Now we want to calculate the power crossing a given area
A
A
.
During a time
Δ
t
Δ
t
an EM wave will pass an amount of energy through
A
A
of
u
c
Δ
t
A
u
c
Δ
t
A
where
u
u
is the energy density of the wave. If we want the
power/
m
2
m
2
then we must divide by
Δ
t
A
Δ
t
A
.
Thus we get
S
=
u
c
Δ
t
A
Δ
t
A
=
u
c
=
1
μ
0
B
2
c
=
1
μ
0
B
E
c
c
=
1
μ
0
B
E
S
=
u
c
Δ
t
A
Δ
t
A
=
u
c
=
1
μ
0
B
2
c
=
1
μ
0
B
E
c
c
=
1
μ
0
B
E
Now we make the reasonable assumption that the energy flows in the direction
of the wave, ie. perpendicular to
E
⃗
E
⃗
and
B
⃗
B
⃗
so we can define a vector that has the power per unit area:
S
⃗
=
1
μ
0
E
⃗
×
B
⃗
S
⃗
=
1
μ
0
E
⃗
×
B
⃗
or
S
⃗
=
c
2
ε
0
E
⃗
×
B
⃗
S
⃗
=
c
2
ε
0
E
⃗
×
B
⃗
This
is the Poynting vector. Thus for a plane EM wave we have three
useful things
E
⃗
(
r
⃗
,
t
)
=
E
⃗
0
e
i
(
k
⃗
⋅
r
⃗
−
ω
t
)
E
⃗
(
r
⃗
,
t
)
=
E
⃗
0
e
i
(
k
⃗
⋅
r
⃗
−
ω
t
)
B
⃗
(
r
⃗
,
t
)
=
B
⃗
0
e
i
(
k
⃗
⋅
r
⃗
−
ω
t
)
B
⃗
(
r
⃗
,
t
)
=
B
⃗
0
e
i
(
k
⃗
⋅
r
⃗
−
ω
t
)
and
S
⃗
=
c
2
ε
0
E
⃗
0
×
B
⃗
0
e
i
2
(
k
⃗
⋅
r
⃗
−
ω
t
)
S
⃗
=
c
2
ε
0
E
⃗
0
×
B
⃗
0
e
i
2
(
k
⃗
⋅
r
⃗
−
ω
t
)