8.
Two identical bi-convex lenses L1 and L2 are arranged 10 cm apart. A collimated,
parallel beam of light strikes L1 from the left and converges 4.3 cm from the left lens.
The emergent rays on the right of L2 will be
a. parallel
b. convergent
c. divergent
d. polarised
Question statement and given data
Two identical bi‑convex lenses L1 and L2 are arranged coaxially and kept
10 cm apart. A collimated (parallel) beam is incident on L1 from the left and
converges to a point 4.3 cm to the right of L1. The nature of the emergent beam
on the right of L2 has to be identified.
Options:
- a. parallel
- b. convergent
- c. divergent
- d. polarised
The correct option is (c) divergent: the emergent rays on the right of lens L2 will be diverging.
Step‑by‑step optical solution
For a parallel beam incident on a thin convex lens, the image of a point at infinity lies at the
focal length of the lens. So the point where rays converge, 4.3 cm from L1, is the
focal point of L1. Hence:
f1 = 4.3 cm.
The lenses are identical, so:
f2 = f1 = 4.3 cm.
The separation between lenses is d = 10 cm.
For the second lens L2, the rays coming from L1 are converging towards the focal
point at 4.3 cm from L1; this point is therefore
10 − 4.3 = 5.7 cm to the left of L2.
Thus the object distance for L2 is:
u2 = −5.7 cm
(negative by the usual sign convention, since the object is on the same side as the incoming light and also within
the focal length).
Apply the thin‑lens formula for L2:
1 / f2 = 1 / v2 − 1 / u2.
Substituting f2 = 4.3 cm and u2 = −5.7 cm,
1 / 4.3 = 1 / v2 − (−1 / 5.7) = 1 / v2 + 1 / 5.7.
Solving gives v2 ≈ −10.0 cm (negative, so the image is virtual, on the
left of L2 and between L1 and L2).
Since the final image formed by L2 is virtual and on the same side as the object, the rays emerging
to the right of L2 diverge as if coming from this virtual point. Therefore the emergent beam is
divergent, which corresponds to option (c).
Why each option is right or wrong
(a) Parallel – incorrect
Two positive lenses make parallel emergent rays only when the separation equals the
sum of their focal lengths (d = f1 + f2). Here
f1 = f2 = 4.3 cm, so
f1 + f2 = 8.6 cm, but the actual separation is 10 cm; hence
the emergent beam cannot be parallel.
(b) Convergent – incorrect
Emergent rays would be converging if the second lens formed a real image on its right side
(positive v2). The calculation instead gives a negative image distance
v2 ≈ −10 cm, indicating a virtual image and therefore divergence on the right.
(c) Divergent – correct
Because the image formed by L1 lies within the focal length of L2
(5.7 cm < 4.3 cm + object‑side sign), L2 behaves like a magnifying glass with
the object inside its focal length and sends out diverging rays. So option (c)
is correct.
(d) Polarised – incorrect
Simple thin lenses made of isotropic glass do not polarise light; they only refract and focus it. Polarisation
requires special devices such as Polaroid sheets, birefringent crystals, or reflection at certain angles, none of
which is given here.
SEO‑friendly introduction
In geometrical optics, problems with two identical bi‑convex lenses 10 cm apart are frequently
asked in competitive exams to test conceptual understanding of focal length, virtual images and ray diagrams. This
article explains step by step how a parallel beam converging 4.3 cm from the first lens interacts with the second
lens and why the emergent rays turn out to be divergent rather than parallel, convergent or polarised.
