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Db I Dl E R R R = ¥ M P4 $ Introduction to the Laws of Electrostatics and Magnetostatics Gauss’ Law Chapter 3 r r m Idl¥ e$ dB= o 11 R 4p R2 The Laws of Electromagnetics Introduction Electromagnetic fields are all around us, both manmade and naturally occurring. This course is concerned with understanding the basics of elec- tromagnetic fields and their interaction with materials. We consider some statics, but most of the course deals with electromagnetic waves. We will postulate the existence of electric and magnetic fields. We can think of these as invisible lines of “flux” or stream lines. Changed particles in these fields will experience forces. Interaction of the electromagnetic fields with matter takes place through these forces. The laws of electromagnetics are summarized in four beautifully elegant vector equations formulated by J. C. Maxwell. They are called Maxwell’s Equations. Electromagnetics is the study and solution of Maxwell’s Equa- tions and of the associated force laws and material constitutive laws. 1. The Electromagnetic Fields [text, Ch. 1-2, 4-3,4-4,5-1,5-2,5-4] MKS units r a. Electric field vector (electric field amplitude: Ch. 4) E Volt/m r 2 b. Magnetic flux density (magnetic field: Ch. 5) B Webers/m The electromagnetic fields are vector functions; i.e., each componentr is a function of space. In general, they are also functions of time, i.e.,bgrt, . In these notes, a vector is denoted by an over-arrow. For example, a space vector is written in the form: r $$$ rexeyez=++xyz, $ where ei is a unit vector in the direction i, and x, y, z are scaler values. A general electric field must be written, for example, as: r $ $ $ E=++ eExxafafaf xyzt,,, eE yy xyzt,,, eE zz xyzt,,, r In this example, each of the components of the vector E is a function of three space variables x, y, z and time t. 2. Force Laws [Ch. 5-1, Lorentz Force Equation] The force on a charged particle due to the electric and magnetic fields is given by: r rr forE field FqEe = r r r r r for B field FqvBm =¥ v = velocity (a vector) The symbol ¥ denotes the vector cross-product. Note that magnetic field only affects a moving charge. r r rrr r If both E and B are present, then FqEvB=+¥ Text eq. 5.5 Other forms of the magnetic force law: r r (i) A moving electronr is ar current:r qv¤ Idl Therefore, dFm =¥ Idl B Text eq. 5-8 (ii) Force on a magnetic monopole (There is no rrsuch thing, but we can think of one pole of a bar magnet): FqB= m Examples: a) cathode ray tube (TV, oscilloscope, computer monitor) Phosphorescent Screen Electric and magnetic fields deflect the moving electron which strikes the screen and causes emission of light. Uniformr fieldsr B1 and E1 (same direction) Electric field accelerates electron VV= toward Vo plate. It travelsr through o the hole with velocityv . r V E = o V = 0 d Uniformr E field Electron is emitted by cathode. Cathode Ther position where ther electron strikes the screen can be controlled by the E 1 field and by the B1 field. 3. The Sources of Electromagnetic Fields Electric and magnetic fields are caused by sources. Maxwell’s equa- tions describe how electromagnetic fields are caused by the sources. (i) Primary sources charges and currents r rr pq, fi E jI, fi B (ii) Secondary sources r r A time change of E field results in B field. r r A time change of B field results in E field. If there is no time dependence, only (i) is present. In elementary (lower division) physics,r your studied the two primary laws in statics which are separate for E and B fields. We quickly repeat these just to remind you, but will use a more general formulation. $ 4. Coulomb’s Law [In vacuum] eR q r 2 R r RR= q1 r qqe$ $ F = 12R andeR is the unit rvector along qq12 4pe R2 R o fi=e$ r . R R 5. Definition of Electric Field rr If q2 is a small test charge, FqE= 2 , and then the field is: r qe$ 1 R ⇐ This is the electric field due to q1 E = 2 4peo R at the position test charge. (Field is force per unit charge.) So far, it is general. To calculate anything, we must use a particular coordi- nate system with an origin. r position vector E origin (observation point) O r rr r Rxx=-¢ x¢ q¢ (source point) Point charge is an approximation, but can be a quite good approximation (e.g., small charged sphere, a single electron, etc.) 6. More than one charge: Use superposition; that is, vector addition. Units E volt/m R 2 m2 q coulombs eo Farads/m Permitivity of Vacuum e r (dimensionless) Relative permitivity Volt/m = Coulomb/(Farad/m)m2 from Coulomb’s Law ⇒ Volt = Coulomb/Farad, which we recognize from circuits: V = Q/C. 7. Ampere’s Experimental Observations The force between two long straight wires carrying parallel currents of I1 and I2 separated by distance d is: rrm II F = o 12L {attractive if currents are in the same direction} 2p d This formula is used for the international standard for current. But this is not a point form law. Infinitely long currents are involved. Therefore, this is not the basic form (not equivalent to Coulomb’s Law). 