EEE 498/598 Overview of Electrical Engineering Lecture 5: Electrostatics: Dielectric Breakdown, Electrostatic Boundary Conditions, Electrostatic Potential Energy; Conduction Current and Ohm’s Law 1

Lecture 5 Objectives To continue our study of electrostatics with dielectric breakdown, electrostatic boundary conditions and electrostatic potential energy. To study steady conduction current and Ohm’s law. 2

Dielectric Breakdown If a dielectric material is placed in a very strong electric field, electrons can be torn from their corresponding nuclei causing large currents to flow and damaging the material. This phenomenon is called dielectric breakdown. 3

Dielectric Breakdown (Cont’d) The value of the electric field at which dielectric breakdown occurs is called the dielectric strength of the material. The dielectric strength of a material is denoted by the symbol EBR. 4

Dielectric Breakdown (Cont’d) The dielectric strength of a material may vary by several orders of magnitude depending on various factors including the exact composition of the material. Usually dielectric breakdown does not permanently damage gaseous or liquid dielectrics, but does ruin solid dielectrics. 5

Dielectric Breakdown (Cont’d) Capacitors typically carry a maximum voltage rating. Keeping the terminal voltage below this value insures that the field within the capacitor never exceeds EBR for the dielectric. Usually a safety factor of 10 or so is used in calculating the rating. 6

Fundamental Laws of Electrostatics in Integral Form Conservative field Gauss’s law Constitutive relation 7

Fundamental Laws of Electrostatics in Differential Form Conservative field Gauss’s law Constitutive relation 8

Fundamental Laws of Electrostatics The integral forms of the fundamental laws are more general because they apply over regions of space. The differential forms are only valid at a point. From the integral forms of the fundamental laws both the differential equations governing the field within a medium and the boundary conditions at the interface between two media can be derived. 9

Boundary Conditions Within a homogeneous medium, there are no abrupt changes in E or D. However, at the interface between two different media (having two different values of e), it is obvious that one or both of these must change abruptly. 10

Boundary Conditions (Cont’d) To derive the boundary conditions on the normal and tangential field conditions, we shall apply the integral form of the two fundamental laws to an infinitesimally small region that lies partially in one medium and partially in the other. 11

Boundary Conditions (Cont’d) Consider two semi-infinite media separated by a boundary. A surface charge may exist at the interface. Medium 1 Medium 2 x rs 12

Boundary Conditions (Cont’d) Locally, the boundary will look planar rs x x x x x x 13

Boundary Condition on Normal Component of D Consider an infinitesimal cylinder (pillbox) with cross-sectional area Ds and height Dh lying half in medium 1 and half in medium 2: Ds Dh/2 rs x x x x x x Dh/2 14

Boundary Condition on Normal Component of D (Cont’d) Applying Gauss’s law to the pillbox, we have 15

Boundary Condition on Normal Component of D (Cont’d) The boundary condition is If there is no surface charge For non-conducting materials, rs = 0 unless an impressed source is present. 16

Boundary Condition on Tangential Component of E Consider an infinitesimal path abcd with width Dw and height Dh lying half in medium 1 and half in medium 2: Dw d a Dh/2 Dh/2 b c 17

Boundary Condition on Tangential Component of E (Cont’d) 18

Boundary Condition on Tangential Component of E (Cont’d) Applying conservative law to the path, we have 19

Boundary Condition on Tangential Component of E (Cont’d) The boundary condition is 20

Electrostatic Boundary Conditions - Summary At any point on the boundary, the components of E1 and E2 tangential to the boundary are equal the components of D1 and D2 normal to the boundary are discontinuous by an amount equal to any surface charge existing at that point 21

Electrostatic Boundary Conditions - Special Cases Special Case 1: the interface between two perfect (non-conducting) dielectrics: Physical principle: “there can be no free surface charge associated with the surface of a perfect dielectric.” In practice: unless an impressed surface charge is explicitly stated, assume it is zero. 22

