Question

Write the formula of refraction of light through a single spherical surface. The radius of curvature of one end of a cylindrical glass (n = 1.5) rod is 2 cm. Find the position and nature of the image of the point source S. See the figure and also draw the ray diagram.
\includegraphics[width=0.5\linewidth]{image9.png}

Hint:

For a spherical surface, the image formed depends on the refractive indices of the media and the radius of curvature. If the object is in air, the refractive index of air is considered as 1.

Answer 1

The refraction of light through a single spherical surface is governed by the formula for refraction at a spherical surface, given by:
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \] Where:
- \( n_1 \) and \( n_2 \) are the refractive indices of the media (in this case, \( n_1 = 1 \) for air and \( n_2 = 1.5 \) for glass),
- \( u \) is the object distance (the distance of the point source \( S \) from the spherical surface),
- \( v \) is the image distance (the distance of the image formed),
- \( R \) is the radius of curvature of the spherical surface.
Given: - Radius of curvature \( R = 2 \, \text{cm} \),
- The object is placed in air, so \( n_1 = 1 \),
- The refractive index of glass \( n_2 = 1.5 \).
Substituting these values into the formula:
\[ \frac{1.5}{v} - \frac{1}{u} = \frac{1.5 - 1}{2} \] \[ \frac{1.5}{v} - \frac{1}{u} = \frac{0.5}{2} = 0.25 \] Thus, the image will be formed at a distance \( v \) from the spherical surface. To find the position and nature of the image, the value of \( u \) (object distance) must be specified, as it determines the final image location.
The ray diagram would show light rays entering the spherical surface from the object at a distance \( u \), refracting and converging at the image point \( v \).