In the figure given below, APB is a curved surface of radius of curvature 10 cm separating air and a transparent material (\( \mu = 4/3 \)). A point object O is placed in air on the principal axis of the surface 20 cm from P. The distance of the image of O from P will be ____________.
Fill in the blank with the correct answer from the options given below.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 16 cm left of P in air
Use the formula for refraction at a spherical surface with the Cartesian sign convention: distances from P, positive to the right (the direction of the incident light).
\( \displaystyle \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \)
The surface is concave towards the object, so its centre of curvature lies on the left, in air.
\( \mu_1 = 1 \)
\( \mu_2 = 4/3 \)
\( u = -20\,\mathrm{cm} \)
\( R = -10\,\mathrm{cm} \)
\( \displaystyle \frac{4/3}{v} - \frac{1}{-20} = \frac{4/3 - 1}{-10} \)
\( \displaystyle \frac{4}{3v} + \frac{1}{20} = -\frac{1}{30} \)
\( \displaystyle \frac{4}{3v} = -\frac{1}{30} - \frac{1}{20} = -\frac{5}{60} = -\frac{1}{12} \)
\( \displaystyle v = -\frac{4 \times 12}{3} = -16\,\mathrm{cm} \)
The negative sign means the image is on the same side as the object: a virtual image 16 cm to the left of P, in air.