Download 20 ElectroMagnetic field: energy, momentum and angular momentum

20
ElectroMagnetic field: energy, momentum and
angular momentum
[] Reference: The Feynman’s lectures on physics (Addison-Wesley, 1964) : {2.27} —
20.1
ElectroMagnetic field energy momentum and angular
momentum
In this section we will always deal with ElectroMagnetic relations in free space, which are the basic law of nature and
always valid, and never ever the description in presence of matter, nor the D and H fields, will be used.
EM fields store and transport energy, momentum and angular momentum.
In the rest of this section we will consider EM fields interacting with charges and currents.
Note that, in order to simplify the formulas, the result:
ε0 µ0 c2 = 1
,
(20.1.1)
which turns out from the solution of Maxwell equations for EM waves, will be often used.
We will use interchangeably the words particles and matter.
20.1.1
Local balance of Conservation Laws
[] Reference: The Feynman’s lectures on physics (Addison-Wesley, 1964) : {2.27.1} —
See section 9.1 and ??.
20.1.2
Energy of the ElectroMagnetic fields
[] Reference: The Feynman’s lectures on physics (Addison-Wesley, 1964) : {} —
[] Reference: D. J. Griffiths, Introduction to Electrodynamics, 3rd Ed., (1999, Prentice Hall) : {Chapter 8} —
See the important comment in section 16.5.4 about the modeling of matter with point charges, dipoles and higher-order
multipoles.
20.1.2.1
Introduction
• Capacitors and inductors demonstrate that ElectroMagnetic fields can store energy.
• Heating by ElectroMagnetic radiation (including light) shows that ElectroMagnetic radiation can transport energy.
• The ElectroMagnetic braking effect shows how kinetic energy can be converted into internal energy and trasferred as
heat via ElectroMagnetic fields (the magnetic field in this case).
20.1.2.2
Expression of the energy of the ElectroMagnetic fields
We know, from the energy equations of mechanics, that energy changes are related to the total power applied by both
internal and external forces.
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20.1: ElectroMagnetic field energy momentum and angular
20: ElectroMagnetic
momentum field: energy, momentum and angular momentum
One needs to start from the expression of the work per unit time (that is power) done by EM fields on matter, in order
to derive the expression of the energy balance, that is the first principle of thermodynamics:
dWEM −→
dt
Π≡
MAT
.
(20.1.2)
The work per unit time (that is power) per unit volume done by the EM fields on charges and currents (that is matter)
is:
dΠ
= ρ v ·(E + v ×B) = j ·E
dV
.
(20.1.3)
By using the fourth Maxwell equation to replace j and the identity:
div ( E ×B) = B · curl E − E · curl B ,
(20.1.4)
one finds Poynting’s theorem which expresses the conservation of energy in local form at any point:
dΠ
∂u
∂Sk
∂u
= j ·E ≡ k 0 = −
−
=−
− div S
dV
∂t
∂xk
∂t
.
(20.1.5)
ElectroMagnetic field energy density (energy per unit volume) is:
u[x] =
ε0 E 2 [x]
B 2 [x]
+
2
2µ0
.
(20.1.6)
The expression for the energy density 20.1.6, which in elementary and static cases is shown to be the energy stored in EM
fields, is assumed to be a valid expression for the energy stored in EM fields in all situations, following Poynting’s theorem.
In static conditions only, it can be shown that an alternative form for the ElectroMagnetic field energy density (energy
per unit volume) can be written in terms of the potentials as:
u[x] =
1y
1y
ρ[r]V [r] dV +
j[r] ·A[r] dV
2
2
in static conditions only
.
(20.1.7)
In presence of matter the above relations apply only in case of a linear relation between the charge and scalar potential
(for the first term) and the current and the vector potential (for the second term).
ElectroMagnetic field energy flux (energy crossing a unit area perpendicular to the direction of energy flow per unit
time) is given by Poynting’s vector:
S[x] =
E[x] ×B[x]
µ0
.
