MAXWELL EQUATIONS IN MATTER

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MAXWELL EQUATIONS IN MATTER
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AMPERES LAW BEFORE MAXWELL
NOT DIVERGENCE FREE REPLACE IT WITH
DIVERGENCE FREE CURRENT
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alternative
AMPERES LAW
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CONSTITUTIVE RELATIONS
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INTERMISSION
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ELECTRODYNAMICS (DRAMATIS PERSONAE)
CHARGED PARTICLES ELECTRIC FIELDS MAGNETIC
FIELDS
ALL POSSES ENERGY
Possiblity of interconversion
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NEWTONS THIRD LAW IN ED
FIELDS OF CHARGE MOVING WITH CONSTANT VELOCITY
E emanates from the present location of
the charge
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B
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Y
EQUAL BUT NOT OPPOSITE!
Q2
V2
X
V1
Q1
Z
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THIS IS ZERO ONLY IF NEWTONS THIRD LAW
IS OBEYED
IN ED THIS IS NOT THE CASE
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ANGULAR MOMENTUM
Line charge
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IF WE DONT CONSIDER THE ELECTROMAGNETIC
FIELDS ENERGY , MOMENTUM AND ANGULAR
MOMENTUM ARE NOT CONSERVED!
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FIELDS SHOULD, THEREFORE, BE TREATED MORE
REALLY! THEY SHOULD BE ATTRIBUTED ENERGY MOME
NTUM ANGULAR MOMENTUM
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POYNTINGS THEOREM (work-energy theorem of
electrodynamics)
dW work done on particles by the
electromagnetic fields in the interval dt
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Energy in the fields
If work is done on the particles their
energy will increase at the expense of the
energy in the fields
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da
S
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PROOF OF POYNTINGS THEOREM
WORK DONE BY THE FIELDS
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Amperes law
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Rule 6
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integrate over volume and convert the last
term into a surface integral
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Ex 8.1 POWER DELIVERED TO A RESISTOR
R
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8.1 (a) POWER TRANSPORTED UP A COAXIAL
CABLE
V
I
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8.1 (b) Ribbon transmission line (7.57)
w
h
Find C and L
7.58
Pd of V
I
l
8.1 b
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CONSERVATION OF MOMENTUM
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MOMENTUM IN ELECTROMAGNETIC FIELDS
total
Per unit volume
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8.6 EM MOMENTUM IN A CAPACITOR IN A
MAGNETIC FIELD
B
d
A
E
(a)
Find em momentum
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E
B
(b)
Total impulse delivered to resistor
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8.6 (c) Instead of turning off the electric
field as in (b) slowly reduce the magnetic
field. This will induce a faraday electric
field, which in turn exerts a force on the
plates. Show that the total momentum is
AGAIN Equal to the momentum originally stored
in the field
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ANGULAR MOMENTUM
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18.4 FEYNMAN VOL II A TRAVELLING FIELD
Y
Z
X
Current sheet switched on at to in XY
plane
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OPPOSITE CURRENT SHEET
K
Z
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SECOND SHEET SWITCHED ON AFTER A DELAY
Y
Z
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Y
Z
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THE CATERPILLAR HAS TURNED INTO A
BUTTERFLY! FEYNMAN VOL II 950
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TRAVELLING FIELDS (continued)
TO GET B
Z gt 0
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TO GET E (side view)
B Region (into)
FARADAYS LAW
X
Z
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B AGAIN
E dn
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Obtained earlier
TWO DIFFERENT ANSWERS?
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ARE ALL MAXWELLS EQUATIONS SATISFIED? CULRY
ONES YES DIVERGENCE TYPES TO BE
VERIFIED
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TOP VIEW (DN THE X AXIS)
Gaussian surface
Z
Y
B
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ELECTROMAGNETIC WAVES IN VACUUM
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Take curl both sides of
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WAVE EQUATION
Similarly!
Not equivalent to all the Maxwell equations
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WHAT ABOUT MAXWELL EQUNS?
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B NOT SPECIFIED YET! THIS E MAY NOT BE
DIVERGENCE FREE!
EM plane waves are transverse!
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DETERMINE B
E along X direction
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Integrating wrt t
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9.9 Write down E an B for a monochromatic
plane wave of amplitude E0, frequency ?,
phase angle 0 that is (a) traveling in the
negative x direction and polarized in the
z direction (b) in the (1,1,1) direction
and polarized parallel to the xz plane.
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8.2 CHARGING CAPACITOR AGAIN
(b) Find energy and the Poynting vector in
the gap
Energy density at s from the axis
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8.2 (c) Determine the total energy in the
gap. Calculate the total power flowing in
the gap by integrating S over the
appropriate surface. Check that the power
input is equal to the rate of increase
of energy.
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