# Electromagnetic Field Theory Lecture Notes

**Topics:**Maxwell's equations, Magnetic field, Electromagnetism

**Pages:**43 (5310 words)

**Published:**January 14, 2013

2) UMRAN S. INAN, AZIZ S. INAN , “ENGINEERING ELECTROMAGNETICS” , ADDISON-WESLEY 1999

3) R.COLLINS, “FOUNDATIONS OF MICROWAVE ENGINEERING” Mc GRAW-HILL 1992

4) D.POZAAR, “MICROWAVE ENGINEERING” ADDISON-WESLEY 1990

FOUR FUNDAMENTAL ELECTROMAGNETIC FUNCTIONS

•

r E

r D

r ( r , t ) - Electric Field Intensity

–

(V/m)

•

r (r , t ) - Electric Flux Density (Displacement vector) – (C/m2)

•

r H (rr, t ) - Magnetic Field Intensity

–

(A/m)

•

r r B (r , t ) - Magnetic Flux Density

r Jc

–

(Wb/m2)

THREE TYPES OF ELECTRIC CURRENTS

•

r (r , t ) - Conduction Current Density – (A/m2)

r Jc=

•

r r J u (r , t ) - Convection Current Density – (A/m2)

σ = Conductivity (S/m)

r σE

r Ju=

r ρu

r ρ = Free Electric Charge Density – (C/m3), u = Velocity Vector – (m/s) •

r J

d

r (r , t ) - Displacement Current Density – (A/m2)

r r ∂D Jd = ∂t

MAXWELL’S EQUATIONS These four fundamental equations of electromagnetics on the basis of three separate experimentally established facts, namely, Coulomb’s law, Ampere’s law (or the Biot-Savart law), Faraday’s law, and the principle of conservation of electric charge. The validity of Maxwell’s equations is based on their consistency with all of our experimental knowledge to date concerning electromagnetic phenomena. The physical meaning of the equations is better perceived in the context of their integral forms, which are listed below together with their differential counterparts: 1. Faraday’s law is based on the experimental fact that time-changing magnetic flux induces electromotive force:

r r r r E . dl = − ∫ ∂ B . d s ∫C s ∂t

r

r ∇ × E = − ∂B

∂t

where the contour C is that which encloses the surface S, and where the direction of the line integration over the contour C (i.e., dl) must be consistent with the direction of the surface vector ds in accordance with the Right-Hand rule. 2. Maxwell’s second equation is a generalisation of Ampere’s law, which states that the line integral of the magnetic field over any closed contour must equal the total current enclosed by the contour. Maxwell’s second equation expresses the fact that time-varying electric fields produce

magnetic fields. The first term of this equation (also referred to as the conduction-current term) is Ampere’s law, which is a mathematical statement of the experimental findings of Oersted, where as the second term, known as the displacement–current term, was introduced theoretically by Maxwell in 1862 and verified experimentally many years later in Hertz’s experiments.

r r r H.dl = ∫s J .ds + ∫s ∂D.ds ∫s ∂t

r r ∇×H = J +

r ∂D ∂t

where the contour C is that which encloses the surface S. 3. Gauss’s law is a mathematical expression of the experimental fact that electric charges attract or repel one another with a force inversely proportional to the square of the distance between them (i.e.,Coulomb’s law):

r r ∫sD.ds = ∫v ρ dv

r ∇ .D = ρ

where the surface S encloses the volume V. The volume charge density is r represented with ρ to distinguish it from the phasor from ρ used in the time-harmonic form of Maxwell’s equations. 4. Maxwell’s fourth equation is based on the fact that there are no magnetic charges (i.e., magnetic monopoles) and that, therefore, magnetic field lines always close on themselves:

r r ∫s B . d s = 0

r ∇ .B = 0

where the surface S encloses the volume V. This equation can actually be derived from the Biot-Savart law, so it is not completely independent. CONSERVATION OF ELECTRIC CHARGE r ds

r r J ( r , t)

Total charge in volume V is,

Q = ∫V ρ . d V

dQ − =I = dt

Lim

∫

S

r r J .d S

V

dQ Lim = ∆V → 0 ∆V → 0 d V

∫

Sv

r r JdS

∆V

r = ∇.J

Conservation of electric charge

r dρ − = ∇.J dt

( CONTINUITY EQUATION )

r r r r ∂D ∇ × H = Jc + J u +...

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