Question

A convex lens of focal length 20 cm and a concave lens of focal length 10 cm are placed 10 cm apart with their principal axes coinciding. A beam of light travelling parallel to the principal axis and having a beam diameter 5.0 mm, is incident on the combination.. Show that the emergent beam is parallel to the incident one. Find the beam diameter of the emergent beam.

Answer 1

Let the parallel beam is first incident on convex lens.

d = diameter of the beam = 5 mm.

Now, the image due to the convex lens should be formed on its focus (point B).

So, for the concave lens,

u = + 10 cm (since, the virtual object is on the right of concave lens)

f = - 10 cm

So,

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

$\Rightarrow \frac{1}{v}=\frac{1}{-10}+\frac{1}{+10}=0$

$\Rightarrow v=0$

So, the emergent beam becomes parallel after refraction in concave lens.

As shown, from the triangle XYB and PQB,

$PQ=\frac{1}{2}\times 5=2.5\text{mm}$

$\frac{PQ}{XY}=\frac{RB}{ZB}=\frac{10}{20}=\frac{1}{2}$

So, the beam diameter becomes 2.5 mm. Similarly, it can be proved that if the light is incident of the concave side, the beam diameter will be 1 cm.