Intermission
Page 343, Griffith
Chapter 8. Conservation Laws
(Page 346, Griffith)
Lecture :
Electromagnetic Power Flow
Flow of Electromagnetic Power
• Electromagnetic waves transport throughout space the energy and momentum arising from a set of charges and currents (the
sources).
• If the electromagnetic waves interact with another set of charges and currents in a receiver, information (energy) can be delivered from the sources to another location in space.
• The energy and momentum exchange between waves and
charges and currents is described by the Lorentz force equation.
• Poynting’s theorem concerns the conservation of energy for a given volume in space.
• Poynting’s theorem is a consequence of Maxwell’s equations.
Poynting’s Theorem is the “Work-energy theorem”
“Conservation of Energy”
“The work done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field,
less the energy that flowed out through the surface” .
1 2 1 2 1
2 2
V V S
dW d
E J dv E B dv E B ds
dt dt
Total energy stored in the EM field
Energy flowed out through the surface Work done by the EM field
1 : Poynting vetor
S E B
em
S
dU
dW S ds
dt dt
Poynting’s Theorem
, .
( ) ( )
V V V
Now q d and v J
dW d E v E v d E J d
dt
( )
dW F dl q E v B vdt qE vdt
Suppose we have some charge and current configuration, produces fields E and B at time t.
Work done on a charge q in the interval dt is
E J is the work done per unit time, per unit volume (or, power/volume) Using the Ampere-Maxwell law,
E J
Proof: Total energy stored in Electromagnetic fields is
2 2
1 1
2 2
em V
U E B dv
E J
1
21
21
2 2
V V S
dW d
E J dv E B dv E B ds
dt dt
Note that
S 1 E B
Poynting Vector in Time Domain
• We define a new vector called the (instantaneous) Poynting vector as
• The Poynting vector has the same direction as the direction of propagation.
• The Poynting vector at a point is equivalent to the power density of the wave at that point.
H E
S ( W/m 2 )
8.2 Momentum of EM fields (Griffiths)
8.2.1 Newton's Third Law in Electrodynamics
Imagine a point charge q traveling In along the x axis at a constant speed v.
Because it is moving, its electric field is not given by Coulomb's law;
nevertheless, E still points radially outward from the instantaneous position of the charge.
If the velocity and acceleration are both zero, Generalized Coulomb field (velocity field)
Radiation field (acceleration field)
Since, moreover, a moving point charge does not constitute a steady current, its magnetic field is not given by the Biot-Savart law.
Nevertheless, it's a fact that B still circles around the axis.
Newton's Third Law in Electrodynamics
Now suppose two identical charges, q1 and q2, moving with a same speed along the x and y axes.
2 1
, ,
e q e q
F F
2 1
, ,
m q m q
F F
The electromagnetic force of q1 on q2 is equal, but not opposite to the force of q1and q2.
violation of Newton’s third law!
In electrostatics and magnetostatics the third law holds, but in electrodynamics it does not.
Is it true?
The third law must be hold ALL THE TIME!
We know that the proof of momentum conservation rest on the cancellation of internal forces.
Therefore, let’s prove momentum conservation in electrodynamics!
The fields themselves carry momentum.
Only when the field momentum is added to the mechanical momentum of the charges, momentum conservation (or, the third law) is restored.
At an instantaneous time t,
this may be regarded as an electromagnetostatic case.
8.2.2 Maxwell’s stress tensor
The fields themselves carry momentum.
To find out the field momentum, let’s calculate the total EM force on the charges in volume V in terms of Poynting vector S and stress tensor T:
The force per unit volume is
It can be simplified by introducing the Maxwell stress tensor T
Maxwell’s stress tensor:
Force density in a symmetry form of E and B.
T
ij
The total EM force on the charges in volume V is
*
In static case
*
ij
T
The force per unit area (or stress) on the surface
Example 8.2 (p. 353)
Find the net force on the northern hemisphere exerted by the southern hemisphere.Q
For the bowl,
In static case
The net force on the bowl is obviously in the z-direction,
Meanwhile, for the equatorial disk,
Combining them, the net force on the northern hemisphere is
8.2.3 Conservation of momentum for EM fields
According to Newton's second law, the force on an object is equal to the rate of change of its momentum:
When the object has an electric charge distribution in V,
The first integral represents momentum stored in the electromagnetic fields:
The second integral represents momentum per unit time flowing in through the surface:
If V is all of space, then no momentum flows in and out,
P
mech P
em constant
In differential form, : densities of mechanical momentum
: densities of field momentum
Solve Problems. 8.5 and 8.6
Example 8.3 (p. 356)
What is the electromagnetic momentum stored in the fields?inner conductor (radius a) outer conductor (radius b) uniform charge per unit length
The momentum in the fields is:
8.2.4 Angular Momentum
Electromagnetic fields carry energy density and momentum density of
The electromagnetic fields also carry density of angular momentum of
[Example 8.4]
Two long cylindrical shells are coaxial with a solenoid carrying current I.When the current in the solenoid is gradually reduced, the cylinders begin to rotate.
Where does the angular momentum comes from?
= constant over the volume of
Total angular momentum in the fields: current I
[Example 8.4] continued.
When the current is turned off, the changing magnetic field induces a circumferential electric field, given by Faraday's law:
Torque on the outer cylinder:
Angular momentum of the outer cylinder:
Torque on the inner cylinder:
Angular momentum of the outer cylinder:
Total angular momentum of the inner and outer cylinders:
Which is the same as the angular momentum of the field:
Angular momentum lost by fields is precisely equal to angular momentum gained by the cylinders,
