CBSE · Class 12 · Physics · Ray Optics and Optical InstrumentsDerive the lens maker's formula for a thin convex lens, clearly stating all the sign conventions and assumptions made during the derivation.

Introduction\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 extensively used by opticians to design lenses of desired focal lengths.

Assumptions

1. The lens is considered thin, meaning the thickness of the lens is negligible compared to the radii of curvature of its surfaces.
2. The aperture of the lens is small, so that paraxial rays are considered.
3. The object is a point object placed on the principal axis.

Step-by-Step Derivation

1. Refraction at the First Surface: Let a point object $O$ be placed in a rarer medium of refractive index $n_1$ in front of a convex spherical surface of radius of curvature $R_1$. The refractive index of the denser lens material is $n_2$. The refraction formula for a spherical surface is given by: $$\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1}$$ Here, $v_1$ is the image distance formed by the first surface alone.

2. Refraction at the Second Surface: The image $I_1$ formed by the first surface acts as a virtual object for the second spherical surface of radius of curvature $R_2$. The light goes from the denser medium ($n_2$) to the rarer medium ($n_1$). Applying the refraction formula for the second surface: $$\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1 - n_2}{R_2}$$ $$\frac{n_1}{v} - \frac{n_2}{v_1} = -\frac{n_2 - n_1}{R_2}$$

3. Combining the Equations: Adding the equations for the first and second surfaces: $$\left(\frac{n_2}{v_1} - \frac{n_1}{u}\right) + \left(\frac{n_1}{v} - \frac{n_2}{v_1}\right) = \frac{n_2 - n_1}{R_1} - \frac{n_2 - n_1}{R_2}$$ $$\frac{n_1}{v} - \frac{n_1}{u} = (n_2 - n_1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

4. Final Formula: Dividing both sides by $n_1$ and setting $u = -f$ when $v = \infty$: $$\frac{1}{v} - \frac{1}{u} = \left(\frac{n_2}{n_1} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ $$\frac{1}{f} = (n_{21} - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ Where $n_{21} = \frac{n_2}{n_1}$ is the refractive index of the lens with respect to the surrounding medium.