Electrostatic Boundary Conditions - Special Cases Special Case 2: the interface between a conductor and a perfect dielectric: Physical principle: “there can be no electrostatic field inside of a conductor.” In practice: a surface charge always exists at the boundary. 23

Potential Energy When one lifts a bowling ball and places it on a table, the work done is stored in the form of potential energy. Allowing the ball to drop back to the floor releases that energy. Bringing two charges together from infinite separation against their electrostatic repulsion also requires work. Electrostatic energy is stored in a configuration of charges, and it is released when the charges are allowed to recede away from each other. 24

Electrostatic Energy in a Discrete Charge Distribution Consider a point charge Q1 in an otherwise empty universe. The system stores no potential energy since no work has been done in creating it. Q1 25

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) Now bring in from infinity another point charge Q2. The energy required to bring Q2 into the system is Q1 Q2 R12 V12 is the electrostatic potential due to Q1 at the location of Q2. 26

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) Now bring in from infinity another point charge Q3. The energy required to bring Q3 into the system is Q1 Q2 R12 Q3 R13 R23 27

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) The total energy required to assemble the system of three charges is 28

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) Now bring in from infinity a fourth point charge Q4. The energy required to bring Q4 into the system is The total energy required to assemble the system of four charges is 29

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) Bring in from infinity an ith point charge Qi into a system of i-1 point charges. The energy required to bring Qi into the system is The total energy required to assemble the system of N charges is 30

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) Note that  Physically, the above means that the partial energy associated with two point charges is equal no matter in what order the charges are assembled. 31

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) 32

Electrostatic Energy in a Discrete Charge Distribution (Cont’d) 33

Electrostatic Energy in a Discrete Charge Distribution (Cont’d)  Physically, Vi is the potential at the location of the ith point charge due to the other (N-1) charges. 34

Electrostatic Energy in a Continuous Charge Distribution 35

Electrostatic Energy in a Continuous Charge Distribution (Cont’d) 36

Electrostatic Energy in a Continuous Charge Distribution (Cont’d) Divergence theorem and 37

Electrostatic Energy in a Continuous Charge Distribution (Cont’d) Let the volume V be all of space. Then the closed surface S is sphere of radius infinity. All sources of finite extent look like point charges. Hence, 38

Electrostatic Energy in a Continuous Charge Distribution (Cont’d) Electrostatic energy density in J/m3. 39

Electrostatic Energy in a Continuous Charge Distribution (Cont’d) energy required to set the field up in free space energy required to polarize the dielectric 40

Electrostatic Energy in a Capacitor V2 V1 + - +Q -Q V12 41

Electrostatic Energy in a Capacitor Letting V = V12 = V2 – V1 42

Electrostatic Forces: The Principle of Virtual Work Electrostatic forces acting on bodies can be computed using the principle of virtual work. The force on any conductor or dielectric body within a system can be obtained by assuming a differential displacement of the body and computing the resulting change in the electrostatic energy of the system. 43

Electrostatic Forces: The Principle of Virtual Work (Cont’d) The electrostatic force can be evaluated as the gradient of the electrostatic energy of the system, provided that the energy is expressed in terms of the coordinate location of the body being displaced. 44

Electrostatic Forces: The Principle of Virtual Work (Cont’d) When using the principle of virtual work, we can assume either that the conductors maintain a constant charge or that they maintain a constant voltage (i.e, they are connected to a battery). 45

Electrostatic Forces: The Principle of Virtual Work (Cont’d) For a system of bodies with fixed charges, the total electrostatic force acting on the body is given by 46

Electrostatic Forces: The Principle of Virtual Work (Cont’d) For a system of bodies with fixed potentials, the total electrostatic force acting on the body is given by 47

Force on a Capacitor Plate Compute the force on one plate of a charged parallel plate capacitor. Neglect fringing of the field. y +Q -Q The force on the upper plate can be found assuming a system of fixed charge. 48

