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.
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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, $\frac{1}{\text{v}}-\frac{1}{\text{u}}=\frac{1}{\text{f}}\Rightarrow\frac{1}{\text{v}}=\frac{1}{-10}+\frac{1}{10}=0\Rightarrow\text{v}=\infty$
So, the emergent beam becomes parallel after refraction in concave lens.
As shown from the triangles XYB and PQB,
$\frac{\text{PQ}}{\text{XY}}=\frac{\text{RB}}{\text{ZB}}=\frac{10}{20}=\frac{1}{2}$
So, $\text{PQ}=\frac{1}{2}\times5=2.5\text{mm}$
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.
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