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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
1
Electromagnetics
• Electromagnetic theory is the study of charges at rest and in motion which produce currents and EM fields.
2
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
3
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
6
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)
7
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
8
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’)
9
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.
10
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
11
0S
dsB
Net magnetic flux leaving a closed surface is zero
dt
dQdv
dt
ddsJ enc
VS
Law of conservation of charge
12
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)
13
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
14
15
16
17
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
18
t
trBtrE
,,
t
trDtrJtrH
,,,
trtrD ,,
0, trB
021 EEn
sJHHn 21
sDDn 21
021 BBn
Maxwell’s Equations Boundary Conditions
19