Question

A light beam converges at a point O. In the path of this beam, a concave lens of focal length 15 cm is placed at a distance of 10 cm before point O. The beam now converges at a point O'. Find the magnitude and the direction of shift OO'.

Hint:

O is a virtual object at u = +10 cm. 1/v = -1/15 + 1/10 gives v = 30 cm, so the shift is 20 cm away from the lens.

Answer 1

Step 1: Treat O as a virtual object:
The beam is converging towards O. After the lens is put in, O is a point the rays would have reached behind the lens. So O acts as a virtual object for the lens.
Take the direction of the light as positive. The object distance is \( u = +10 \) cm (virtual object on the right of the lens). The focal length of a concave lens is \( f = -15 \) cm.

Step 2: Apply the lens formula:
\( \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f} \). So \( \dfrac{1}{v} = \dfrac{1}{f} + \dfrac{1}{u} = -\dfrac{1}{15} + \dfrac{1}{10} = \dfrac{-2+3}{30} = \dfrac{1}{30} \).
So \( v = +30 \) cm. The beam now converges at O', 30 cm from the lens, on the same side as O.

Step 3: Shift:
\( OO' = 30 - 10 = 20 \) cm.
O' is farther from the lens than O, so the shift is in the direction of the beam, away from the lens. A concave lens diverges the beam a little, so it converges later.

Final Answer:
The shift is OO' = 20 cm, away from the lens (along the direction of propagation of light).