Biomedical Engineering Reference
In-Depth Information
A
B
d
1
d
2
h
h
θ
1
θ
2
x
L
-
x
L
FIGuRE 2.1
Relection. of. the. light. from. a. mirror. surface.. he. light. travels. from. point.
A
. to. point.
B
. by. being.
relected. of. the. mirror. surface. at. a. horizontal. distance.
x
. from. point.
A
.. Point.
B
. is. a. horizontal. distance.
L
. from.
point.
A
..he.distance.that.the.light.travels.from.point.
A
.to.the.mirror.is.
d
1
.and.the.distance.that.it.travels.from.the.
mirror.to.point.
B
.is.
d
2
..To.ind.the.angle.of.relection,.θ
2
,.given.the.angle.of.incidence,.θ
1
,.we.minimized.the.travel.
time.along.this.path.according.to.the.Fermat's.principle.of.least.time.
horizontal.distance.between.points.
A
.and.
B
.is.
L
,.then.at.what.point.
x
.on.the.mirror.will.the.light.beam.
be.relected?.We.see.that.the.light.will.take.a.path.that.is.a.distance.
d
1
.from.point.
A
.to.the.mirror.and.a.
path.that.is.a.distance.
d
2
.from.the.mirror.to.point.
B
..We.assume.that.the.light.is.traveling.in.a.vacuum.
at.a.speed.
c
.
he.time.
t
.required.for.the.light.to.travel.from.point.
A
.to.
B
.along.this.path.is.given.by.the.following.
equation:
d x
c
( )
d L x
c
(
−
)
1
1
2
t x
( )
=
+
=
(
d x
( )
+
d L x
(
−
))
1
2
c
(
)
1
2
(
)
=
h
+
x
+
h
+
L x
−
2
2
2
c
.
To.ind.the.minimum.time,.we.take.the.derivative.of.
t
.with.respect.to.
x
.and.set.it.equal.to.zero,.which.
gives
=
d
t x
x
( )
1 1
2
1
2
(
)
−
1
2
−
1
2
(
)
(
)
+
(
)
2
(
)
=
h
2
+
x
2
2
x
h
2
+
L x
−
−
2
(
L
−
x
)
0
d
c
x
L x
−
=
−
h
2
+
x
2
(
)
2
h
2
+
L x
−
(2.2)
x
d
L
−
x
=
−
d
2
1
=
sin(
θ
)
−
sin(
θ
)
1
2
sin(
θ
)
=
sin(
θ
)
→ =
θ
θ
.
.
.
1
2
1
2
We.see.that.for.the.light.to.take.the.path.of.least.time.to.get.from.point.
A
.to.
B
.by.relecting.of.the.
mirror.surface.requires.that.the.angle.of.incidence,.θ
1
,.is.equal.to.the.angle.of.relection,.θ
2
..his.simple.
formula.allows.us.to.calculate.the.path.light.will.take.when.relected.from.a.mirror..As.we.shall.see.in.
Chapter.9,.Section.3,.it.is.interesting.to.consider.mirrors.that.are.not.lat..In.this.case,.the.angle.of.inci-
dence.is.equal.to.the.angle.of.relection,.where.the.angles.of.incidence.and.relection.are.deined.by.the.
local.normal.to.the.curved.surface.
