Derivation of Lens Maker's Formula
Step 1 (Formula / Governing Law & Given Data): [1 Mark]
For a single spherical refracting surface, the relation between object distance $u$, image distance $v$, and radius of curvature $R$ is given by:
$$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$$Where $n_1$ is the refractive index of the surrounding medium and $n_2$ is the refractive index of the lens material.
Step 2 (Step-by-Step Mathematical Working): [1.5 Marks]
- For the first surface (refraction from $n_1$ to $n_2$): $$\frac{n_2}{v'} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1} \quad \dots (1)$$
- For the second surface (refraction from $n_2$ to $n_1$, where the image $v'$ acts as a virtual object): $$\frac{n_1}{v} - \frac{n_2}{v'} = \frac{n_1 - n_2}{R_2} = -\frac{n_2 - n_1}{R_2} \quad \dots (2)$$
- Adding equations (1) and (2): $$\left( \frac{n_2}{v'} - \frac{n_1}{u} \right) + \left( \frac{n_1}{v} - \frac{n_2}{v'} \right) = (n_2 - n_1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$ $$\frac{n_1}{v} - \frac{n_1}{u} = (n_2 - n_1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$
- Dividing by $n_1$ and setting $n = \frac{n_2}{n_1}$: $$\frac{1}{v} - \frac{1}{u} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$
Step 3 (Final Concluded Answer): [0.5 Mark]
Since $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$, we obtain the Lens Maker's Formula:
$$\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$💡 EL10 Concept Insight:
Think of a lens as two separate curved mirrors working together; the first surface bends the light to start forming an image, and the second surface bends it again to finish the job, with the lens's material and shape deciding exactly how much that light gets "bent."