Q.(i) If $f = 0.5\ \text{m}$ for a glass lens, what is the power of the lens?

Concept: Lens Maker’s Formula and Power of a Lens.

(i) Power $P = \frac{1}{f}$ with $f$ in metres.

$P = \frac{1}{0.5} = 2\ \text{D}$.

(ii) Lens Maker’s formula: $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$.

For double convex, $R_1 = +10\ \text{cm}$, $R_2 = -15\ \text{cm}$, $f = 12\ \text{cm}$.

$\frac{1}{12} = (\mu - 1)\left(\frac{1}{10} + \frac{1}{15}\right) = (\mu - 1)\left(\frac{5}{30}\right) = (\mu - 1)\frac{1}{6}$.

So $\mu - 1 = \frac{6}{12} = 0.5$, hence $\mu = 1.5$.

(iii) In air: $\frac{1}{f_a} = (\mu_g - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$.

In water: $\frac{1}{f_w} = \left(\frac{\mu_g}{\mu_w} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$. …

(i) Power of the lens

The power $P$ of a lens is defined as the reciprocal of its focal length in metres:

Given $f = 0.5\ \text{m}$:

$P = \frac{1}{0.5} = 2\ \text{D}$ …

Method: Lens Maker’s Formula & Power of a Lens

This method connects the geometry (radii of curvature) and material (refractive index) of a lens to its focal length and power.

(i) Power from focal length

Formula:

where $f$ is in metres.

Steps:

1. Given $f = 0.5\ \text{m}$.
2. Apply formula:

Answer:

(ii) Refractive index from radii and focal length

Formula (Lens Maker’s Formula):

Steps:

1. For a double convex lens:
   • $R_1 = +10\ \text{cm}$ (first surface convex toward object)
   • $R_2 = -15\ \text{cm}$ (second surface convex, but sign convention makes it negative)
2. Given $f = 12\ \text{cm}$.
3. Substitute:

4. Simplify bracket:

5. So:

6. Multiply both sides by 6:

7. Therefore:

Answer:

(iii) Focal length in water

Key idea: The lens maker’s formula uses relative refractive index of lens material with respect to surrounding medium.

Formula:

Steps:

1. In air:

Given $f_{\text{air}} = 20\ \text{cm}$, $\mu_{\text{glass}} = 1.5$.

2. In water:

3. Take ratio: …

Common Mistakes & How to Avoid Them: Refraction at Spherical Surfaces & Lens Formula

Here are the most frequent errors students make on these three classic lens problems, along with the correct reasoning to avoid them.

(i) Power from Focal Length

Mistake 1: Forgetting the unit conversion

• Students often write $P = \frac{1}{0.5} = 2\ \text{D}$ without checking units.
• Why it's wrong: The formula $P = \frac{1}{f}$ requires $f$ in metres. If $f$ is given in cm, you must convert first.

Mistake 2: Confusing sign convention

• Some students write $P = -\frac{1}{f}$ for a convex lens.
• Why it's wrong: For a convex (converging) lens, $f$ is positive, so $P$ is positive. The negative sign is for concave lenses.

✓ How to avoid:

• Always convert $f$ to metres before plugging into $P = \frac{1}{f}$.
• Remember: Convex lens → positive $f$ → positive $P$.

Correct solution:

(ii) Refractive Index from Radii & Focal Length

Mistake 1: Using the wrong sign for radii

• Students often take $R_1 = +10\ \text{cm}$ and $R_2 = +15\ \text{cm}$ for a double convex lens.
• Why it's wrong: By the Cartesian sign convention, for a double convex lens:
  • First surface (left side): centre of curvature is to the right → $R_1 = +10\ \text{cm}$
  • Second surface (right side): centre of curvature is to the left → $R_2 = -15\ \text{cm}$

Mistake 2: Forgetting the sign in the lens maker's formula

• The formula is:

• Students often write $\frac{1}{R_1} + \frac{1}{R_2}$ instead of the subtraction.

Mistake 3: Unit inconsistency

• Using $f = 12\ \text{cm}$ but $R$ in metres, or vice versa.

✓ How to avoid:

• Always apply the sign convention systematically:
  • Light travels from left to right.
  • $R$ is positive if the centre of curvature is to the right of the surface.
  • $R$ is negative if the centre of curvature is to the left.
• For a double convex lens: $R_1 > 0$, $R_2 < 0$.
• Keep all lengths in the same unit (cm or m).

Correct solution:

(iii) Focal Length Change from Air to Water

Mistake 1: Using the wrong refractive index ratio

• Students often write:

but forget that the formula uses relative refractive indices.

Mistake 2: Confusing the lens maker's formula for different media

• The correct relation is:

• Students often invert this ratio or use absolute indices incorrectly.

Mistake 3: Sign errors in the final answer …