Question

Write down the formula for refraction of light from a spherical surface and with the help of this, derive the relation \( \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \) for a thin lens. Also show that, the focal length for concave lens will be negative.

Hint:

For a concave lens, the focal length is negative because the image formed is virtual and diminished. For a convex lens, the focal length is positive, and the image formed can be real or virtual depending on the object distance.

Answer 1

Refraction of Light from a Spherical Surface:
When light passes from one medium to another at a spherical surface, the relation between the object distance \( u \), image distance \( v \), and the radius of curvature \( R \) of the spherical surface is given by the lens-maker's formula. The formula for refraction at a spherical surface is: \[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}, \] where:
- \( n_1 \) and \( n_2 \) are the refractive indices of the first and second mediums,
- \( u \) is the object distance,
- \( v \) is the image distance,
- \( R \) is the radius of curvature of the surface.
For a thin lens, we have two spherical surfaces with radii \( R_1 \) and \( R_2 \), and the light is refracted twice. The lens-maker's equation for a thin lens in air is: \[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right), \] where:
- \( f \) is the focal length of the lens,
- \( n \) is the refractive index of the material of the lens,
- \( R_1 \) is the radius of curvature of the first surface, and
- \( R_2 \) is the radius of curvature of the second surface.
Focal Length for Concave Lens:
For a concave lens, the radii of curvature are negative (\( R_1 \) is negative and \( R_2 \) is positive). Substituting these values into the lens-maker's equation, we get: \[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right). \] Since \( R_1 \) is negative for a concave lens, the focal length \( f \) will also be negative. This confirms that the focal length of a concave lens is negative.