today in physics 218: the classic conservation laws in ...dmw/phy218/lectures/lect_45b.pdf · in...

23 January 2004 Physics 218, Spring 2004 1

Today in Physics 218: the classic conservation laws in electrodynamics

Poynting’s theoremEnergy conservation in electrodynamicsThe Maxwell stress tensor (which gets rather messy)Momentum conservation in electrodynamics

Electromagnetism on the sun, doing work on matter and emitting radiation. (TRACE satellite; Stanford U./Lockheed/NASA.)


Poynting’s theorem

Suppose a collection of charges and currents lies entirely within a volume V. If they are released at some point in time, electromagnetic forces will begin to do work on them. Consider, for instance, the work done by the forces on charges and currents in an infinitesimal volume dτ during a time dt:

2mech.

mechThus,

d W d dq dtc

dqdtd dt

dW ddt

τ

τ

⎛ ⎞= ⋅ = + × ⋅⎜ ⎟⎝ ⎠

= ⋅= ⋅

= ⋅∫

vF E B v

E vE J

E JV

i.e. B does no work.using , dq dρ τ ρ= =J v


Poynting’s theorem (continued)

We’ve seen this before, of course; it’s just another way to write P = VI. But we will learn something by elimination of Jusing Ampère’s law:

According to Product Rule #6,

( )mech.

1 :4 41 .

4

ct

dW d cdt t

π π

τπ

∂= × −

∂∂⎛ ⎞= ⋅ × − ⋅⎜ ⎟∂⎝ ⎠∫

EJ B

EE B EV

—

—

( ) ( ) ( )

( ) 1 .c t

⋅ × = − ⋅ × + ⋅ ×

∂= − ⋅ × − ⋅

∂

E B E B B EBE B B

— — —

—

Use Faraday’s law:



The time-derivative terms in all of these expressions can be written as

Using the last three results in the expression for dW/dt gives us

( ) 2

2

1 1 ,2 21 and, similarly, .2

Et t t

Bt t

∂ ∂ ∂⋅ = ⋅ =∂ ∂ ∂∂ ∂⋅ =∂ ∂

EE E E

BB

( )

( ) ( )

mech.

2 2

14

1 1 .4 2

dW d cdt t t

d c B Et

τπ

τπ

∂ ∂⎛ ⎞= − ⋅ × − ⋅ − ⋅⎜ ⎟∂ ∂⎝ ⎠

∂⎛ ⎞= − ⋅ × + +⎜ ⎟∂⎝ ⎠

∫

∫

B EE B B E

E B

V

V

—

—



V is bounded by surface S. Apply the divergence theorem to the first term in the integral, converting it to a surface integral over S:

In the last step we have used the fact that time is the only variable that survives the integration.

( )

( ) ( )

( ) ( )

mech.

2 2

2 2

14

14 8

1 .4 8

dW d cdt t t

c d d B Et

c dd B E ddt

τπ

τπ π

τπ π

∂ ∂⎛ ⎞= − ⋅ × − ⋅ − ⋅⎜ ⎟∂ ∂⎝ ⎠

∂= − × ⋅ − +

∂

= − × ⋅ − +

∫

∫ ∫

∫ ∫

B EE B B E

E B a

E B a

V

S V

S V

—

Poynting’stheorem


Energy conservation in electrodynamics

Poynting’s theorem of course expresses energy conservation.

We have long known that the energy density stored in electric and magnetic fields – that is , the work required to assemble the charges and currents in the configuration they’re in – is

So the integral of u is evidently the energy within V stored in the E and B fields, and the term containing it just gives the rate at which the stored energy changes.

( )2 21 .8EBu E Bπ

= +


Energy conservation in electrodynamics (continued)

The surface integral term contains a vector field that deserves its own name:

Thus,

This means that the rate at which work is done on the charges and currents by the fields is balanced not just by the rate at which the stored energy decreases, but also by a new term – which, since it involves a flux integral over the surface bounding V, must be the rate at which the fields carry energy out of V.

.4cπ

= ×S E B Poynting vector

mech. .EBEB

dW dWdd u d ddt dt dt

τ= − ⋅ − = − ⋅ −∫ ∫ ∫S a S aS V S



Put another way: if the stored energy decreases, not all of the resulting work goes into kinetic energy of the charges – some is radiated away. The Poynting vector tells us the energy per unit area and time that is radiated.

Rewrite the last result as

We may equate the integrands now, and obtain:

mech.

mech. .

EB

EB

d du d d u ddt dt

u ud d dt t

τ τ τ

τ τ τ

= − ⋅ −

∂ ∂= − ⋅ −

∂ ∂

∫ ∫ ∫

∫ ∫ ∫

S

S

V V V

V V V

—

—



an expression that invites comparison with the (charge-current) continuity equation,

Think of the Poynting vector S therefore as the “current density” of energy.

