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Energy Density and the Poynting Vector
Overview and Motivation : We saw in the last lecture that electromagnetic waves are
one consequence of Maxwell's (M's) equations. With electromagnetic waves, as with
other waves, there is an associated energy density and energy flux. Here we introduce
these electromagnetic quantities and discuss the conservation of energy in the
electromagnetic fields. Further, we see how the expressions for the energy density
and energy flux can be put into a form that is similar to expressions for the same
quantities for waves on a string.
Key Mathematics : We will gain some more practice with the "del" operator ∇. We
will also discuss what is meant by a time-averaged quantity.
I. Energy Density and Energy-current Density in EM Waves
Recall from the last lecture the basic Maxwell's equations,
( )
( )
0
ρ t
t
r
∇ ⋅ E r = , (1)
∇ ⋅ B ( r , t ) = 0 , (2)
( )
( )
t
t t ∂
∇ × =−
Br
E r , (3)
( ) ( )
( )
t
t t t ∂
∇ × = +
E r
Br μ jr μ ε. (4)
As we discussed last time, for ρ ( r , t ) = 0 and j ( r , t ) = 0 , M's equations imply the wave
equation for both E ( r , t )and B ( r , t ). We know that waves transport energy. So how
is the energy in an electromagnetic wave expressed? Well, you should have learned in
your introductory physics course that the energy density contained in the electric field
is given by
1
( ) ( ) ( ) [ ( )] 0 0 2 , 2
u (^) el r , t Er t Er t Er t
Typically this energy density is introduced in a discussion of the energy required to
charge up a capacitor (which produces an electric field between the plates). Similarly,
the energy density contained in the magnetic field is given by
(^1) In keeping with standard EM notation, we use u for the energy density and S for the energy flux.
( ) ( ) ( ) [ ( )]
2
0 0
u (^) mag r , t Br t Br t Br t
Typically this relationship is introduced in a discussion of the energy required to
establish a current in a toroid (which produces a magnetic field inside the toroid).
Notice again that the two fundamental constants of E and M, ε 0 and μ 0 , appear in Eq.
(5) and Eq. (6), respectively. Thus the total energy u ( r , t )contained in a region of
space with both electric and magnetic fields is
( ) [ ( )] [ ( )]
2
0
2 0 ,
u r , t Er t Br t μ
Because ( 0 0 )
2
c = 1 μ ε , this can also be written as
( )
( ) [ ( )]
2
2
0
, t c
t u t Br
Er r
Recall, for a traveling EM wave in vacuum the electric and magnetic field amplitudes
are related by B = E c. Equation (8) thus shows that equal amounts of energy are
contained in the electric and magnetic fields in such a wave.
What about the energy current density (also known as the energy flux)? Well, another
basic fact about electromagnetic radiation (that you may or may not have learned in
your introductory physics course) is that the energy flux in a particular region of space
is equal to
( t ) ( , t ) ( , t )
0
S r = Er × B r
As we learned in the last lecture, the direction of propagation of an electromagnetic
plane wave is in the direction of E (^ r , t )^ × B ( r^ , t ). As expected, Eq. (9) indicates that the
energy flux points in this same direction. In E and M the energy flux is known as the
Poynting vector (convenient because it points in the direction of the energy flow).
Obviously, if the homogeneous M's equations apply [ ρ ( r , t ) = 0 and j ( r , t ) = 0 ], then
Eq. (10), the standard continuity equation is indeed valid.
III. The densities u and S for an EM Plane Wave
In the last lecture we looked at the plane-wave solution
E ( r , t ) = E 0 cos( k ⋅ r −ω t +φ) (15)
( ) = ( × ) ( ⋅ −ω t +φ) c
B r t k E cos kr
to the homogeneous Maxwell's equations. Let's calculate u and S
r
for these fields.
Substituting Eqs. (15) and (16) into Eqs. (8) and (9) produces
( ) [ ( )]
2 0 0
0 cos
= E ⋅ − t + c
u r t kr , (17)
and
S ( r , ) [ cos( k r )] k ˆ
2 0 0
0
t = E ⋅ − t + , (18)
respectively. Comparing Eqs. (17) and (18) we see that
S ( r , t ) = cu ( r , t ) k ˆ. (19)
The agrees with the general expectation for a traveling wave that the energy current
flux j ε ( r , t ) is related to its associated energy density ρ ε ( r , t ) via j (^) ε( r , t ) = ρ ε( r , t ) v ,
where v is the velocity of ρ (^) ε( r , t ).
The Poynting vector expressed in Eq. (18) is a space and time dependent quantity.
Often, however, often we are more interested in the time-averaged value of this
quantity. In general, the time-averaged value of a periodic function with period T is
given by
( ) ( ) ∫
T
t
At dt T
At
0
With this definition the time-averaged value of S is
( ) [ ( )]
ω π
2
0
2
0
2 0 0 cos 2
t E ˆ^ t dt t
S k k r (21)
Because the average value of any harmonic function squared is simply 1 2 , we have
S ( ) k ˆ (^2 )
0
2 0
E ε
t t
On last remark about ( ) t
S r , t. In the optics world ( ) t
S r , t is known as the intensity
associated with the electromagnetic wave. Its dot product with a normal vector to
some surface gives the average power per unit area incident on that surface.
IV. An Analogy Between Mechanical and EM Waves
We previously studied the energy contained in mechanical waves. In particular, we
looked at transverse waves on a string, which have an energy density and energy-
current density that were essentially expressed as
( )
( ) ( )
2 2 1 , ,
2
x
qxt
t
qxt
c
xt
( )
( ) ( )
x
qxt
t
qxt j xt
ε ,^ τ.^ (24)
As they stand, these equations do not look particularly like Eqs. (8) and (9) for the
corresponding electromagnetic quantities.
The mechanical-waves expressions are written in terms of derivatives of the
displacement while the electromagnetic quantities are written in terms of the fields.
However, in the theory of electricity and magnetism we can introduce a quantity
known as the vector potential A ( r , t )that, in the absence of ( r , t )
r ρ and j ( r , t ), can be
defined such that it is related to the electric and magnetic fields via
( )
( )
t
t t ∂
Ar
E r (25)
and
( A ) x
A
x
A
t
A
t
y y
∂
×∇× =
With these last three expressions we can express u and S
r
for our plane wave solution
[Eqs. (31) and (32)] as
( )
2 2
0
x
A
t
A
c
u xt
y y
S ( ) x ˆ
0 x
A
t
A
x t
y y
∂
These expression are now essentially identical to Eqs. (23) and (24), the analogous
expressions for mechanical waves on a string if the following correspondences are
made: q ↔ Ay and τ ↔ 1 μ 0.
Exercises
* 32.1 Show that Eqs. (29) and (30) follow from Eq. (31).
** 32.2 A traveling-wave solution to Maxwell's equations. Consider the electric
field E (^ r , t )^ = E 0 x ˆcos(^ kz − kct )
( a ) What is the corresponding magnetic field?
( b ) Calculate the energy density u ( z , t )associated with each of these fields.
( c ) Calculate the Poynting vector S ( z , t )associated with these fields.
( d ) Show that u (^ z , t )and S (^ z , t )satisfy the appropriate continuity equation.
* 32.3 Show in the absence of charge and current densities that – in general – the
vector potential A ( r , t )satisfies the wave equation. In addition to equations in the
notes, you will need to use the fact that the vector potential satisfies ∇ ⋅ A = 0.