(20.1.8)
The Poynting vector describes the energy flux: energy per unit time per unit area perpendicular to the direction of
energy flow. This is evident from the form of equation 20.1.5, which is a balance equation, reducing to the equation for the
conservation of ElectroMagnetic energy in case no work is done on matter by ElectroMagnetic fields. In fact equation 20.1.5
has the form of a typical balance law, with a non zero source term: − j ·E .
The time-averaged value of the Poynting vector thus describes the time-averaged energy flux: average energy per unit
time per unit area perpendicular to the direction of energy flow); this is called the intensity of EM radiation.
Note that only the entire surface-integral of the Poynting vector, on a closed surface, has a clear physical meaning. In
fact one has to be careful with the strict identification of the Poynting vector at a point with the local energy flux per unit
time per unit area perpendicular to the direction of energy flow at that point. In fact the Poynting vector is defined to
within a divergence-less field, by Poynting’s theorem. No troubles arise when one uses the theorem 20.1.5 in integral form,
applied to an arbitrary volume, without attempting to localise the energy flux in space. On the other hand the differential
form is correct at any point. In fact Poynting’s theorem defines, in a unambiguous way, the divergence of S, that is the
infinitesimal flux per unit volume, not S itself.
It must be emphasised that equation 20.1.5 implies that energy conservation is a local energy conservation, that is
whenever there is a change of energy inside a certain volume the energy must be either given to (received from) matter or
has to flow across the boundaries of the surface.
Equation 20.1.5 can be written, in integral form and for fixed integration domains (Poynting theorem for the energy),
as:
y
{
{
d y
u dV +
j ·E dV = −
Sk dAk = −
S ·dA ,
(20.1.9)
dt
V
V
∂V
with
UEM ≡
y
u dV
∂V
.
(20.1.10)
V
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December 23, 2011
20: ElectroMagnetic field: energy, momentum and angular
20.1: ElectroMagnetic
momentum
field energy momentum and angular momentum
Equation 20.1.9 represents the energy balance for the system made of the ElectroMagnetic fields inside volume V . It
shows that, in non-stationary situations, EM energy can be stocked in EM fields and not only transformed into other forms
of energy via its transformation into mechanical energy. In fact the immediate effect of the EM fields on matter is to
transform EM energy into kinetic energy of the particles. This energy can then be transformed into other forms of energy.
The balance is only the balance for EM fields inside volume V : if other sources of energies or work are present as well
as other parts, they should be taken into account in the balance of the overall system by writing the energy balance for any
component.
20.1.2.3
Some special cases
• Stationary conditions:
∂u
= 0.
∂t
• No work by EM fields: j ·E = 0.
• Localised fields (see the above discussion):
20.1.2.4
v
∂V
S ·dA = 0.
Poynting theorem and the first principle of thermodynamics
Poynting theorem is the energy balance applied to the system made of the EM fields (the system). When comparing
with the first principle of thermodynamics 39.1 one identifies the expression 20.1.6 of the energy density u[x] with the total
energy density of the EM fields, the term j ·E as the work per unit volume done by the EM fields on matter and the flux
of Poynting vector as a radiation term, for exchanging EM energy across the boundaries of the system.
Note that when writing the first principle of thermodynamics to some piece of matter (now the system) the term j ·E
becomes the work per unit volume done by the EM fields on the system.
20.1.2.5
Matter and EM fields only
Note that:
y
V
j ·E dV ≡
dWEM −→
dt
MAT
≡
dUMAT
dt
.
(20.1.11)
Therefore the integrated form of equation 20.1.9 can be also written as:
{
{
d
(UEM + UMAT ) = −
Sk dAk = −
S ·dA ,
dt
∂V
(20.1.12)
∂V
showing that the energy (of the ElectroMagnetic fields plus matter) inside the volume can change because energy flows into
or out of the volume across the surface.
If the fields are localised inside a finite spatial region the volume can be taken as large as it is required to contain all
the fields. This implies the conservation of the total energy:
d
(UEM + UMAT ) = 0
dt
(localised fields)
.