( )

mech.

mech.

, or

0 ,

EB

EB

u ut t

u ut

∂ ∂= − ⋅ −

∂ ∂∂

+ + ⋅ =∂

S

S

—

—

0 .tρ∂+ ⋅ =

∂J—


The Maxwell stress tensor

We won’t use tensors very much in this class, and you won’t haveto cope with this particular tensor after today. But the path toexpressions for momentum conservation goes straight through the Maxwell stress tensor, and this tensor does turn out to be a valuable tool in more-advanced courses, so it won’t hurt to introduce it here. Return to the Lorentz force law, and make a new definition:

is the force per unit volume exerted by E and B.

1 1 , where

1 1

q d dc c

c c

τ ρ τ

ρ ρ

⎛ ⎞ ⎛ ⎞= + × ⇒ = + ×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= + × = + ×⎜ ⎟⎝ ⎠

∫ ∫F E v B E v B

E v B E J B

V V

F

F ( )ρ=J v


The Maxwell stress tensor (continued)

Using Gauss’s law and Ampère’s law, in the forms

we can eliminate the source terms, in favor of the fields:

The very last term can be put into a more useful form by noting that

1 1, ,4 4 4

ct

ρπ π π

∂= ⋅ = × −

∂EE J B— —

( )1 1 1 .4 4 c tπ π

∂⎛ ⎞= + −⎜ ⎟∂⎝ ⎠EE E B B— —F ◊ ¥ ¥

( )

( )

1 1 1

1 .

c t c t c t

c t

∂ ∂ ∂= +

∂ ∂ ∂∂

= −∂

E BE B B E

E B E E—

¥ ¥ ¥

¥ ¥ ¥

Use Faraday’s law:



Thus,

Since it changes nothing if we add to the second square brackets to make the whole thing look more symmetrical:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 14 4 41 1 1 .

4 4 4

t

c t

π π π

π π π

∂⎛ ⎞= + + − −⎜ ⎟∂⎝ ⎠∂

= ⎡ − ⎤ − ⎡ ⎤ −⎣ ⎦ ⎣ ⎦ ∂

E E B B E B E E

E E E E B B E B

— — —

— — —

F ◊ ¥ ¥ ¥ ¥ ¥

◊ ¥ ¥ ¥ ¥ ¥

0,=B—◊ ( )B B—◊

( ) ( )

( ) ( ) ( )

14

1 1 .4 4 c t

π

π π

= ⎡ − ⎤⎣ ⎦

∂+ ⎡ − ⎤ −⎣ ⎦ ∂

E E E E

B B B B E B

— —

— —

F ◊ ¥ ¥

◊ ¥ ¥ ¥



We can “simplify” the terms in square brackets if we reintroduce tensor notation. Consider the three Cartesian directions to be represented by x, y, z = 1, 2, 3; for example,

Then the x-component of the first term in [] can be written as

1 1

2 2

3 3

, or .x

y

z

x r E Ey r E Ez r EE

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = = =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

r E

( ) ( ) 31 211 1 2 3

32 1 12 3

1 2 3 1

EE EEr r r

EE E EE Er r r r

⎛ ⎞∂∂ ∂⎡ − ⎤ = + +⎜ ⎟⎣ ⎦ ∂ ∂ ∂⎝ ⎠

⎛ ⎞⎛ ⎞ ∂∂ ∂ ∂− − + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

E E E E— —◊ ¥ ¥



It will pay to multiply this out and regroup:

[ ]

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

31 21 1 11

1 2 3

32 1 12 2 3 3

1 2 3 12

1 1 2 1 31 2 3

222 3

1 12 2 2 2

1 1 2 1 3 1 2 31 2 3 1

...

12

1 12 2

12

EE EE E Er r r

EE E EE E E Er r r r

E E E E Er r r

E Er r

E E E E E E E Er r r r

∂∂ ∂= + +

∂ ∂ ∂∂∂ ∂ ∂

− + + −∂ ∂ ∂ ∂

∂ ∂ ∂= + +

∂ ∂ ∂∂ ∂

− −∂ ∂

∂ ∂ ∂ ∂= + + − + +∂ ∂ ∂ ∂



where we have reintroduced the Kronecker delta,

This is for component 1; for component i, we’d get

[ ] ( ) ( ) ( ) ( )

( )

