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 electromagnetic 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 equations 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 approximations. (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 electromagnetic 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 coordinates), then:

QV= rv Note: For volume charge distributions, the subscript v is often omitted. (b.) Surface charge distribution (surface charge density) DQ then Qds=Æ r rs = lim zz s DsÆ0 DS s

and dQ= rS ds

If rS is constant on surface, then Qs= rS (c.) Line charge distribution DQ rl = lim and Qdl= r DlÆ0 Dl z l l

and dQ1 = r l dl

If r l is constant along the line, the Q = r l ◊1. (d.) Surface integral of a vector flux [Text 5-8] r r z Dds◊ ∫ Total flux crossing the surface s Surface can be open or closed: r Think of flux as an invisible stream. D is the amount crossing perpen- r dicular to the surface. ds r The vectords points normally to the actual surface. r Trivial caser ifD is constant on the surface and is normal to it. Then the value is Ddsz . s Gauss’ Law

r Q D = e$ 4p R 2 R

r r Q Q Dds==4p RQ2 zz ◊ 2 s 4p R

Sphere of radius R

In detail: r 2 pp Q Q Dds=== djqqsin d eer$$ 2 rQ2 4p zz ◊◊zz 2 rr 2 s 00 44ppr r

12. Current Distributions I : total current $ r area s es D I $ F A I (a.) Current density j = lim es G J D sÆ 0 D s H m 2 K

r number of free electrons/unit volume j is also a vector flux B B r rr r jnevv=-af =r U rr r r r jneE==ms E Ijds= ◊ V e zz But vE=-m e W s A mobility If j is constant (not a function of coordinates), then Isj= ⋅ . total current

∆d r D I $ F A I (b.) Surface current density js = lim e1 G J D d = 0 D d H m K (Current distribution in a thin layer on a surface. Surface can be straight or curved.) Ijdl= If j is constant, Ijl= ⋅ z s ◊ s s l Example: coaxial line at high frequencies inner conductor radius = a outer conductor radius = b Ijasa= ◊ 2p a I Ijbsb= ◊ 2p b I

In reality, such currents are not truly surface currents, but have a finite penetration thickness into the conductor. However, if this penetration thickness is small, we approximate it as a surface current. Later, we shall prove that at high frequencies the current in a conductor is distributed as an exponential penetration into the surface.

For example: x Conductor

13. Displacement Current In electromagnetics, we need to extend our definition of current. In addition to the usual current which is due to the flow of charged particles (usually free electrons), there exists another kind of current, known as displacement current. rr r ∂ D ∂ E Definition: j ==e D ∂ t ∂ t This looks like something very new, but actually we already know and use this in a.c. circuit theory. The following discussion explains what displacement current is physically. Consider a parallel plate capacitor in a circuit: By Kirchoff’s current law, the R C same current i flows in the Vtoocosw i resistanceless (negligible R) wires, in the resistor R, and in the capacitor.

But what about inside the capacitor? It is obvious that no charges flow across the plates since there is air or another insulator (dielectric, σ = 0 ). Instead, as i is positive in the direction of current flow, positive charges accumulate on the plate marked + , and - charges on the other. When the current reverses, the charge accululations also reverse. dv The current inside the capacitor is iC= , where v is the voltage across the capacitor. dt

ButCAd= e /.o Thus: 1 dV d L V O d iA==eeA M P area A o do dt dt N d Q However, Vd/ = E. d d \=i ae EAf = D dt dt d dD i But iAjAjA== D.Then, by comparison, ==j D dt D r dt A r ∂ D For a more general configuration: j = D ∂ t So, in reality, the current in an ideal capacitor is a displacement current, which is simply a changing electric field. As we shall see shortly, this displacement current can exist even in free space without any capacitor. But the capacitor current well illustrates what it is! 14. Line Integral and Definition of the Electric Potential r r N Ei P2 Edl◊ ∫ lim Eiii cosa D l zP Â DDlNÆÆ•0, 1 i 1 = ∆l ai r i ∆ where Eiicosa is the tangentialr r component of Ei along the path li seg- ment. Mathematically, Edl◊ is the scalar product. Detailed derivation is given in the text. The line integral of the electric field is an important physical and measurable quantity. 1.) The electric potential Φ rr E P2 Force on the charge due to field FqE= TT =− External force needed to move charge FqEext 2 r r External work done WqEdl21 =- ◊ z1 P 2 r r 1 W21 V21 ==-Edl◊ q z1 Algebraic quantity. May be positive or negative, depending on whether we move the charge against the field or with the field.

If V21 is positive, we have done work, and this is converted into a higher potential energy for the charge. To move a positive test charge from the arrow side to the tail side requires work, so the potential is higher at the tail side. Moving from the tail side to the arrow side, the positive test charge gains kinetic energy, so potential is lower at arrow side. Simple example: - - - - - xba=> r $ EeE= xo xa= + + + + + r r b b $$ b VEdlEeedxExEbaVVba=-◊◊ =- o xx =- o =-obab -g = - <0 a zz a a

a is at higher potential than b [Therefore, VVba< . Field lines go from higher to lower potential. r Obviously, if the entire path of integration is perpendicular to E , then: r r z Edl◊ = 0. This shows that the equipotential surfaces are perpendicular to the field lines.

+ + + +

This definition of the electric potential is the same as you have encountered in circuit theory. Kirchoff’s voltage law states that around a loop: ∑V = e.m.f.

If there is no e.m.f. in the loop, then ∑V = 0. Therefore, in electromagnetic theory, too, if there is no e.m.f. in the loop: r r z Edl◊ = 0 Sources of e.m.f.: (i) Electrochemical (e.g., battery). We do not deal with these in electromagnetic theory. (ii) Changing magnetic fields. ( Faraday’s law) This is another law of electromagnetic theory incorporated in Maxwell’s equations. It is the physical basis of an a.c. generator and will be given shortly in the notes “Fara- day’s Law - Maxwells Integral Equations.”