A point object is placed at a distance of 20 cm from a spherical lens of refractive index \( n = \displaystyle\frac{3}{2} \) and the radius of curvature of each surface 10 cm. The distance of the image formed from the lens is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 20 cm
The lens has two convex surfaces of equal radius (a thin double-convex lens). Its focal length follows from the lens maker's formula, and the image from the thin lens formula. With the Cartesian sign convention (light travelling left to right), \( R_1 = +10\,\mathrm{cm} \) and \( R_2 = -10\,\mathrm{cm} \).
\( \displaystyle \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = \left(\frac{3}{2} - 1\right)\left(\frac{1}{10} + \frac{1}{10}\right) = \frac{1}{10} \)
\( \displaystyle f = 10\,\mathrm{cm} \)
Putting \( u = -20\,\mathrm{cm} \) in the lens formula,
\( \displaystyle \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \;\Rightarrow\; \frac{1}{v} = \frac{1}{10} - \frac{1}{20} = \frac{1}{20} \)
\( \displaystyle v = +20\,\mathrm{cm} \)
The image is real and 20 cm from the lens on the other side.
Shortcut: The object is at \( 2f \) (20 cm, with \( f = 10\,\mathrm{cm} \)). A convex lens forms the image of an object at \( 2f \) at \( 2f \) on the other side, so \( v = 20\,\mathrm{cm} \) at once.