MP Board · Class 12 · Physics · Ray Optics and Optical InstrumentsDerive the Lens Maker's formula for a thin convex lens, stating clearly the assumptions made. Also, define the power of a lens and write its SI unit.

Introduction to Lens Maker's Formula\nThe Lens Maker's formula relates the focal length of a lens to the refractive index of the lens material and the radii of curvature of its two surfaces. It is used by manufacturers to design lenses of desired focal lengths.

Assumptions Made

• The lens is thin, so the distance measured from the pole can be taken from the optical center.
• The aperture of the lens is small.
• The object is a point object placed on the principal axis.
• All angles made by the rays with the principal axis are small.

Mathematical Derivation

1. Let us consider a thin convex lens of refractive index $n_2$ placed in a rarer medium of refractive index $n_1$. Let $R_1$ and $R_2$ be the radii of curvature of surface 1 and surface 2 respectively.
2. When refraction takes place at the first spherical surface (facing the object), the light goes from medium $n_1$ to $n_2$. The image formed by the first surface acts as a virtual object for the second surface.
3. Using the refraction formula for a single spherical surface: $$\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1}$$
4. For the second surface, refraction takes place from medium $n_2$ to $n_1$. The light emerges into the medium $n_1$, forming the final real image at distance $v$: $$\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1 - n_2}{R_2}$$
5. Adding the above two equations: $$\frac{n_1}{v} - \frac{n_1}{u} = (n_2 - n_1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$
6. Dividing throughout by $n_1$ and substituting $n_{21} = \frac{n_2}{n_1}$: $$\frac{1}{v} - \frac{1}{u} = (n_{21} - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$
7. When the object is at infinity ($u = \infty$), the image is formed at the focus ($v = f$): $$\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$\nThis is the required Lens Maker's formula.

Power of a Lens

• Definition: The power of a lens is defined as the tangent of the angle by which it converges or diverges a beam of light parallel to the principal axis at unit distance. In simple terms, it is the reciprocal of the focal length in meters.
• Formula: $P = \frac{1}{f \text{ (in meters)}}$
• SI Unit: Dioptre (D). One dioptre is the power of a lens whose focal length is one meter.