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Lecture 4 Electric Potential Conductors Dielectrics Electromagnetics Prof. Viviana Vladutescu Electric Potential Electric Potential The electric field intensity is acting as a force on any charges it arrives upon. Therefore in moving a unit charge from P1 to P2, work must be done against the field. When force is applied to move an object, work is the product of the force and the distance the object travels in the direction of the force W P2 P2 P2 P1 P1 P1 F d l Q E d l Q E d l but since t heforcem ovest hecharge P2 against t he field W Q E d l P1 P2 Therefore W E dl Q P1 without specifying the path P1 E The scalar line integral of an Irrotational (conservative) E field is path-independent P2 E d l 0 Equipotential surfaces A set of points with same potential forms equipotential surface. For a point charge, equipotentials are spheres at fixed radius r. Consider the plot of the electrostatic potential contours forming equipotential surfaces around the point charge superimposed over the field lines for the point charge As we can notice the field goes into the direction of decreasing potential If the behavior of the potential is unknown, the electric intensity field can be determined by finding the maximum rate and direction of the spatial change of the potential field E V P2 W E dl By using the above in the following equation Q P1 we get P2 P2 P1 P1 E d l V d l P2 P2 (V ) a dl dV V l P1 2 P1 Potential difference V1 P2 P2 V21 E d l P1 P2 Q 4 0R P1 V Q 4 0R W V Q P1 Q 4 0R 2 a R dRa R Q 1 1 VP1 VP2 4 0 R2 R1 Absolute potential at some finite radius from a point charge fixed at the origin (reference voltage of zero at an infinite radius) Work per Coulomb required to pull a charge from infinity to the radius R For a collection of charges of continuous distribution V V dQ 4 0R V V 1 4 0 1 4 0 1 4 0 v R dv (V) v s R ds (V) s l R dl (V) l Review If the electrical force moves a charge a certain distance, it does work on that charge. The change in electric potential over this distance is defined through the work done by this force: Work done=Charge on Q*Potential where potential is shorthand for change in electric potential, or potential difference. This is analogous to the definition of the gravitational potential energy through the work done by the force of gravity in moving a mass through a certain distance. The units of potential difference, or simply potential, are Joules / Coulomb, which are called Volts (V). Physically, potential difference has to do with how much work the electric field does in moving a charge from one place to another. • Batteries, for example, are rated by the potential difference across their terminals. In a nine volt battery the potential difference between the positive and negative terminals is precisely nine volts. On the other hand the potential difference across the power outlet in the wall of your home is 110 volts. Conductors Are caractherized by ε, μ and σ The conductivity σ (S/m or 1/Ω*m or mhos/m) -depends on the charge density ρ -depends on the temperature Ex of superconductors: yttrium-barium-copper-oxide Current and Current Density • Current • Current density amountof charge(C) 1C given time(s) 1s current(A) A J 2 2 area(m ) m I J ds Types of current -conduction currents: present in conductors and semiconductors and caused by drift motion of conduction e- or holes in a media in response to an applied field ex: J=σ* E (conduction current density) -displacement or electrolytic currents: is the result of migration of positive and negative ions as well known as time-varying field phenomenon that allows current to flow between plates of a capacitor. -convection currents: involve the movement of charged particles through vacuum, air or other nonconductive media (e- in a cathode ray tube) J&E V=I*R V R I j k k j Conservation of charge J t (V) k I j j 0 (A) Conduction currents Q Nqu an st Q I Nqu s t A J Nqu 2 m For most conducting materials the average drift velocity is directly proportional to el field intensity u e Em / s J e e E E Conductors in static electric field Inside a conductor ρ=0 E=0 Under static conditions the E field on a conductor surface is everywhere normal to the surface (the surface of a conductor is an equipotential surface under static conditions) Charactheristics of E on conductor /free space interfaces -The tangential component of the E field on a conductor surface is zero -The normal component of the E field at a conductor /free space boundary is equal to the surface charge density on the conductor divided by the permittivity of free space Boundary Conditions at a Conductor/ /Free Space Interface Et=0 En=ρs/ε0 E d l E w 0 E 0 t t abcda s S s s E d s En S 0 or En 0 Dielectrics -Ideal dielectrics do not contain free charges -contain bound charges Induced electric dipoles The material is polarized Polar molecules (Permanent dipole moment) Nonpolar molecules Ex: By aligning the molecules during the fabrication of a material (use E field when the material is melted and maintain it until it solidifies) we can obtain electrets The volume density of the electric dipole moment nv P lim v 0 p k 1 k v Polarization vector n-#of molecules per unit volume Vector sum of the induced dipole moments Homogeneous & linear & isotropic media D = εE D=ε0E+P Polarization charge densities ps P an -surface p P -volume A polarized dielectric may be replaced by an equivalent polarization surface charge density and an equivalent polarization volume charge density for field calculation V 1 4 0 s ps R ds 1 4 0 v p R dv Total Charge Q ps ds p dv s v P an ds Pdv o s v