(20.1.13)
Note that in general, for radiation fields, the surface contribution cannot be neglected, even taking the surface at infinity,
because for radiation fields the Electric and Magnetic fields decrease only as r−1 at infinity.
20.1.2.6
Ohmic conductors
Note that in the case of a Ohmic conductor the power per unit volume transferred by the fields to matter is the positive
quantity (Joule effect):
dΠ
j2
= j ·E =
dV
σ
.
(20.1.14)
This is the Joule effect. See section 20.2.1 for the description of the energy transfers.
20.1.2.7
Energy balance with external generators (batteries)
[] Reference: Bobbio-Gatti, Elettromagnetismo e Ottica (1◦ ed., Bollati-Boringhieri, 1991) : {9.10} —
Consider the case when a battery, providing a linear electromotive field E 0 , is feeding energy to matter. One has:
j = σ E + E0
=⇒
j ·E =
j2
− j ·E 0 .
σ
(20.1.15)
The power per unit volume transferred by the EM fields to matter is: j ·E .
The power per unit volume transferred by the electromotive fields to matter is: j ·E 0 .
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20.1: ElectroMagnetic field energy momentum and angular
20: ElectroMagnetic
momentum field: energy, momentum and angular momentum
The energy balance becomes:
y j2
{
y
d y
S ·dA +
u dV +
dV =
j ·E 0 dV
dt
σ
V
V
∂V
.
(20.1.16)
V
The above equation means that EM energy is transferred out of the EM fields via the Joule effect, taking out energy from
the EM fields, and that the linear electromotive field feeds energy into the EM fields.
Let’s consider an infinitesimal time interval dt.
Poynting theorem for EM fields gives:
dUEM + dWEM
→ PART
+ Φ∂V [S] dt = 0 .
(20.1.17)
Note that Poynting theorem involves the work done by EM fields only on matter, not the work done by anything else.
Note that for a closed fixed circuit under static conditions dW EM → PART = 0.
Energy conservation (first principle of ThermoDynamics) applied to matter gives:
dUPART = dQ + dWEM
→ PART
+ dWGEN
→ PART
.
(20.1.18)
Finally one obtains:
dWGEN
→ PART
= − dQ + dUPART + dUEM + Φ∂V [S] dt
.
(20.1.19)
Take as an example a piece of ohmic conductor outside the batteries in stationary conditions. One has:
dQ = Φ∂V [S] dt < 0 .
(20.1.20)
The above equation means that heat is given to the external world whose amount is measured by the flux of the Poynting
vector.
One could also add the energy balance for the generator, if required.
20.1.3
Momentum of the ElectroMagnetic fields
[] Reference: The Feynman’s lectures on physics (Addison-Wesley, 1964) : {} —
[] Reference: D. J. Griffiths, Introduction to Electrodynamics, 3rd Ed., (1999, Prentice Hall) : {Chapter 8} —
See the important comment in section 16.5.4 about the modeling of matter with point charges, dipoles and higher-order
multipoles.
20.1.3.1
Introduction
ElectroMagnetic fields can carry momentum.
The following example motivates the above statement. Consider two identical point charges constrained to move at
constant speed along the x and y axis, both of them moving in the negative direction and approaching the origin, that is
still at x > 0 and y > 0. It can be shown that the electric field is radial and that the magnetic field is purely tangential
to circumferences perpendicular and concentric to the velocity. The electromagnetic force between the charges would tend
to drive them off the axes but we can assume that the charges are mounted on tracks and forced to maintain the same
direction and the same speed by some external agent. The electric force between them is repulsive. The magnetic field
produced by the charge on the x axis points in the negative direction of the z axis, on the y at y > 0, so that the force
is in the positive direction of the x axis. The magnetic field produced by the charge on the y axis points in the positive
direction of the z axis, on the x at x > 0, so that the force is in the positive direction of the y axis.
The electromagnetic forces on the two charges are equal but not opposite, in violation of Newton’s third law. As
Newton’s third law is linked to momentum conservation this results challenges momentum conservation.
The problem is solved once one realizes that the ElectroMagnetic field carry a momentum.
