Lets remember what Irradiance is: We have the Poynting vector
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
)
But
the problem is that this varies rapidly in time. So we define the Irradiance
I
=
〈
S
〉
T
I
=
〈
S
〉
T
which
has units Watts per meter squared and can also be called the radiant flux
density. We showed in lecture that this is (where I drop the T because it is
tiresome to write)
<
S
>=
c
2
ε
0
E
0
B
0
/
2
<
S
>=
c
2
ε
0
E
0
B
0
/
2
which
can also be written
<
S
>=
c
ε
0
E
0
2
/
2
<
S
>=
c
ε
0
E
0
2
/
2
We
need to make one minor modification though, the above presumes we are in free
space. So we modify the irradiance to take into account that different media
have different speeds of light. (Now we write
v
v
instead of
c
c
because we are not assuming free space)
I
=
<
S
>=
v
ε
2
E
0
2
I
=
<
S
>=
v
ε
2
E
0
2
The
reflectance is the ratio of the reflected power to the incident power is
R
=
I
r
A
cos
θ
r
I
i
A
cos
θ
i
R
=
I
r
A
cos
θ
r
I
i
A
cos
θ
i
A
A
is the area illuminated by the electomagnetic radiation and
θ
r
,
θ
i
θ
r
,
θ
i
are the reflected and incident angles. But we know that
θ
r
=
θ
i
θ
r
=
θ
i
so
R
=
v
r
ε
r
2
E
0
r
2
v
i
ε
i
2
E
0
i
2
R
=
v
r
ε
r
2
E
0
r
2
v
i
ε
i
2
E
0
i
2
and
the media are the same so
R
=
(
E
0
r
E
0
i
)
2
=
r
2
.
R
=
(
E
0
r
E
0
i
)
2
=
r
2
.
Likewise the transmittance (using
μ
0
≈
μ
t
≈
μ
i
μ
0
≈
μ
t
≈
μ
i
)
is
T
=
I
t
A
cos
θ
t
I
i
A
cos
θ
i
=
I
t
cos
θ
t
I
i
cos
θ
i
=
v
t
ε
t
2
E
0
t
2
cos
θ
t
v
i
ε
i
2
E
0
i
2
cos
θ
i
=
μ
0
v
t
ε
t
E
0
t
2
cos
θ
t
μ
0
v
i
ε
i
E
0
i
2
cos
θ
i
.
T
=
I
t
A
cos
θ
t
I
i
A
cos
θ
i
=
I
t
cos
θ
t
I
i
cos
θ
i
=
v
t
ε
t
2
E
0
t
2
cos
θ
t
v
i
ε
i
2
E
0
i
2
cos
θ
i
=
μ
0
v
t
ε
t
E
0
t
2
cos
θ
t
μ
0
v
i
ε
i
E
0
i
2
cos
θ
i
.
Now
note
μ
0
ε
v
=
μ
ε
v
=
μ
ε
1
μ
ε
c
c
=
μ
ε
μ
0
ε
0
1
c
=
n
c
.
μ
0
ε
v
=
μ
ε
v
=
μ
ε
1
μ
ε
c
c
=
μ
ε
μ
0
ε
0
1
c
=
n
c
.
So
now we can write
T
=
μ
0
v
t
ε
t
E
0
t
2
cos
θ
t
μ
0
v
i
ε
i
E
0
i
2
cos
θ
i
=
n
t
E
0
t
2
cos
θ
t
n
i
E
0
i
2
cos
θ
i
=
n
t
cos
θ
t
n
i
cos
θ
i
(
E
0
t
E
0
i
)
2
=
n
t
cos
θ
t
n
i
cos
θ
i
t
2
.
T
=
μ
0
v
t
ε
t
E
0
t
2
cos
θ
t
μ
0
v
i
ε
i
E
0
i
2
cos
θ
i
=
n
t
E
0
t
2
cos
θ
t
n
i
E
0
i
2
cos
θ
i
=
n
t
cos
θ
t
n
i
cos
θ
i
(
E
0
t
E
0
i
)
2
=
n
t
cos
θ
t
n
i
cos
θ
i
t
2
.
This
is a more complicated expression than
R
R
because1)The speed of energy transmission is affected by the
medium2)
θ
i
≠
θ
t
θ
i
≠
θ
t
so the projected areas normal to the propagation direction are different.