Question

A convex lens of focal length $20cm$ and a concave lens of focal length $10cm$ are placed $10cm$ apart with their principal axes coinciding. A beam of light travelling parallel to the principal axis and having a beam diameter $5.0mm$, 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 = $5mm$
Now, the image due to the convex lens should be formed on its focus (point B)
So, for the concave lens,
$u=+10cm$ (since, the virtual object is on the right of concave lens)
$f=10cm$

So $,\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\Rightarrow \frac{1}{v}=\frac{1}{-10}+\frac{1}{10}=0\Rightarrow v=\infty$

So, the emergent beam becomes parallel after redfraction in concave lens.
As shown from the triangles XYB and PQB,

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

$PQ=\frac{1}{2}\times 5=2.5cm=25\text{mm}$

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