Biomedical Engineering Reference
In-Depth Information
∇ ⋅
E
B
=
πρ
4
∇ ⋅
=
0
.
∂
B
1
∇
×
= −
E
c
∂
t
∂
E
π
1
4
∇
×
B
=
+
J
c
∂
t
c
where.
E
.and.
B
.are.the.electric.and.magnetic.vector.ields,.respectively,.
c
.is.the.speed.of.light,.and.ρ.and.
J
.
are.the.source.terms'.charge.and.current.density,.respectively..In.the.absence.of.source.terms,.the.equa-
tions.have.a.free-space.propagation.solution,.where.E.and.B.components.are.normal.to.the.direction.of.
propagation,.as.depicted.in.Figure.1.1..he.waves.obey.the.Helmholtz.equation,
1
∂
∂
2
2
.
∇
2
u r t
(
,
)
=
u r t
(
,
)
c
2
t
where.
u
(
r
,
t
).represents.any.of.the.transverse.components.of.E.or.B.
We.detect.the.presence.of.the.electromagnetic.ield.(photon).because.it.interacts.with.the.electron..
Classically,.this.is.through.the.force.equation:
v
c
+
×
.
F
=
q
E
B
where.
F
.is.the.vector.force.on.the.electron,.
q
.is.the.charge.of.the.electron,.and.
v
.is.its.velocity.
One. free-space. traveling. wave. solution. to. the. Helmholtz. equation. is. the. plane. wave. depicted. in.
Figure 1.1.and.expressed.mathematically.as
ω
.
E t
(x, )
=
E
(k)e
(
i
t
− ⋅
k x)
where.
k
.=.2π/λ.is.the.spatial.wavenumber.(as.a.vector,.
k
.has.amplitude.
k
.and.points.in.the.direction.of.
propagation),.ω.is.the.spatial.frequency.in.radians.per.second,.ω.=.2πν,.and.ν.is.the.frequency.expressed.
in.hertz.
E
B
FIGuRE 1.1
Electromagnetic.wave.propagation.
