9.2 Lens Maker's Formula

9.2 Lens Maker's Formula

For a thin lens of refractive index $n_2$ in a medium of refractive index $n_1$, with radii of curvature $R_1$ and $R_2$:

$$\frac{1}{f} = \left(\frac{n_2}{n_1} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

For a lens in air ($n_1 = 1$, $n_2 = n$):

$$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Thin Lens Formula

$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$

Power and Combination of Lenses

Power of a lens:

$$P = \frac{1}{f\,(\text{in metres})} \quad (\text{unit: dioptre, D})$$

For thin lenses in contact:

$$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \cdots$$ $$P = P_1 + P_2 + \cdots$$

Example

Two thin lenses of focal lengths $+20\,\text{cm}$ and $-40\,\text{cm}$ are placed in contact. Find the effective focal length and power.

$$\frac{1}{f} = \frac{1}{20} + \frac{1}{-40} = \frac{2 - 1}{40} = \frac{1}{40}$$ $$f = +40\,\text{cm} = 0.4\,\text{m}$$ $$P = \frac{1}{0.4} = +2.5\,\text{D}$$