Question

Using the relation for refraction at a curved spherical surface, derive the expression for lens maker's formula.

Hint:

Lens Maker's Formula: \[ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} -\frac{1}{R_2} \right) \] For a convex lens: \[ R_1>0, \qquad R_2<0. \] For a concave lens: \[ R_10. \]

Answer 1

Concept: A thin lens consists of two refracting spherical surfaces. The focal length of the lens depends upon:
• Refractive index of the lens material,
• Refractive index of the surrounding medium,
• Radii of curvature of the two refracting surfaces. The relation connecting these quantities is known as the Lens Maker's Formula because it enables a lens manufacturer to determine the focal length of a lens from its shape and material. The derivation is based on the formula for refraction at a spherical surface: \[ \frac{n_2}{v}-\frac{n_1}{u} = \frac{n_2-n_1}{R} \] where
• \(n_1\) = refractive index of first medium,
• \(n_2\) = refractive index of second medium,
• \(u\) = object distance,
• \(v\) = image distance,
• \(R\) = radius of curvature.

Step 1: Refraction at the first spherical surface. Consider a thin convex lens of refractive index \(n\) placed in air. Let \[ \mu_a=1 \] and \[ \mu_l=n. \] An object is placed at distance \(u\) from the first surface of radius \(R_1\). Applying refraction formula, \[ \frac{n}{v_1} -\frac{1}{u} = \frac{n-1}{R_1}. \] This gives \[ \boxed{ \frac{n}{v_1} = \frac{1}{u} +\frac{n-1}{R_1} } \]

Step 2: Refraction at the second spherical surface. The image formed by the first surface acts as a virtual object for the second surface. For the second surface, \[ n_1=n, \qquad n_2=1. \] Applying the refraction formula again, \[ \frac{1}{v} -\frac{n}{u_2} = \frac{1-n}{R_2}. \] For a thin lens, \[ u_2=v_1. \] Therefore, \[ \frac{1}{v} -\frac{n}{v_1} = \frac{1-n}{R_2}. \] or \[ \boxed{ \frac{1}{v} = \frac{n}{v_1} -\frac{n-1}{R_2} } \]

Step 3: Substitute the value of \(\frac{n}{v_1}\). Using the result from the first surface, \[ \frac{1}{v} = \left( \frac{1}{u} +\frac{n-1}{R_1} \right) -\frac{n-1}{R_2}. \] Therefore, \[ \frac{1}{v} -\frac{1}{u} = (n-1) \left( \frac{1}{R_1} -\frac{1}{R_2} \right). \] Hence the lens formula becomes \[ \boxed{ \frac{1}{v} -\frac{1}{u} = (n-1) \left( \frac{1}{R_1} -\frac{1}{R_2} \right) } \]

Step 4: Obtain the focal length of the lens. For focal length, \[ u=\infty. \] Thus, \[ \frac{1}{u}=0. \] and \[ v=f. \] Therefore, \[ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} -\frac{1}{R_2} \right). \] Hence, \[ \boxed{ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} -\frac{1}{R_2} \right) } \] This is the required Lens Maker's Formula for a thin lens in air. Final Result: \[ \boxed{ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} -\frac{1}{R_2} \right) } \]