8. Biot-Savart’s Law (Text 5-2) Magnetic field caused by an elemental current: r r r m Idl¥ e$ I r r dB= o 11 R 1 r 4p R 2 dB1 r (into paper) 0 r r R r¢ r rr dl1 Rrr=-¢ This is also an inverser square law, but, due to crossr product, much more complicated. Also, dB must ber integrated to giveB because there is no isolated elemental current Idl in reality. Combining with force law will give force between two elemental currents: rr r $ m I12 I dl 2¥¥di dl 1 eR dF = o 4p R 2 To get the force between two finite current loops, we would have to perform a double integration over the two. Because of the triple cross-product, this is pretty messy! Units r B Webers/m2 or Tesla Weber is unit of flux: r r R 2 m2 Y=zz BdS◊ I Ampere mo Henrys/m permeability of vacuum From Law of Biot and Savart, we have the following dimensional analysis: Tesla = (Henry/m) ¥ (Ampere·m/m2) = Henry Amp/m2 So Weber = Henry Ampere Now let’s devide by seconds: Weber/sec = Henry/sec Ampere But a Henry/sec = ohm dI Why? vL= or volt = Henry Amp/sec, or V = I R, where R = Henry/sec dt Weber/sec = W Amp = volt r r r From the force law dF=¥ Idl dB, we have: Newton = Ampere m Tesla = Amp m Henry Amp/m2 or a Newton meter = Joule = Henry (Ampere)2 1 Recall that LI2 = W (the energy stored in an energized inductor) 2 m 9. The Presence of Materials The above primary source laws are in a vacuum. Since all matter con- sists of change particles, in principle, we could just count the effect of all particles. This is, however, clearly impractical. What we can do instead is make some useful and practical approximations and take into account the presence of materials in a “gross” way. This is enormously difficult and complex. We will study it later in a very elementary fashion only. For now, we simply use the following approx- imations. (i) In certain materials and relatively weak fields (homogeneous, linear, isotropic dielectric material) r $ qer E = , where eee= or 4peR 2 and where er is a constant property of the material. It is called the relative permittivity of the material. This is a relatively good, and often valid, approximation for many materials. (ii) In certain materials and relatively weak fields (homogeneous, linear, isotropic magnetic materials) r r m Idl¥ e$ again, a constant property of the dB = r mmm= 4p R 2 ormaterial called relative permeability This is a rather poor approximation for almost all materials which have mr significantly different from 1. (But we use it in thisr form anyway, with the understanding that mr is, in fact, a function of B itself.) Most dielectrics for which circumstance (i) holds will have mr ª1 (very close). 10. Two New Field Quantities r rr r B DE= e and H = (Text Ch. 4-8, 5-6.2) m These are linear approximations only, a more general form will be dis- cussed later. r r qe$ r 1 Idl¥ e$ D = R and H = R 4p R 2 4p R 2 r r 2 Units of D arer Coulombr /m and H are Amp/m. Remember: E and B are the fundamental quantities.r r The force laws are based on these, but it will be easier to relate D and H to the primary sources. r r Although E and B are the fundamental quantities,r r in Engineering, for elec- tromagnetic waves, we tend to use mostly E and H . [Note that the ratio of has E/H units of Ω (impedance), thus they are analogous to voltage and current respectively.] 11. Charge and Current Densities, Volume, Surface and Line Integrals Charges and currents are often not point charges and filament currents. Instead they are distributed in some way. Continuous charge distributions (Text 4-2) (a.) Volume charge distribution (or charge density) Q D Qdv= r rv = lim then zzz v DvÆ0 DV v and dQ= rv dv If (and only if!) r is a constant, i.e., not a function of position (of coordi- nates), then: QV= rv Note: For volume charge distributions, the subscript v is often omitted.
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