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Question: A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12 c...

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?

Explanation

Solution

In this question we have been given about a beam of light that converges at point P. A lens is placed in the path of a convergent beam 12cm from P. Now the question demands that at what point does the beam converge if the lens is a convex lens of focal length 20 centimeter and a concave lens of focal length 16 centimeter. The convex lens is a converging lens while a concave lens is a diverging lens.

Formula Used:
Thin lens formula 1f=1v−1u\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}
where f is the focal length, u is the object distance and v is the image distance.

Complete step by step answer:
(a) The lens is placed in the path of the convergent beam and when the convex lens is placed, we know the convex lens is a converging lens. So, all the light rays will converge at a point behind P. So, the object in the first case is placed at the right side of the lens, thus object distance u=12cmu=12cm and we need to find where the image formed. Using the lens formula:
⇒1f=1v−1u\Rightarrow \dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}
⇒120=1v−112\Rightarrow \dfrac{1}{20}=\dfrac{1}{v}-\dfrac{1}{12}
⇒1v=120+112\Rightarrow \dfrac{1}{v}=\dfrac{1}{20}+\dfrac{1}{12}
⇒1v=3+560\Rightarrow \dfrac{1}{v}=\dfrac{3+5}{60}
⇒1v=860\Rightarrow \dfrac{1}{v}=\dfrac{8}{60}
∴v=7.5cm\therefore v=7.5cm

So, the beam converges at a point 7.5 cm from the lens and it is a real image.

(b) Now the lens is a concave lens and the focal length of such lens is always negative, thus f=-16cm.
Again using same formula,
\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u} \\\ \Rightarrow \dfrac{-1}{16}=\dfrac{1}{v}-\dfrac{1}{12} \\\ \Rightarrow \dfrac{1}{v}=\dfrac{1}{16}-\dfrac{1}{12} \\\ \Rightarrow \dfrac{1}{v}=\dfrac{-3+4}{48} \\\ \Rightarrow \dfrac{1}{v}=\dfrac{1}{48} \\\ \therefore v=48cm \\\
Thus, the beam of light in this case will converge at a distance of 48cm from the lens.

Note: Convex lens is a converging lens and its focal length is positive.While doing problems in optics, we have to keep in mind the sign convention. For convex lenses take the focal length as positive and for concave lenses take the focal length as negative. Convex lens converges parallel rays of light while a concave lens diverges parallel rays of light. The mirror formula and the thin lens formula are identical except for the change of sign.