Even perfectly static ElectroMagnetic fields can harbor momentum and angular momentum, as long as E ×B is nonzero. It is only when these contributions from ElectroMagnetic fields are included that the classical conservation laws
hold.
Momentum conservations for insulated systems is manteined if and only if the momentum carried by the ElectroMagnetic
fields is accounted for.
Experiments have shown that ElectroMagnetic waves can carry and transport momentum.
20.1.3.2
Expression of the momentum of the ElectroMagnetic fields
Momentum is a vector, its volume density must be a vector. A local conservation law implies that one must take
the divergence of the flux. But the flux must be a quantity capable to provide the flux of all the three components of the
momentum, that is we need one flux for each component of the momentum. In other words the divergence of the momentum
flux must be a vector, it cannot be a scalar. Therefore the flux of momentum must be a second-rank tensor.
Consider the momentum balance in vacuum in presence of a charge density ρ and current density j. The ElectroMagnetic
fields loose momentum which is transferred to the charges. We know, from the first cardinal equation of mechanics, that
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20: ElectroMagnetic field: energy, momentum and angular
20.1: ElectroMagnetic
momentum
field energy momentum and angular momentum
momentum changes are connected to the total external force. The momentum per unit volume per unit time given by the
ElectroMagnetic fields to matter can be written, after using the Maxwell equations, as:
dF
∂g
∂g
∂Θik ei
∂Θk
=−
= (ρE + j ×B) ≡ k = −
−
−
k
dV
∂t
∂x
∂t
∂xk
,
(20.1.21)
in terms of the force F applied by the ElectroMagnetic fields to matter.
ElectroMagnetic field momentum density (momentum per unit volume) and momentum flux (momentum crossing a
unit area perpendicular to the direction of momentum flow per unit time) are:
S[x]
= ε0 E ×B[x]
c2
g[x] = ε0 E[x] ×B[x] =
1
Bi Bj
Θij [x] = − ε0 Ei Ej +
− δij
µ0
2
B2
ε0 E +
µ0
2
!!
.
(20.1.22)
Bi Bj
= − ε0 Ei Ej +
− δij u[x]
µ0
!
.
(20.1.23)
The vector g is the volume density of the ElectroMagnetic momentum and it describes the momentum density (momentum per unit volume). EM radiation (see section 24) is given by the propagation of EM fields. The EM fields in EM
radiation are time-varying. The time-averaged value of the g vector thus describes the momentum density (momentum per
unit volume) of EM radiation.
The tensor Θij is a tensor describing the flux of momentum out of the volume defining the system, that is the momentum
per unit area perpendicular to the direction of the flow per unit time. It is analogous of the Poynting vector for momentum
and as the Poynting vector it conventionally describe a positive flux when the flux is out-going. The tensor −Θij (note
the minus sign) is the ElectroMagnetic stress tensor (Maxwell stress tensor). The momentum flux per unit time out any
surface Σ can thus be written as:
{
dP
= ei
Θij dAj .
(20.1.24)
dt
Σ
Note that only the entire surface-integral of the ElectroMagnetic stress tensor has a clear physical meaning. One has
to be careful with the identification of the ElectroMagnetic stress tensor with the local momentum flux per unit area per
unit time. In fact the ElectroMagnetic stress tensor is defined to within a divergence-less field. No troubles arise when one
uses the theorem 20.1.21 in integral form, applied to an arbitrary volume without attempting to localise the momentum
flow in space. On the other hand the differential form is correct at any point, as the derivation provide the divergence of
the momentum flow.
It must be emphasised that equation 20.1.21 implies that momentum conservation is a local momentum conservation,
that is whenever there is a change of momentum inside a certain volume the momentum must be either given to (received
from) matter or has to flow across the boundaries of the surface.
Note also the equation 20.1.21 has the form of a typical balance law, with a non zero source term.
Equation 20.1.21 can be written, in integral form (Poynting theorem for the momentum), as:
y
{
d y
g[x] dx +
k[x] dx = −
ei Θij [x] dAj
dt
V
V
with
P EM ≡
,
(20.1.25)
∂V
y
g dV
.
