ENE/EIE 325 Electromagnetic Fields and Waves Lecture 3 Coulomb’s law, Static Electric Fields and Electric Flux density
Outline Coulomb’s law Electric field intensity in different charge configurations Electric flux density
Electric charge A fundamental quantity that is responsible for all electric phenomena. 1e = 1.6x10-19 Coulomb Law of attraction: positive charge attracts negative charge Same polarity charges repel one another http://www.school-for-champions.com/science/images/electrical_charges-field_lines.gif http://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/VFPt_charges_plus_minus_thumb.svg/300px-VFPt_charges_plus_minus_thumb.svg.png
Coulomb’s Experimental Law + Q1 Q2 R F Force of repulsion, F, occurs when charges have the same sign. Charges attract when of opposite sign where
Free Space Permittivity with which the Coulomb force becomes:
Coulomb Force with Charges Off-Origin
Electric Field Intensity Consider the force acting on a test charge, Qt , arising from charge Q1: where a1t is the unit vector directed from Q1 to Qt The electric field intensity is defined as the force per unit test charge, or N/C A more convenient unit for electric field is V/m, as will be shown.
Electric Field of a Charge Off-Origin
Superposition of Fields From Two Point Charges For n charges:
Ex1 A charge Q is 310-9 C produces the electric field E. Find E at P, using First, find the vectors: Then:
Ex1 (continued) Find E at P, using Now: so that: where
Volume Charge Density Given a charge Q within a volume , the volume charge density is defined as: ….so that the charge contained within a volume is
Ex2 Find the charge contained within a 2-cm length of the electron beam shown below, in which the charge density is
Ex2 (continued) Q
Electric Field from Volume Charge Distributions Next, sum all contributions throughout a volume and take the limit as approaches zero, to obtain the integral:
Line Charge Electric Field Line charge of constant density L Coul/m lies along the entire z axis. At point P, the electric field arising from charge dQ on the z axis is: where so that Therefore and
Line Charge Field (continued) We have: By symmetry, only a radial component is present:
Line Charge Field Results
Ex3: Off-Axis Line Charge With the line displaced to (6,8), the field becomes: where Finally:
Sheet Charge Field or The total field on the x axis is: Uniform surface charge of density s covers the entire y-z plane. We begin by writing down the line charge field on the x axis for a strip of differential width dy´, where we consider the x component of that field: The total field on the x axis is: or
Types of Streamline Sketches of Fields:
Methodology of Streamline Construction The ratio of the y and x field components gives the slope of the field plot in the x-y plane
Ex4: Line Charge Field Then whose solution is Finally Begin with the normalized line charge field in cylindrical coordinates: Convert to rectangular components: Then whose solution is Finally
Michael Faraday Michael Faraday (22 September 1791 – 25 August 1867) was an English scientist who contributed to the fields of electromagnetism and electrochemistry. http://youtube/YHNMMdGswwg
Faraday Experiment He started with a pair of metal spheres of different sizes; the larger one consisted of two hemispheres that could be assembled around the smaller sphere
Faraday Experiment Illustration Grounded outer conducting sphere +q + Insulating or dielectric material +q + Charged conducting sphere +q + Guess what will happen?
Observations Charge transfers from inner to outer sphere without contact. Charge on outer sphere is of the same magnitude but opposite sign –q. Charge on outer sphere is the same regardless of the insulating material used. Charge on outer electrode is the same regardless of electrode’s shape. +q + - –q
Faraday Apparatus, Before Grounding +Q The inner charge, Q, induces an equal and opposite charge, -Q, on the inside surface of the outer sphere, by attracting free electrons in the outer material toward the positive charge. This means that before the outer sphere is grounded, charge +Q resides on the outside surface of the outer conductor.
Faraday Apparatus, After Grounding q = 0 ground attached Attaching the ground connects the outer surface to an unlimited supply of free electrons, which then neutralize the positive charge layer. The net charge on the outer sphere is then the charge on the inner layer, or -Q.
Interpretation of the Faraday Experiment q = 0 Faraday concluded that there occurred a charge “displacement” from the inner sphere to the outer sphere. Displacement involves a flow or flux, existing within the dielectric, and whose magnitude is equivalent to the amount of “displaced” charge. Specifically:
Electric Flux Density q = 0 The density of flux at the inner sphere surface is equivalent to the density of charge there (in Coul/m2)
Vector Field Description of Flux Density q = 0 A vector field is established which points in the direction of the “flow” or displacement. In this case, the direction is the outward radial direction in spherical coordinates. At each surface, we would have:
Radially-Dependent Electric Flux Density q = 0 D(r) At a general radius r between spheres, we would have: Expressed in units of Coulombs/m2, and defined over the range (a ≤ r ≤ b) r
Point Charge Fields C/m2 (0 < r <∞ ) V/m (0 < r <∞ ) If we now let the inner sphere radius reduce to a point, while maintaining the same charge, and let the outer sphere radius approach infinity, we have a point charge. The electric flux density is unchanged, but is defined over all space: C/m2 (0 < r <∞ ) We compare this to the electric field intensity in free space: V/m (0 < r <∞ ) ..and we see that:
Finding E and D from Charge Distributions We learned in Chapter 2 that: It now follows that:
Displacement Flux Concept Faraday concluded that there was some sort of "displacement" from the inner sphere to the outer which was independent of the medium. It is called displacement flux (aka electric flux, ). Displacement flux makes outer sphere charged with –q. Inside the outer sphere shell The inner sphere Charge in E + - - +q + –q +q + +