2 2 2 21 1 2 1 3 1 2 31

1 2 3 1

3 32 2

1 1 111 1

1...2

1 1 ,2 2j j j

j jj j

E E E E E E E Er r r r

E E E E E Er r r

δ= =

∂ ∂ ∂ ∂= + + − + +∂ ∂ ∂ ∂

⎡ ⎤∂ ∂ ∂ ⎛ ⎞⎢ ⎥= − = −⎜ ⎟⎢ ⎥∂ ∂ ∂ ⎝ ⎠⎣ ⎦∑ ∑

1, 0, ij

i ji j

δ=⎧

= ⎨ ≠⎩

( ) ( )3

2

1

1 .2i j iji jj

E E Er

δ=

∂ ⎛ ⎞⎡ − ⎤ = −⎜ ⎟⎣ ⎦ ∂ ⎝ ⎠∑E E E E— —◊ ¥ ¥



Similarly,

Let us now define a nine-component object,

with which we can re-write the ith component of both of the ugly terms in []:

( ) ( )3

2

1

1 .2i j iji jj

B B Br

δ=

∂ ⎛ ⎞⎡ − ⎤ = −⎜ ⎟⎣ ⎦ ∂ ⎝ ⎠∑B B B B— —◊ ¥ ¥

( )2 21 1 ,4 2ij i j i j ijT E E B B E B δπ⎛ ⎞= + − +⎜ ⎟⎝ ⎠

Maxwell stress tensor

( ) ( )

( ) ( )3

1

14

1 .4

i

iji jjT

r

π

π =

⎡ − ⎤⎣ ⎦

∂+ ⎡ − ⎤ =⎣ ⎦ ∂∑

E E E E

B B B B

— —

— —

◊ ¥ ¥

◊ ¥ ¥



The best way to envision the sum on the right-hand side is as matrix multiplication:

The result is represented by a three-component object; that is, a vector. We can use a vector-algebra-like symbolism for this operation, by using to denote the second-rank tensorwhose nine components are The sum is represented thus:

since it now looks so much like a divergence.

11 21 313

12 22 321 2 31 13 23 33

.ijjj

T T TT T T T

r r r rT T T=

⎡ ⎤⎡ ⎤∂ ∂ ∂ ∂ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎢ ⎥⎣ ⎦

∑

T.ijT

3

1,ij

jjT

r=

∂↔

∂∑ T—◊



Finally, the force per unit volume becomes, in terms of the Maxwell stress tensor,

where

( ) ( )

( ) ( ) ( )

14

1 14 4

,EB

c t

t

π

π π

= ⎡ − ⎤⎣ ⎦

∂+ ⎡ − ⎤ −⎣ ⎦ ∂

∂= −

∂

E E E E

B B B B E B

T g

— —

— —

—

F ◊ ¥ ¥

◊ ¥ ¥ ¥

◊

21 1 .

4EB c cπ≡ =g E B S¥ Momentum

density


Momentum conservation in electrodynamics

The total force on the charges within volume V can be found by integrating

We can now identify as the total momentum

stored in the fields E and B, so that is the

momentum density in the fields.

:Fmech.

,

EB

EB

d d ddt t

dd ddt

τ τ

τ

∂⎛ ⎞= = −⎜ ⎟∂⎝ ⎠

= −

∫ ∫

∫ ∫

p T g

T a g

V V

S V

—F ◊

◊

Use divergencetheorem:

EB EBdτ≡ ∫p gV 1

4EB cπ=g E B¥


Momentum conservation in electrodynamics (continued)

Rearranging slightly, we have

Interpretation:Changes in mechanical momentum (mvs for all the charges) and momentum stored in fields within V are caused by the pressure and shear distributed over the boundary surface S, and vice versa. The terms on the diagonal of

represent pressures in the E and B fields.

( )mech. .EBd ddt

+ = ∫p p T aS

◊ Momentum conservation in electrodynamics

( )T,T

( )2 2 2 21 1 ,4 2i iE B E Bπ⎛ ⎞+ − +⎜ ⎟⎝ ⎠


Momentum conservation in electrodynamics (continued)

The off-diagonal terms,

represent shear in the fields – the kind of stress that causes strain in a direction different than the stress.

One can also write the momentum-conservation relation in differential form:

where is the mechanical (mv-type) momentum per unit volume for the charges within V.

( )1 ,4 i j i jE E B Bπ

+

( )

mech.mech.

mech.

, or

0 ,

EB

EB

d d ddt t t

t

τ τ∂ ∂⎛ ⎞= = −⎜ ⎟∂ ∂⎝ ⎠

∂+ − =

∂

∫ ∫p g T g

g g T

V V

—

—

◊

◊

mech.g


Summary

Electric and magnetic fields can store energy and momentum:

or, for that matter, even angular momentum:

and can transport energy (and thus all the other quantities):

Electromagnetic waves are the normal method of transport.

( )2 21 ,8

1 ,4

EB

EB

u E B

c

π

π

= +

≡g E B¥

energy/volume

momentum/volume

( )1 ,4EB EB cπ

= =r g r E BL ¥ ¥ ¥ angular momentum/volume

.4cπ

= ×S E B energy/area/time

today in physics 218: the classic conservation laws in ...dmw/phy218/lectures/lect_45b.pdf · in...

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