(20.1.26)
V
Equation 20.1.25 represents the momentum balance for the system made of the ElectroMagnetic fields inside volume V .
The left side gives the rate of change of EM momentum inside the volume V .
Note that:
y
dP MAT
(ρE + j ×B) dV ≡ F EM −→ MAT =
.
(20.1.27)
dt
V
Therefore the integral form of equation ?? can be also written as:
{
d
(P EM + P MAT ) = −
ei Θij dAj
dt
,
(20.1.28)
∂V
showing that the momentum (of the ElectroMagnetic fields plus matter) inside the volume can change because momentum
flows into or out of the volume across the surface.
If the fields are localised inside a finite spatial region the volume can be taken as large as it is required to contain all
the fields. This implies the conservation of the total momentum:
d
(P EM + P MAT ) = 0
dt
337
(localised fields)
.
(20.1.29)
December 23, 2011
20.2: Some Examples and Physical Applications
20: ElectroMagnetic field: energy, momentum and angular momentum
Note that in general, for radiation fields, the surface contribution cannot be neglected, even taking the surface at
infinity, because for radiation fields the Electric and Magnetic fields decrease only as r−1 at infinity. Compare the analogous
behaviour of the Poynting vector.
20.1.3.3
Examples of ElectroMagnetic momentum in action
• Tail of dust of comets: it is produced by the Sun radiation pressure.
• Radiation pressure has a relevant role in the internal equilibrium of a star.
• Radiation pressure has a relevant role the dynamics of very small bodies in solar system, with effects such as the
Poynting-Robertson effect, the Yarkovsky effect and the YORP effect).
• Radiation pressure is used in laser cooling (http://en.wikipedia.org/wiki/Laser_cooling).
20.1.4
Angular momentum of the ElectroMagnetic fields
[] Reference: The Feynman’s lectures on physics (Addison-Wesley, 1964) : {} —
[] Reference: D. J. Griffiths, Introduction to Electrodynamics, 3rd Ed., (1999, Prentice Hall) : {Chapter 8} —
See the important comment in section 16.5.4 about the modeling of matter with point charges, dipoles and higher-order
multipoles.
20.1.4.1
Introduction
ElectroMagnetic fields can carry angular momentum.
Even perfectly static ElectroMagnetic fields can harbor momentum and angular momentum, as long as E ×B is nonzero. It is only when these contributions from ElectroMagnetic fields are included that the classical conservation laws
hold.
The Feynman Cylinder paradox is solved by taking into account the angular momentum of ElectrMagnetic fields.
Angular momentum conservations for insulated systems is manteined if and only if the anglular momentum carried by the
ElectroMagnetic fields is accounted for.
Experiments have shown that ElectroMagnetic waves can carry and transport angular momentum.
20.1.4.2
Expression of the angular momentum of the ElectroMagnetic fields
The density of angular momentum and the angular momentum of the ElectroMagnetic fields are:
dJ
[x] = x ×g
dV
20.1.4.3
J=
y
x ×g dx
.
(20.1.30)
V
Examples of ElectroMagnetic angular momentum in action
• The Feynman cylinder paradox 1 .
• A variation on the Feynman cylinder paradox.
Consider a cylindrical capacitor immersed in a magnetic field parallel to the axis.
First case. If the capacitor is discharged there is an average current in the radial direction which interacts with
the magnetic field. This generates an overall torque on the capacitor which tends to put it into rotation: angular
momentum of the EM field is transferred into mechanical angular momentum.
Second case. If the magnetic field is reduced to zero a circular electric field is induced, whose modulus depends on
the distance form the axis. Therefore the torques acting on the two charged faces of the capacitor are not balanced
and the capacitor acquires angular momentum: angular momentum of the EM field is transferred into mechanical
angular momentum.
