SME
University Physics III: Waves, Optics & Modern · Problem type
Two lenses in a row
Use this for “An object is 20 cm before a 10 cm lens; a 20 cm lens stands 50 cm behind it. Where is the final image and how big?”
Diagram
The first lens’s image (violet) is the second lens’s object.
Final image: real, upright, m = 2.
This is an example. Type your own values to start a new problem.
1
f₁ (cm)
=
1
dₒ (cm)
+
1
dᵢ₁ (cm)
dₒ₂ (cm)
=
s (cm)
−
dᵢ₁ (cm)
1
f₂ (cm)
=
1
dₒ₂ (cm)
+
1
dᵢ₂ (cm)
m₁
First magnification
worked out (not typed)
m₂
Second magnification
worked out (not typed)
m
Total magnification
worked out (not typed)
Formulas
1
f₁
=
1
dₒ
+
1
dᵢ₁
1
10
=
1
20
+
1
20
dₒ₂
=
s
−
dᵢ₁
30 = 50 − 20
1
f₂
=
1
dₒ₂
+
1
dᵢ₂
1
20
=
1
30
+
1
60
m₁
=
−
dᵢ₁
dₒ
−1
=
−
20
20
m₂
=
−
dᵢ₂
dₒ₂
−2
=
−
60
30
m
=
m₁
×
m₂
2 = −1 × (−2)
Assumptions
•
Two thin lenses on one axis, s apart: the image the first lens forms is the object for the second.
•
f₁ may have either sign (− for a diverging lens); the second lens converges (f₂ > 0) and its object lies in front of it (dₒ₂ > 0).
•
The total magnification is m = m₁m₂: m > 0 means the final image stands upright.
Step-by-step
WE KNOW
f₁ = 10 cm, f₂ = 20 cm, s = 50 cm, dₒ = 20 cm
FIND
first image distance (dᵢ₁), second object distance (dₒ₂), final image distance (dᵢ₂), first magnification (m₁), second magnification (m₂), total magnification (m)
Step 1 · Find first image distance (dᵢ₁)
Use 1 ÷ f₁ = 1 ÷ dₒ + 1 ÷ dᵢ₁
The
first
lens
alone:
take
1
dₒ
from
1
f₁
and
turn
the
result
over.
dᵢ₁
=
f₁
×
dₒ
dₒ −
f₁
dᵢ₁
=
10
×
20
20 − 10
dᵢ₁
=
10
×
20
10
dᵢ₁
=
200
10
dᵢ₁ = 20 cm
Step 2 · Find second object distance (dₒ₂)
Use dₒ₂ = s − dᵢ₁
The
first
image
is
dᵢ₁
past
the
first
lens,
so
it
is
s
−
dᵢ₁
in
front
of
the
second.
dₒ₂ = 50 − 20
dₒ₂ = 30 cm
Step 3 · Find final image distance (dᵢ₂)
Use 1 ÷ f₂ = 1 ÷ dₒ₂ + 1 ÷ dᵢ₂
The second lens images the first image: the thin-lens equation again, with dₒ₂.
dᵢ₂
=
f₂
×
dₒ₂
dₒ₂ −
f₂
dᵢ₂
=
20
×
30
30 − 20
dᵢ₂
=
20
×
30
10
dᵢ₂
=
600
10
dᵢ₂ = 60 cm → real: behind the second lens.
Step 4 · Find first magnification (m₁)
Use m₁ = −dᵢ₁ ÷ dₒ
The
first
lens’
s
magnification,
the
image
distance
over
the
object
distance
with
a
minus
sign.
m₁
=
−
20
20
m₁
=
−1
Step 5 · Find second magnification (m₂)
Use m₂ = −dᵢ₂ ÷ dₒ₂
The
second
lens’
s
magnification,
in
the
same
way.
m₂
=
−
60
30
m₂
=
−2
Step 6 · Find total magnification (m)
Use m = m₁ × m₂
The
second
lens
scales
the
first
image
again,
so
the
magnifications
multiply;
m
>
0
means
the
final
image
is
upright,
m
<
0
inverted.
m
=
−1
×
(−2)
m
=
2
Check
Put the numbers back in. Both sides should match.
1
10
=
1
20
+
1
20
✓
30 = 50 − 20 ✓
1
20
=
1
30
+
1
60
✓
−1
=
−
20
20
✓
−2
=
−
60
30
✓
2 = −1 × (−2) ✓
Topic check
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