Question

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

Hint:

Lens Maker’s Formula: \[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] Focal length increases if curvature decreases or refractive index decreases.

Answer 1

Concept: A convex lens has two spherical surfaces. The focal length depends on:

• Refractive index of lens material (\( \mu \))
• Radii of curvature of its surfaces (\( R_1, R_2 \))

Lens maker’s formula gives relation between focal length and lens geometry. We derive it using refraction at spherical surfaces.
Refraction Formula at a Spherical Surface: \[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \]
Step 1: Refraction at First Surface Let:

• Object at infinity → \( u = \infty \)
• Refractive index of air = 1
• Refractive index of lens = \( \mu \)
• Radius of first surface = \( R_1 \)

Using refraction formula: \[ \frac{\mu}{v_1} - \frac{1}{\infty} = \frac{\mu - 1}{R_1} \] \[ \frac{\mu}{v_1} = \frac{\mu - 1}{R_1} \] \[ v_1 = \frac{\mu R_1}{\mu - 1} \] This image acts as virtual object for second surface.
Step 2: Refraction at Second Surface Now light goes from lens to air:

• Object distance = \( u_2 = -v_1 \) (sign convention)
• Final image at focal point → \( v_2 = f \)
• Radius of second surface = \( R_2 \)

Applying refraction formula: \[ \frac{1}{f} - \frac{\mu}{-v_1} = \frac{1 - \mu}{R_2} \] \[ \frac{1}{f} + \frac{\mu}{v_1} = \frac{1 - \mu}{R_2} \]
Step 3: Substitute \( \frac{\mu}{v_1} \) From first surface: \[ \frac{\mu}{v_1} = \frac{\mu - 1}{R_1} \] Substitute: \[ \frac{1}{f} + \frac{\mu - 1}{R_1} = \frac{1 - \mu}{R_2} \] Rearranging: \[ \frac{1}{f} = \frac{1 - \mu}{R_2} - \frac{\mu - 1}{R_1} \] Factor out \( (\mu - 1) \): \[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] \[ \boxed{\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)} \] This is the Lens Maker’s Formula.
Special Cases:

• For symmetric biconvex lens: \( R_1 = R, R_2 = -R \) \[ \frac{1}{f} = (\mu - 1)\left(\frac{2}{R}\right) \]
• If lens in medium of refractive index \( \mu_m \): \[ \frac{1}{f} = \left(\frac{\mu}{\mu_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

Sign Convention:

• \( R_1>0 \) for convex surface
• \( R_2<0 \) for convex lens second surface

Importance:

• Used in lens design
• Helps control focal length
• Basis for optical instruments