20.2
20.2.1
Some Examples and Physical Applications
Energy balance for a resistive wire
Consider a circular cylindrical wire of radius R (not to be confused with the resistance of the wire, which will not be
used), whose axis is the z axis, where a constant and uniform current I flows. Consider a finite part of the wire, of length
L, and apply the energy balance to it.
1 The
Feynman’s lectures on physics (Addison-Wesley, 1964), 17.4
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December 23, 2011
20: ElectroMagnetic field: energy, momentum and angular momentum
20.2: Some Examples and Physical Applications
Consider the microscopic picture for a typical solid conductor, where free electrons move inside a fixed lattice of positive
ions. The power per unit volume given by the EM fields to free electrons first goes into kinetic energy of the free electrons.
The kinetic energy gained by the free electrons is then given to the lattice of ions, as potential and kinetic energy of the
ions in the lattice, thanks to collisions. Both the kinetic energy of the free electrons and the total energy of the lattice
contribute to the internal energy of the resistive material. All in all the EM field transfers some energy into internal energy
of the material, which is eventually given to the surrounding ambient as heat.
We will first consider the integral balance for a whole piece of wire and later the differential balance at the internal
points.
The equipotential surfaces and the electric field for the very idealised situation of a long rectilinear and cylindrical
coaxial wire with circular cross-section are shown in figures 20.1 and 20.2. The external shell is kept at zero potential and
it is assumed to have zero resistance, while the difference of potential is applied at the two bases of the wire. Note that in
general at the surface of the wire there is both a normal component of the electric field and a surface charge.
A Toy Coaxial Cable
10
5
0
-5
-10
-10
-5
0
5
10
Figure 20.1: FIGURE
A Toy Coaxial Cable
Figure 20.2: FIGURE
20.2.1.1
Integral balance
Under stationary conditions the Poynting theorem shows that the energy given by the ElectroMagnetic fields into
internal energy of the resistor (the Joule effect) can be considered to enter from the lateral perimeter of the wire, feeded
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December 23, 2011
20.3: Exercises, problems and physical applications 20: ElectroMagnetic field: energy, momentum and angular momentum
by the ElectroMagnetic fields. However it should be kept in mind that the Poynting vector is not unique and therefore
the spatial localisation of the energy flux is not relevant, but only the divergence of the Poynting vector is as well as the
integral on a closed surface are.
It is important to stress that the electromotive field of the battery does not intervene here, because the battery is outside
the volume where the energy balance is applied.
Consider, in fact, a circular cylinder surface, Σ, coaxial with the wire and with radius r > R, just slightly larger than
R. Inside the wire the electric field is parallel to the axis of the wire, thanks the stationary situation, and therefore, its
tangential component, thanks to its continuity, has the same value just outside the wire. Note that, a priori, a normal
component of the electric field is present at the surface of the wire, as well as a corresponding surface charge density. The
normal component of the electric field, only present externally to the wire, is not relevant because it does not affect the
energy balance as it provides a component of the Poynting vector parallel to the wire, whose flux across the cylindrical
surface is zero. To avoid complications with this normal component at the tow bases of the cylinder it is necessary to
choose the surface Σ just slightly bigger that the wire in such a way that the part of the cylinder basis outside the wire has
a negligible area, tending to zero, as the normal component has a non-zero contribution there. In fact, a priori the normal
components of the electric field at the two bases are different because ρS may be different.
The radial component of the Poynting vector is directed towards the interior of the wire. It has the value, at r = R:
!
!
µ
|I|
j2R
j2R
|j|
0
=
and
S=−
er .
(20.2.1)
S = ε0 c2 E ×B
=⇒
|S| = ε0 c2
σ
2πR
2σ
2σ
The negative flux of the Poynting vector across the surface Σ is thus:
y
{
πj 2 R2 L
j2R
(2πRL) =
= |I ∆V | =
j ·E dV
−
S ·dA =
2σ
σ
The negative flux of the Poynting vector implies that EM energy is entering the volume.
20.2.1.2
.
(20.2.2)
Differential balance at the internal points
The internal electric/magnetic fields and the Poynting vector are:
E = Ez e3
B=
µ0 jz r
eφ
2
S=−
E z jz r
er .
