A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12 cm from P. At what point does the beam converge if the lens is (a) a convex lens of focal length 20 cm, and (b) a concave lens of focal length 16 cm?
In the given situation, the object is virtual and the image formed is real.
Object distance, u = +12 cm
(a) Focal length of the convex lens, f = 20 cm
Image distance = v
According to the lens formula, we have the relation:
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\frac{1}{v}-\frac{1}{12}=\frac{1}{20}$
$\frac{1}{v}=\frac{1}{20}+\frac{1}{12}=\frac{3+5}{60}=\frac{8}{60}$
$\therefore v=\frac{60}{8}=7.5 \mathrm{~cm}$
Hence, the image is formed 7.5 cm away from the lens, toward its right.
(b) Focal length of the concave lens, f = −16 cm
Image distance = v
According to the lens formula, we have the relation:
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\frac{1}{v}=-\frac{1}{16}+\frac{1}{12}=\frac{-3+4}{48}=\frac{1}{48}$
$\therefore v=48 \mathrm{~cm}$
Hence, the image is formed 48 cm away from the lens, toward its right.
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