Question

Derive the Lens Maker’s Formula for a thin convex lens.

Hint:

Lens Maker’s Formula gives focal length from lens geometry. Use it when refractive index and radii of curvature are known.

Answer 1

Concept: The Lens Maker’s Formula relates the focal length of a lens to: 
Refractive index of lens material 
Radii of curvature of its two surfaces 
It is derived using refraction at two spherical surfaces. 
Assumptions: 
Thin convex lens 
Refractive index of lens = \( \mu \) 
Lens in air (\( \mu_{\text{air}} = 1 \)) 
Radii of curvature = \( R_1 \) and \( R_2 \) 
Step 1: Refraction at first spherical surface Using refraction formula at a spherical surface: \[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \] For first surface: 
Light travels from air to glass 
\( \mu_1 = 1, \quad \mu_2 = \mu \) 
Object at distance \( u \), image formed at \( v_1 \) 
\[ \frac{\mu}{v_1} - \frac{1}{u} = \frac{\mu - 1}{R_1} \quad \cdots (1) \] Step 2: Refraction at second spherical surface Now the image formed by the first surface acts as the object for the second surface. For second surface: 
Light travels from glass to air 
\( \mu_1 = \mu, \quad \mu_2 = 1 \) 
Object distance = \( v_1 \) 
Final image distance = \( v \) 
\[ \frac{1}{v} - \frac{\mu}{v_1} = \frac{1 - \mu}{R_2} \quad \cdots (2) \] Step 3: Add equations (1) and (2) \[ \left( \frac{\mu}{v_1} - \frac{1}{u} \right) + \left( \frac{1}{v} - \frac{\mu}{v_1} \right) = \frac{\mu - 1}{R_1} + \frac{1 - \mu}{R_2} \] Cancel \( \frac{\mu}{v_1} \): \[ \frac{1}{v} - \frac{1}{u} = (\mu - 1)\left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] Step 4: For focal length For focal length, object at infinity: \[ u = \infty \Rightarrow \frac{1}{u} = 0, \quad v = f \] \[ \frac{1}{f} = (\mu - 1)\left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] Lens Maker’s Formula: \[ \boxed{ \frac{1}{f} = (\mu - 1)\left( \frac{1}{R_1} - \frac{1}{R_2} \right) } \] Notes: 
Valid for thin lenses. 
Sign convention must be used carefully. 
For convex lens in air, \( R_1>0 \), \( R_2<0 \).