Chapter 8 Conservation Laws

We study the conservation of energy, momentum, and angular momentum, in electrodynamics.

8.1 Charge and Energy

8.1.1 The Continuity Equation Review the conservation of charge.

The charge in the volume v is ( ) ( )∫=v

dtrtQ τρ ,r .

The current flow out through the boundary S is ∫ ⋅S

danJ ˆr

. S

v

QJ

The local conservation of charge is ∫ ⋅−==Surface

adJdtdQI rr

.

∫∫∫ ⋅∇−=⋅−=∂∂ ττρ dJadJd

t Surface

rrr J

tr

⋅−∇=∂∂ρ

1. Conservation of charge can be derived from Maxwell’s equations. 2. Conservation of charge is a consequence of the laws of electrodynamics.

8.1.2 Poynting’s Theorem (Conservation of energy) The work necessary to assemble a static charge and a static current is:

∫=SpaceAll

e dEW_

20

2τε ( ∫∫∑ ∑ ===

−= >=

τε

τρ dEVdVqWni ijj

ij2

01 ,1 21

21 , Ch2.4.3),

and ∫=SpaceAll

m dBW_

2

021 τµ

( ∫∫∫ =⋅=⋅== τµ

τ dBdJAdlIALIW 2

0

2

21

21

21

21 rrrr

,

Ch7.2.4). This suggests that the total energy stored in electromagnetic fields is:

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

SpalceAllem dBEU

_

2

0

20

121 τ

µε

According to the Lorentz force law, the work done on a moving charge q is

( ) ( )( )dtvqEdtvEqldEqldBvqEqldFdW rrrrrrrrrrrr⋅=⋅=⋅=⋅×+=⋅= .

Change the view of point charges to volume charge, we have

( ) ∫∫ ⋅=⋅=⋅=VV

dJEdvEvqEdt

dW ττρrrrrrr

.

( vqE )rr⋅ is the work done per unit time.

( )( ) ( )(AlJEElJAIVP )=== is the work done per unit time.

JErr

⋅ is the work done per unit time, per unit volume. That means JErr

⋅ may be

related to the volume energy stored in electromagnetic fields 0

220

22 µε BE

+ .

Since tEJB∂∂

+=×∇r

rr000 εµµ , we can replace J

r with

tEBJ∂∂

−×∇=r

rr0

0

1 εµ

.

tEEBEJE∂∂⋅−×∇⋅=⋅r

rrrrr0

0

1 εµ

( ) BEEBBErrrrrr

×∇⋅−×∇⋅=×⋅∇ ( )BEEBBErrrrrr

×⋅∇−×∇⋅=×∇⋅

( )( )tEEBEEBJE∂∂⋅−×⋅∇−×∇⋅=⋅r

rrrrrrr0

0

1 εµ

( )tEEBE

tBBJE

∂∂⋅−⎟

⎟⎠

⎞⎜⎜⎝

⎛×⋅∇−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−⋅=⋅r

rrrr

rrr0

0

1 εµ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−×⋅∇−=⋅ 20

0

2

0 221 EB

tBEJE ε

µµ

rrrr

( )∫ ⋅===V

dJEIVPdt

dW τrr

( )∫ ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−×⋅∇−=V

dEBt

BEdt

dW τεµµ

20

0

2

0 221 rr

∫∫ ⋅×−⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

SurfaceV

adBEdEBdtd

dtdW rrr

0

20

0

2 122 µ

τεµ

(Poynting’s theorem)

Energy stored in the system: 20

0

2

22EB ε

µ+ , the decrease of the system energy means

the power consumption ( EB uudtdJE +−=⋅ )

rr, if the integral volume is extending to

infinity. If we have a finite volume with a bounding surface, we may have energy

transported through the surface. The energy per unit time, per unit area, transported by the fields is called the Poynting

vector: ( BES )rrr×=

0

1µ

(energy flux density)

∫ ⋅−−=S

em adSdt

dUdt

dW rr

The work done on charges will increase their mechanical energy.

∫=V

mechdudtd

dtdW τ & ∫∫ ⋅−−=

SVem adSdu

dtd

dtdW rr

τ

( ) ( )∫∫∫ ⋅∇−=⋅−=+VSV

emmech dSadSduudtd ττ

rrr

( ) Suut emmech

r⋅−∇=+

∂∂ -- conservation of energy

Jt

r⋅−∇=

∂∂ρ -- conservation of charge

( BES )rrr×=

0

1µ

flow of energy

Jr

flow of charge Example: When current flows down a wire, work is done, which shows up as Joule heating of the wire. Though there are certainly easier ways to do it, the energy per unit time delivered to the wire can be calculated using the Poynting vector.

LVE = ,

aIB

πµ2

0= raL

IVraLIVEBrS ˆ

2ˆ

211ˆ 0

00 ππµ

µµ−=⎟

⎠⎞

⎜⎝⎛−=−=

r

IVadSS

−=⋅∫rr

, IVadSdt

dW

S

=⋅−= ∫rr

Energy transported into the wire: BESrrr

×=0

1µ

being used for electron’s scattering

with phonons and impurities (resistance) The direction of Poynting vector represents the way that energy is transported through.

Exercise: 8.1, 8.2