Question

A glass spherical paper weight of refractive index $\frac{3}{2}$ and radius 15 cm is placed on a page of a book (choose a letter of the print vertically below the centre of flat base.) The position of image of the print from top of sphere is

• A. 60 cm below
• B. 60 cm above
• C. 30 cm below
• D. 15 cm below

Answer 1

Answer: (D)

15 cm below

Answer 2

The problem can be solved using the formula for refraction at a spherical surface. Let the print be the object. The light travels from air ($n_1 = 1$) to glass ($n_2 = \frac{3}{2}$). The print is located vertically below the center of the sphere. Since the sphere is placed on a book page and we are looking from the top, the print is at the bottom of the sphere. This means the object is at a distance equal to the radius from the center of the sphere.

We use the lens maker's formula for a single spherical surface: $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ Here:

• $n_1 = 1$ (refractive index of air)
• $n_2 = \frac{3}{2}$ (refractive index of glass)
• $R = +15$ cm (radius of curvature of the spherical surface. It is positive because the surface is convex towards the object, which is in air).
• $u$ is the object distance from the surface. Since the print is at the bottom of the sphere, let's consider the distance from the center of the sphere. The print is at a distance $d = R = 15$ cm from the center of the sphere.

Using the formula with distances measured from the center of the sphere: $\frac{n_2}{v} - \frac{n_1}{d} = \frac{n_2 - n_1}{R}$ Substituting the values: $\frac{3/2}{v} - \frac{1}{15} = \frac{3/2 - 1}{15}$ $\frac{3}{2v} - \frac{1}{15} = \frac{1/2}{15}$ $\frac{3}{2v} - \frac{1}{15} = \frac{1}{30}$ $\frac{3}{2v} = \frac{1}{30} + \frac{1}{15}$ $\frac{3}{2v} = \frac{1 + 2}{30} = \frac{3}{30} = \frac{1}{10}$ $2v = 30$ $v = 15 \text{ cm}$ This distance $v$ is the image distance from the center of the sphere. Since $v$ is positive, the image is formed inside the glass at a distance of 15 cm from the center. This means the image is formed exactly at the center of the sphere.

The question asks for the position of the image from the top of the sphere. The top of the sphere is at a distance $R$ above the center. If the image is at the center of the sphere, its distance from the top of the sphere is $R = 15$ cm. Since the image is at the center and the top is above the center, the image is located 15 cm below the top of the sphere.