Force on a Capacitor Plate (Cont’d) The capacitance can be written as a function of the location of the upper plate: The electrostatic energy stored in the capacitor may be evaluated as a function of the charge on the upper plate and its location: 49

Force on a Capacitor Plate (Cont’d) The force on the upper plate is given by Using Q = CV, 50

Force on a Capacitor Plate (Cont’d) Compute the force on one plate of a charged parallel plate capacitor. Neglect fringing of the field. y V = V12 V = 0 The force on the upper plate can be found assuming a system of fixed potential. 51

Force on a Capacitor Plate (Cont’d) The capacitance can be written as a function of the location of the upper plate: The electrostatic energy stored in the capacitor may be written as a function of the voltage across the plates and the location of the upper plate: 52

Force on a Capacitor Plate (Cont’d) The force on the upper plate is given by Manipulating, we obtain 53

Steady Electric Current Electrostatics is the study of charges at rest. Now, we shall allow the charges to move, but with a constant velocity (no time variation). “steady electric current” = “direct current (DC)” 54

Conductors and Conductivity A conductor is a material in which electrons are free to migrate over macroscopic distances within the material. Metals are good conductors because they have many free electrons per unit volume. Other materials with a smaller number of free electrons per unit volume are also conductors. Conductivity is a measure of the ability of the material to conduct electricity. 55

Semiconductor A semiconductor is a material in which electrons in the outermost shell are able to migrate over macroscopic distances when a modest energy barrier is overcome. Semiconductors support the flow of both negative charges (electrons) and positive charges (holes). 56

Conduction Current When subjected to a field, an electron in a conductor migrates through the material constantly colliding with the lattice and losing momentum. The net effect is that the electron moves (drifts) with an average drift velocity that is proportional to the electric field. electron mobility 57

Conduction Current (Cont’d) Consider a conducting wire in which charges subject to an electric field are moving with drift velocity vd. current electron E vd Ds cross-section 58

Conduction Current (Cont’d) If there are nc free electrons per cubic meter of material, then the charge density within the wire is Consider an infinitesimal volume associated with Ds: Ds Dl 59

Conduction Current (Cont’d) The total charge contained within Dv is This charge packet moves through the surface Ds with speed The amount of time it takes for the charge packet to move through Ds is 60

Conduction Current (Cont’d) Current is the rate at which charges passes through a specified surface area (such as the cross-section of a wire). The incremental current through Ds is given by 61

Current Density The component of the current density in the direction normal to Ds is In general, the current density is given by 62

Current Density (Cont’d) The constant of proportionality between the electric field and the conduction current density is called the conductivity of the material: Ohm’s law at a point: 63

Current Density (Cont’d) The conductivity of the medium is the macroscopic quantity which allows us to treat conduction current without worrying about the microscopic behavior of conductors. In semiconductors, we have both holes and electrons hole mobility hole density 64

Current Density (Cont’d) The total current flowing through a cross-sectional area S may be found as If the current density is uniform throughout the cross-section, we have cross-sectional area 65

Current Flow Consider a wire of non-uniform cross-section: E A1 A2 66

Current Flow (Cont’d) To maintain a constant electric field and a steady current flow, both E and J must be parallel to the conductor boundaries. The total current passing through the cross-section A1 must be the same as through the cross-section A2. So the current density must be greater in A2. 67

Ohm’s Law and Resistors Consider a conductor of uniform cross-section: Let the wires and the two exposed faces of the “resistor” be perfect conductor. l A s E V2 F1 + V - I In a perfect conductor: J is finite s is infinite E must be zero. 68

Ohm’s Law and Resistors (Cont’d) To derive Ohm’s law for resistors from Ohm’s law at a point, we need to relate the circuit quantities (V and I) to the field quantities (E and J) The electric field within the material is given by The current density in the wire is 69

Ohm’s Law and Resistors (Cont’d) Plugging into J = sE, we have Define the resistance of the device as Thus, Ohm’s law for resistors 70