MAXWELL’S EQUATIONS (1873)

James Clerk Maxwell (1831-1879)

Michael Faraday 1791 –1867

André-Marie Ampère 1775 – 1836

Charles Augustin de Coulomb 1736 -1806 Johann Carl

Friedrich Gauss 1777 - 1855

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Electromagnetics

• Electromagnetic theory is the study of charges at rest and in motion which produce currents and EM fields.

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Electromagnetic field vectors

E : Electric field intensity vector, Volt/m

H : Magnetic field intensity vector, Ampere/m

D : Electric flux density vector, Coul/m2

B : Magnetic flux density vector, Weber/m2=Tesla

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Sources of an EM field

J : Electric current density, Ampere/m2

: Electric charge density, Coul/m3

distribution Charge density Current density

volume (Coul/m3) (Amp/m2)

surface s (Coul/m2) (Amp/m)

along a line l (Coul/m) I (Amp)

point charge q (Coul.)

J

sJ

Direction is specified by a vector ld 4

Maxwell’s Equations in differential form

describes and relates the field vectors, current densities and charge densities at any point in space at any time

t

trBtrE

,,

t

trDtrJtrH

,,,

trtrD ,,

0, trB

(1) (Faraday’s induction law)

(2) (Generalized Ampere’s circuital law)

(3) (Gauss’ law)

(4) (Conservation of magnetic flux)

There are no magnetic charges in nature

Maxwell’s equations as given above are in the most general form, in the sense that they are valid in any kind of medium. 5

t

trDtrJtrH

,,,

Displacement current in Amp/m2

Remark: For Maxwell’s equations expressions to be valid, it is assumed that the field vectors are single valued, bounded, continuous functions of position and time and exhibit continuous derivatives. EM field vectors possess these characteristics except where there exist abrupt changes in charge and current densities.

include impressed sources, as well as induced ones

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Maxwell’s equations include the information contained in the

continuity equation

t

trtrJ

,,

(5) (Conservation of charge)

Continuity equation is not an independent relation, it can be obtained from Maxwell’s 2nd and 3rd equations. (show this as an exercise)

Alternatively, Maxwell’s two divergence equations can be deduced directly from curl relations with the aid of continuity equation. (derivation in class)

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In addition to Maxwell’s equations, the following force law holds concerning the force on a charge q moving with velocity through an electric field and a magnetic field

v

E B

BvEqF Lorentz Force Equation

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Maxwell’s Equations in integral form describes the relations of the field vectors, current densities and

charge densities over an extended region of space. These are more general than the equations in differential form, since the fields and their derivatives do not need to possess continuous distributions.

SC

dsBt

dlE

SSC

dst

DdsJdlH

0S

dsB

VS

dvdsD

VS

dvdt

ddsJ

C

S

dl

dsnds ˆ

dsnds ˆn̂

S V

(1’)

(2’)

(3’)

(4’)

(5’)

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SC

dsBdt

ddlE

dt

dv m

ind

S

m dsBWhere magnetic flux linking C is

Faraday’s law: emf appearing at the open circuited terminals of a loop is equal to time rate of decrease of magnetic flux linking the loop.

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enc,denc

I

S

I

SC

IIdst

DdsJdlH

enc,denc

Ampere’s law: Line integral of magnetic field over closed path is equal to the current enclosed

enc

Q

VS

QdvdsD

enc

Gauss’ law: Total electric flux through a closed surface is equal to the total charge enclosed

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0S

dsB

Net magnetic flux leaving a closed surface is zero

dt

dQdv

dt

ddsJ enc

VS

Law of conservation of charge

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Differential and integral forms of Maxwell’s equations can be obtained from each other by using Stoke’s Theorem and Divergence Theorem. (Show as an exercise)

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t

trBtrE

,,

t

trDtrJtrH

,,,

trtrD ,,

0, trB

SC

dsBdt

ddlE

SSC

dsDdt

ddsJdlH

0S

dsB

VS

dvdsD

VS

dvdt

ddsJ

t

trtrJ

,,

MAXWELL’S EQUATIONS

In Differential Form In Integral Form

Continuity Equation

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15

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BOUNDARY CONDITIONS Region 1

Region 2

n̂

S

21 En̂En̂

s21 JHHn̂

s21 DDn̂

where is surface current density flowing on S

where is surface charge density on S

sJ

s

21 Bn̂Bn̂ (derivation in class)

11, HE

22 , HE

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t

trBtrE

,,

t

trDtrJtrH

,,,

trtrD ,,

0, trB

021 EEn

sJHHn 21

sDDn 21

021 BBn

Maxwell’s Equations Boundary Conditions

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