Two thin lenses in contact
In many optical instruments there may be
compound lenses, that is, two or more lenses in contact. We will first deal with the case of
two thin lenses in contact.
In Figure 1, let the be focal lengths of the two
lenses be f
1 and f
2.
u
1 = OC
1 and v
1 =
I
1C
1u
2= -C
1I
1 which is approximately equal to C
2 I
1
and v
2 = C
2I which is approximately equal to C
1I
Therefore
1/u
1 +
1/v
1 = 1/f
1 and 1/u
2 + 1/v
2 =
1/f
21/OC
1 + 1/I
1C
1 = 1/f
1 and 1/-
C
1I
1 + 1/C
1I = 1/f
2Therefore:
1/OC
1 +
1/I
1C
1 = 1/f
1 + 1/f
2 = 1/F
Combined focal length
of two thin lenses in contact is given by:
1/F = 1/f1 + 1/f2
Combined focal length (F): =f1f2/[f1 + f2]
where F is the focal length of the
combination.
Example problem
A bi-convex lens of focal length 10 cm is fixed to a plano concave lens of focal length 20 cm made of glass of the same refractive index. What is the focal length of the combination?
Combined focal length (F): =f1f2/[f1 + f2]
F = 10x(-20)/[10-20] = +20 cm
The focal length of the combination is positive and so it acts as a convex lens.
The combined focal length for two thin lenses separated by a distance
a (Figure 2) is given by the equation:
1/F = 1/f1 + 1/f2 - a/f1f2
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