2
(20.2.3)
It follows:
∂Sr
Sr
jz2
−
= +Ez jz = +
>0 .
(20.2.4)
∂r
r
σ
This shows clearly in a non-ambiguous way the meaning of Poynting theorem in differential form: at any point internal
to the wire the EM fields are feeding a power per unit volume to matter, energy which is then dissipated by the Joule effect.
− div S = −
20.2.1.3
Differential balance at the external points
The calculation of the Poynting vector outside the wire is difficult but Poynting’s theorem allows to state that, in
stationary conditions, div S = 0 because j = 0.
20.2.1.4
Energy transfers
Inside the wire the electric field gives energy to the free charges (electrons), which becomes kinetic energy of the free
charges. The free charges (electrons) collide with the fixed particles (positive ions) giving them mechanical energy. The
energy of all the particles contributes to the internal energy. In stationary conditions the work done by the electric field is
thus totally transferred into internal energy, which is then possibly given to the external world as heat.
Poynting theorem, in stationary conditions, reads:
y
{Work done by EM fields on free electrons inside the wire} =
j ·E dV
(20.2.5)
z
{EM Energy entering the system from outside} = S ·dA
(20.2.6)
y
z
j ·E dV + S ·dA = 0 .
(20.2.7)
In fact in stationary conditions the energy of the EM fields inside the system does not change and therefore any energy
must come from outside.
20.3
Exercises, problems and physical applications
Problem - 20.1
Energy absorption versus momentum absorption
[] Reference: Halliday-Resnick-Krane (4◦ ed., 1994, Casa Editrice Ambrosiana) : {2.41 - Q-15} —
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December 23, 2011
20: ElectroMagnetic field: energy, momentum and angular momentum 20.3: Exercises, problems and physical applications
Can any object absorb EM energy in the form of light without absorbing any EM momentum ? If yes do some examples;
if not explain why.
Can any object absorb EM momentum in the form of light without absorbing any EM energy ? If yes do some examples;
if not explain why.
Problem - 20.2
Distance dependence
[] Reference: Halliday-Resnick-Krane (4◦ ed., 1994, Casa Editrice Ambrosiana) : {2.41.17} —
Walk toward an isotropically emitting lamp by D = 162 m. You find that the intensity rises by a factor 1.5. What was
the initial distance ? Can one determine the power radiated by the lamp? If yes, find it out. If not explain why.
Problem - 20.3
Solar light
[] Reference: Halliday-Resnick-Krane (4◦ ed., 1994, Casa Editrice Ambrosiana) : {2.41.20} —
The intensity of solar light just ouside the Earth atmosphere is about 1.4 kW/m2 . Find the average values of the EM
fields.
Problem - 20.4
I. E. Irodov, Problems in General Physics, ISBN 5-03-000800-4, MIR publishers Moscow, (1988) 4.210
Demonstrate that at the boundary between two media the normal components of the Poynting vector are continuous.
Problem - 20.5
I. E. Irodov, Problems in General Physics, ISBN 5-03-000800-4, MIR publishers Moscow, (1988) 4.224
Gravitational attraction versus sun radiation pressure on interplanetary particles.
Problem - 20.6
A charging capacitor
[] Reference: Halliday-Resnick-Krane (4◦ ed., 1994, Casa Editrice Ambrosiana) : {2.41.31} —
Consider a cylindrical capacitor with parallel, plane and circular plates, neglecting edge effects, while it is charging via
two long rectilinear wires coaxial to the plates.
Show that the Poynting vector is everywhere radial entering the capacitor on the lateral surface of the cylinder.
Show that the rate of increase of the electric energy stored in the capacitor can be calculated using Poynting vector.
Problem - 20.7
♠
The Feynman’s cylinder paradox
[] Reference: The Feynman’s lectures on physics (Addison-Wesley, 1964) : {17.4} —
[] Reference: http://www.physics.princeton.edu/~mcdonald/examples/feynman_cylinder.pdf: {} —
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December 23, 2011