lecture 3 47 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) Notice that for r > b Notice that for r > b Total charge contained in spherical shell

Lecture 3 48 Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (Cont’d) R D r (C/m)

Lecture 3 49 Electric Flux Density of an Infinite Line Charge Using Gauss’s Law Consider a infinite line charge carrying charge per unit length of q el : z

Lecture 3 50 Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d) (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces cylinders of radius  where

Lecture 3 51 Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d) (3) Evaluate the total charge within the volume enclosed by each Gaussian surface cylinder is infinitely long!

Lecture 3 52 Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d) (4) For each Gaussian surface, evaluate the integral magnitude of D on Gaussian surface. surface area of Gaussian surface.

Lecture 3 53 Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (Cont’d) (5) Solve for D on each Gaussian surface

Lecture 3 54 Gauss’s Law in Integral Form V S

Lecture 3 55 Recall the Divergence Theorem Also called Gauss’s theorem or Green’s theorem. Also called Gauss’s theorem or Green’s theorem. Holds for any volume and corresponding closed surface. Holds for any volume and corresponding closed surface. V S

Lecture 3 56 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

Lecture 3 57 Fields in Materials Materials contain charged particles that respond to applied electric and magnetic fields. 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. Materials are classified according to the nature of their response to the applied fields.

Lecture 3 58 Classification of Materials Conductors Conductors Semiconductors Semiconductors Dielectrics Dielectrics Magnetic materials Magnetic materials

Lecture 3 59 Conductors A conductor is a material in which electrons in the outermost shell of the electron migrate easily from atom to atom. 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. Metallic materials are in general good conductors.

Lecture 3 60 Conduction Current In an otherwise empty universe, a constant electric field would cause an electron to move with constant acceleration. In an otherwise empty universe, a constant electric field would cause an electron to move with constant acceleration. -e e =  C magnitude of electron charge

Lecture 3 61 Conduction Current (Cont’d) In a conductor, electrons are constantly colliding with each other and with the fixed nuclei, and losing momentum. 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 v d which is proportional to the electric field. The net macroscopic effect is that the electrons move with a (constant) drift velocity v d which is proportional to the electric field. Electron mobility

Lecture 3 62 Conductor in an Electrostatic Field To have an electrostatic field, all charges must have reached their equilibrium positions (i.e., they are stationary). 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.) Under such static conditions, there must be zero electric field within the conductor. (Otherwise charges would continue to flow.)

Lecture 3 63 Conductor in an Electrostatic Field (Cont’d) 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. 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 s - an extraordinarily short amount of time indeed. In a metallic conductor, the establishment of equilibrium takes place in about s - an extraordinarily short amount of time indeed.

Lecture 3 64 Conductor in an Electrostatic Field (Cont’d) There are two important consequences to the fact that the electrostatic field inside a metallic conductor is zero: 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. A corollary of the above is that the electric field just outside the conductor must be normal to its surface.

Lecture 3 65 Conductor in an Electrostatic Field (Cont’d)

Lecture 3 66 Macroscopic versus Microscopic Fields In our study of electromagnetics, we use Maxwell’s equations which are written in terms of macroscopic quantities. 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 m = 100 angstroms. For smaller dimensions, quantum mechanics is needed. The lower limit of the classical domain is about m = 100 angstroms. For smaller dimensions, quantum mechanics is needed.

Lecture 3 67 Boundary Conditions on the Electric Field at the Surface of a Metallic Conductor E = 0

Lecture 3 68 Induced Charges on Conductors The BCs given above imply that if a conductor is placed in an externally applied electric field, then 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 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 a surface charge is induced on the conductor to support the electric field

Lecture 3 69 Applied and Induced Electric Fields The applied electric field (E app ) is the field that exists in the absence of the metallic conductor ( obstacle ). The applied electric field (E app ) is the field that exists in the absence of the metallic conductor ( obstacle ). The induced electric field (E ind ) is the field that arises from the induced surface charges. The induced electric field (E ind ) is the field that arises from the induced surface charges. The total field is the sum of the applied and induced electric fields. The total field is the sum of the applied and induced electric fields.