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Lecture 3: Electrostatics: Electrostatic Potential; Charge Dipole; Visualization of Electric Fields; Potentials; Gauss’s Law and Applications; Conductors and Conduction Current Lecture 3 1 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 2 r r V r E d l aˆ r Q 40 r 2 aˆ r dr dr Q 2 40 r r 40 r Q P r spherically symmetric Q Lecture 3 3 R2 Q2 r 2 r r 1 P(R,q,f) R1 Q1 n Qk V r k 1 40 Rk O No longer spherically symmetric! Lecture 3 4 qel r dl V r 40 L R 1 qes r ds V r 40 S R 1 qev r dv V r 40 V R 1 line charge surface charge volume charge Lecture 3 5 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). -Q +Q d Lecture 3 6 • Dipole moment p is a measure of the strength of the dipole and indicates its direction +Q p Qd d p is in the direction from the negative point charge to the positive point charge -Q Lecture 3 7 P observation point z R +Q R r d/2 d/2 -Q p aˆ z Qd q Lecture 3 8 Q Q V r V r , q 4 0 R 4 0 R cylindrical symmetry Lecture 3 9 P R d/2 q r R d/2 R r 2 ( d / 2) 2 rd cos q R r 2 ( d / 2) 2 rd cos q Lecture 3 10 • assume R>>d • zeroth order approximation: R R R R V 0 not good enough! Lecture 3 11 • first order approximation from geometry: R d/2 d/2 d R r cos q 2 d R r cos q 2 q r R lines approximately parallel Lecture 3 12 • Taylor series approximation: 1 1 d 1 d r cos q 1 cos q R 2 r 2r 1 d 1 cos q Recall : r 2r 1 1 1 d 1 cos q R r 2r x 1 1 x n 1 nx, Lecture 3 13 d cos q V r , q 1 40 r 2r Qd cos q 2 40 r Q • In d cos q 1 2r terms of the dipole moment: 1 p aˆ r V 40 r 2 Lecture 3 14 1 V V E V aˆ r aˆq r q r Qd ˆ ˆ a 2 cos q a sin q r q 3 40 r Lecture 3 15 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. Lecture 3 16 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. Lecture 3 17 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. Lecture 3 18 charged sphere (+Q) + + + + metal insulator Lecture 3 19 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. Lecture 3 20 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. Lecture 3 21 +Q -Q Lecture 3 22 The density of electric displacement is the electric (displacement) flux density, D. In free space the relationship between flux density and electric field is D 0 E Lecture 3 23 The electric (displacement) flux density for a point charge centered at the origin is Lecture 3 24 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.” D d s Q encl S Lecture 3 25 S By convention, ds is taken to be outward from the volume V. ds V Qencl qev dv V Since volume charge density is the most general, we can always write Qencl in this way. Lecture 3 26 Gauss’s law is an integral equation for the unknown electric flux density resulting from a given charge distribution. D d s Q encl S known unknown Lecture 3 27 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. Lecture 3 28 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. Lecture 3 29 Consider a point charge at the origin: Q Lecture 3 30 (1) Assume from symmetry the form of the field D aˆ r D r r spherical symmetry (2) Construct a family of Gaussian surfaces spheres of radius r where 0r Lecture 3 31 (3) Evaluate the total charge within the volume enclosed by each Gaussian surface Qencl qev dv V Lecture 3 32 Gaussian surface R Q Qencl Q Lecture 3 33 (4) For each Gaussian surface, evaluate the integral surface area D d s DS S magnitude of D on Gaussian surface. of Gaussian surface. D d s D r 4 r 2 r S Lecture 3 34 (5) Solve for D on each Gaussian surface Qencl D S Q D aˆ r 4 r 2 D Q E aˆ r 0 4 0 r 2 Lecture 3 35 Consider a spherical shell of uniform charge density: q0 , a r b qev 0, otherwise a b Lecture 3 36 (1) Assume from symmetry the form of the field D aˆ r D r R (2) Construct a family of Gaussian surfaces spheres of radius r where 0r Lecture 3 37 Here, we shall need to treat separately 3 sub-families of Gaussian surfaces: 1) 0ra 2) arb 3) rb a b Lecture 3 38 Gaussian surfaces for which 0ra Gaussian surfaces for which arb Gaussian surfaces for which rb Lecture 3 39 (3) Evaluate the total charge within the volume enclosed by each Gaussian surface Qencl qev dv V Lecture 3 40 For For 0ra Qencl 0 arb r Qencl 4 3 4 3 q0 dv q0 r q0 a 3 3 a 4 3 3 q0 r a 3 Lecture 3 41 For rb b Qencl 4 3 4 3 qev dv q0 b q0 a 3 3 a 4 3 3 q0 b a 3 Lecture 3 42 (4) For each Gaussian surface, evaluate the integral surface area D d s DS S magnitude of D on Gaussian surface. of Gaussian surface. D d s D r 4 r 2 r S Lecture 3 43 (5) Solve for D on each Gaussian surface Qencl D S Lecture 3 44 0r a 0, 4 3 3 a r q 3 0 q a 3 0 ar b aˆ r r 2 , D aˆ r 2 r 3 4 r 4 3 3 a b q 3 3 0 q a b 0 aˆ r 3 ˆ r b , a r 2 2 r 3 4 r Lecture 3 45 Notice that for r > b Total charge contained in spherical shell Qtot D aˆ r 2 4 r Lecture 3 46 0.7 0.6 q0 1 C/m 3 0.5 a 1m 0.3 b2m Dr (C/m) 0.4 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 R Lecture 3 47 Consider a infinite line charge carrying charge per unit length of qel: qel z Lecture 3 48 (1) Assume from symmetry the form of the field D aˆ D (2) Construct a family of Gaussian surfaces cylinders of radius where 0 Lecture 3 49 (3) Evaluate the total charge within the volume enclosed by each Gaussian surface Qencl qel dl L cylinder is infinitely long! Q encl q el l Lecture 3 50 (4) For each Gaussian surface, evaluate the integral surface area D d s DS S magnitude of D on Gaussian surface. of Gaussian surface. D d s D 2 l S Lecture 3 51 (5) Solve for D on each Gaussian surface Qencl D S qel D aˆ 2 Lecture 3 52 D d s Q q dv encl ev S V ds V S Lecture 3 53 Also called Gauss’s theorem or Green’s theorem. Holds for any volume and corresponding closed surface. D d s D dv S V ds V S Lecture 3 54 D d s D dv q S V ev dv V Because the above must hold for any volume V, we must have D q ev Differential form of Gauss’s Law Lecture 3 55 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. Lecture 3 56 Conductors Semiconductors Dielectrics Magnetic materials Lecture 3 57 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. Lecture 3 58 In an otherwise empty universe, a constant electric field would cause an electron to move with constant acceleration. E a -e e = 1.602 10-19 C eE a me magnitude of electron charge 59 Lecture 3 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. vd e E Electron mobility 60 Lecture 3 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.) Lecture 3 61 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. Lecture 3 62 • 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. Lecture 3 63 - - - - + + + + + Lecture 3 64 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. Lecture 3 65 Et 0 Dn aˆ n D qes ân - - - E=0 + + + + + Lecture 3 66 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 Lecture 3 67 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. E E app E ind Lecture 3 68