Δ
t
=
t
OPO

t
OO
=
t
OP
+
t
PO

t
OD
(
n
1
)

t
DO

(
t
V W
(
n
2
)

t
V W
(
n
1
))
=
(
t
OP

t
OD
(
n
1
)) + (
t
PO

t
DO
)

(
t
V W
(
n
2
)

t
V W
(
n
1
))
=
n
1
h
2
2
s
+
n
1
h
2
2
s

(
n
2

n
1
)
V W
and we still need to evaluate the distance
V W
.
To evaluate this distance, notice that it
is the distance on the axis between two spheres with radii
R
1
and
R
2
(non–concentric!)
that intersect. From
PDC
2
,
R
2

DC
2
=
h
2
2
R
2
. But
R
2

DC
2
is also the distance
WD
.
Therefore,
WD
=
h
2
2
R
2
. Similarly, from
PDC
1
,
R
1

DC
1
=
h
2
2
R
1
=
V D
. Next,
V W
=
V D

WD
=
h
2
2
R
1

h
2
2
R
2
.
We now require that Fermat’s Principle holds, or
Δ
t
= 0. Therefore,
n
1
h
2
2
s
+
n
1
h
2
2
s
= (
n
2

n
1
)
h
2
2
R
1

h
2
2
R
2
n
1
s
+
n
1
s
= (
n
2

n
1
)
1
R
1

1
R
2
.
Now take the limit
s
→ ∞
, which would give us an
s
value we identify with
f
. Therefore,
n
1
f
= (
n
2

n
1
)
1
R
1

1
R
2
,
58
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1
f
=
n
2
n
1

1
1
R
1

1
R
2
.
We now define
n
:=
n
2
/n
1
, so that
1
f
= (
n

1)
1
R
1

1
R
2
.
This last relation is sometimes called the lens maker’s formula
. (Notice, it is a formula, not
an equation.) If we now take
s
→ ∞
we identify
s
with
f
. Then,
n
1
f
= (
n
2

n
1
)
1
R
1

1
R
2
1
f
= (
n

1)
1
R
1

1
R
2
which is exactly what we found for 1
/f
. Therefore,
1
f
= (
n

1)
1
R
1

1
R
2
=
1
f
Notice that we find that
f
=
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 Spring '09
 LIORBURKO
 Physics, R1 R2, ray tracing, SS 1

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