Q.Double-convex lenses are to be manufactured from a glass of refractive index $1.55$, with both faces of the same radius of curvature. What is the radius of curvature required if the focal length is to be $20\ \text{cm}$?

Concept: Refraction at Spherical Surface — Lens Maker’s Formula for a thin lens in air.

For a thin lens in air, the focal length $f$ is given by the Lens Maker’s formula:

where $\mu = 1.55$ is the refractive index of the glass, and $R_1$, $R_2$ are the radii of curvature of the two surfaces.

Since the lens is double-convex with both faces having the same radius $R$, we take $R_1 = +R$ and $R_2 = -R$ (using the sign convention: convex surface facing the incident light is positive, the other is negative). Substituting: …

The problem asks for the radius of curvature of each face of a double-convex lens, given the refractive index of the glass ($n = 1.55$) and the desired focal length ($f = 20\ \text{cm}$). Both faces have the same radius, so the lens is symmetric.

The key tool here is the lens maker’s formula, which connects the focal length of a thin lens to the refractive index of the material and the radii of curvature of its two surfaces. The formula comes from applying the refraction formula at a spherical surface twice — once for each surface — and then combining the results. For a thin lens, the thickness is negligible, so we can treat the two refractions as happening at the same point.

The sign convention for radii is crucial. For a double-convex lens, the first surface is convex toward the incoming light, so its centre of curvature lies on the opposite side of the incident light. By the Cartesian sign convention (commonly used in Indian exams), this means $R_1$ is positive. The second surface is also convex, but now the centre of curvature lies on the same side as the incident light (since light has already passed through the lens), so $R_2$ is negative.

Let’s work through the calculation step by step.

1. Identify the given values.

   Refractive index of glass: $n = 1.55$

   Focal length: $f = 20\ \text{cm}$ (positive, since it’s a converging lens)

   Both radii have the same magnitude: let $R_1 = R$ and $R_2 = -R$.

2. Substitute into the lens maker’s formula. …

Method: Lens Maker’s Formula

This is the standard formula that relates the focal length of a thin lens to the radii of curvature of its two surfaces and the refractive index of the lens material.

Steps

Step 1: Write the Lens Maker’s Formula

For a thin lens in air:

where:

• $f$ = focal length of the lens
• $n$ = refractive index of the lens material (relative to air)
• $R_1$ = radius of curvature of the first surface
• $R_2$ = radius of curvature of the second surface

Sign convention: For a double-convex lens, $R_1$ is positive and $R_2$ is negative.

Step 2: Apply the given conditions

• Both faces have the same radius of curvature, say $R$.
• For a double-convex lens: $R_1 = +R$ and $R_2 = -R$
• Given: $n = 1.55$, $f = +20\ \text{cm}$ (positive for converging lens) …

Common Mistakes & How to Avoid Them

This is a classic lens-maker’s formula problem. Here are the most frequent errors students make:

1. Using the wrong sign convention

Mistake: Students often plug $R_1 = +R$ and $R_2 = +R$ (both positive) without considering the sign convention for the second surface.

Why it happens: The problem says "both faces have the same radius," so it feels natural to give them the same sign.

How to avoid:

• For a double-convex lens in air, the first surface (where light enters) has centre of curvature on the right → $R_1 = +R$
• The second surface (where light exits) has centre of curvature on the left → $R_2 = -R$
• Always draw a quick ray diagram to decide signs.

2. Forgetting the lens-maker’s formula structure

Mistake: Using $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} + \frac{1}{R_2}\right)$ instead of the correct form with subtraction.

Correct formula:

How to avoid:

• Memorise the subtraction form: $R_1$ is the first surface, $R_2$ is the second surface.
• The minus sign accounts for the direction of bending at each surface.

3. Sign error in the second radius

Mistake: Substituting $R_2 = +R$ leads to:

This gives an infinite focal length — clearly wrong.

How to avoid:

• For a double-convex lens: $R_1 = +R$, $R_2 = -R$
• Then:

4. Unit mismatch

Mistake: Using $f = 20$ without converting to metres (or forgetting that $R$ comes out in the same unit as $f$).

How to avoid:

• Keep everything in cm or everything in m — but be consistent. …