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Chapter 8 Conservation Laws
8.1 Charge and Energy
8.2 Momentum
8.1 Charge and Energy
8.1.1 The continuity equation: conservation of charge
8.1.2 Poynting’s theorem and conservation of energy
8.1.1 The continuity equation: conservation of charge
Global conservation of charge: the total charge in the universe is constant.
Local conservation of charge: the change of the total charge in some volume exactly equals to the amount of charge passing in or out through the surface.
Jt
( ) ( , )V
Q t r t d
V Vs
dQd J da Jd
t dt
8.1.2 Poynting’s theorem and conservation of energy
The work done on charge
The rate at which work is done on all charges in a volume
Ampere-Maxwell law
( )F d q E v B vdt qE vdt
( )V
dW E J d
dt
00
1( )
EE J E B E
t
00
1 EJ B
t
00
1( )
EE J E B E
t
8.1.2 (2)
21
2E
t
( ) ( )
( ) ( )
B E E B
BB E B
t
21
2B
t
2 20
0 0
1 1 1( ) ( )
2E B E B
t
( )V s
E B d E B da
8.1.2 (3)
Poynting theorem :
Uem, the total energy stored in electromagnetic field.
Poynting vector )(1
0
BES
energy flux density
2 20
0 0
1 1 1( ) ( )
2V
dW dE B d E B da
dt dt
The work done on the charges by the EM force is equal to the decrease in energy stored in the field, less the energy flowed out through the surface.
8.1.2 (4)
Poynting theorem in differential form: continuity eq.
ems
dUdWS da
dt dt
mechV
dW du d
dt dt
( ) ( )mech emV s V
du u d S da S d
dt
( )mech emu u St
J
t
8.1.2 (5)
answer:
Ex 8.1
?S da
VE
L 0
2
IB
a
0
0
1
2 2
IV VIS
L a aL
2S da S aL VI
V
8.2 Momentum
8.2.1 Newton’s third law in electrodynamics
8.2.2 Maxwell’s stress tensor
8.2.3 Conservation of momentum
8.2.1 Newton’s third law in electrodynamics
Newton’s third law in trouble ?The fields themselves carry momentum.
2112
12
21 1 2 2 1
1 2 2 1
e e
m m
F F
F F
8.2.2 Maxwell’s stress tensor
The total electromagnetic force on the charge in volume V
The force per unit volume
0 00
1( ) ( )
f E J B
EE E B B
t
)()()( EEBEtt
BEBE
t
( ) ( )V V
F E v B d E J B d
0 00
1[( ) ( )] [ ( )] ( )E E E E B B E B
t
S
8.2.2 (2)
BBBBB
EEEEE
EEEEEE
)()(21
)(
)()(21
)(
)(2)(2)(
2
2
)]1
([]21
)()[(1
]21
)()[(
)()1
(21
])()[(1
])()[(
000
2
0
20
02
0
20
00
BEt
BBBBB
EEEEE
BEt
BE
BBBBEEEEf
Maxwell stress tensor, :T�
)21
(1
)21
( 220
0
BBBEEET ijjiijjiij
)(2
1)(
21 222
0
2220 zyxzyxxx BBBEEET
e.g.,
shear pressure
iji
ij TaTa ).(�
]21
)()[(1
]21
)()[(
]}21
)()[(1
21
)()([{)(
2
0
20
2
0
20
EBBBB
EEEEE
BBBBB
EEEEET
jjj
jjj
ijijiijii
ijijiijiii
j
�
)(1
)(0
0 yxyxxy BBEET
8.2.2 (3)
The total force on the charges in V
8.2.2 (4)
0 0f T St
�
0 0V s V
dF fd T da Sd
dt
�
Ex.8.2 net force on the northern hemisphere of a uniformly charged solid sphere?
[Problem 2.43]
solution: The net force is on zFor the bowl
8.2.2 (5)
RQ
2 ˆsinda R d d r
20
1ˆ ˆ ˆ ˆ ˆsin cos sin sin cos
4
QE r r x y z
R
20 0 2
0
( ) sin cos cos4
zx z xQ
T E ER
20 0 2
0
( ) sin cos sin4
zy z yQ
T E ER
2 2 2 2 2 20 02
0
( ) ( ) (cos sin )2 2 4
zz z x yQ
T E E ER
8.2.2 (6)
2 20
0
2 2 2
2 3 2 20
0
20
0
( )
( ) [2sin cos cos sin cos2 4
2sin cos sin sin sin (cos sin )sin cos ]
( ) [2sin cos (cos sin )sin cos ]2 4
( ) sin cos2 4
z zx x zy y zz zT da T da T da T da
Q
R
d d
Qd d
R
Qd d
R
�
11 120 0
1sin cos
2d d x
220 2
200 0
1( ) 2 sin cos
2 4 4 8bowl
Q QF d
R R
For the equatorial disk
8.2.2 (7)y
x
E
E
E
E
Inside sphere
ˆda rdrd z3
2 30
1 1ˆ( )
4
rE Q r
r R
30
1ˆ ˆ(cos sin )
4
Qr x y
R
2 2 20 ( )2zz z x yT E E E
2 203
0
( )2 4
Qr
R
2 303
0
( ) ( )2 4
zQ
T da r drdR
�
22 30
3 20 00
1( ) 2
2 44 16
Rdisk
Q QF r dr
R R
For the r>R and z=0 area,
8.2.2 (8)
the net force
diskF
bowlF
RrF
>
2
20
1 3
4 64
bowl diskF F F
Q
R
2 20 04 3
0 0
1 1( ) ( ) ( )
2 4 2 4zz zQ Q
T T da drdr r
�
220
3 20 0
1 1( ) 2
2 4 4 8r R R
bowl
Q QF dr
r R
F
8.2.3 Conservation of momentum
mechmech mechV
pdP
F P ddt
Vs
dSdtd
adT �
00
emP ,the momentum stored in the electromagnetic fields.
The density of momentum in the fields
conservation of momentum
TPPt emmech
�
)(
T� is the electromagnetic stress (force per unit area)
T�
is the momentum flux density
0 0emP S
Ex. 8.3
What is the electromagnetic momentum stored in the fields? solution:
Power transported from the battery to the resistor
8.2.3 (2)
0
0
1ˆˆ
2 2
IE s B steady
s s
20 0
1ˆ( )
4
IS E B z
s
2 200
0
12 ln( )
24
ln( )4
b
a
I I bP S da sds
as
I bIV
a
The momentum in the fields
(There is a hidden mechanic momentum to cancel this EM momentum to maintain the motionless cable and the static fields.This hidden momentum is due to a relativistic effect as discussed in chapter 12 ,Ex.12.12 )
8.2.3 (3)
00 0 2 2
0
1ˆ 2
4
ˆln( )2
bem a
Ip Sd z l s ds
SIl b
za
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