Q.Will the focal length of a lens for red light be more, same or less than that for blue light?

The key idea is that the refractive index of glass is slightly higher for blue light than for red light (dispersion).

Reasoning:

1. The lens maker's formula is $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$.
2. For blue light, $n$ is larger than for red light, so $(n-1)$ is larger. …

The key to this question lies in understanding dispersion — the fact that the refractive index of a transparent material changes with the wavelength (colour) of light. A lens works by bending light at its two surfaces, and the amount of bending depends entirely on the refractive index of the lens material relative to the surrounding medium.

For most optical glasses, the refractive index is higher for shorter wavelengths. Blue light has a shorter wavelength than red light, so:

• $n_{\text{blue}} > n_{\text{red}}$

Now, the lens maker’s formula tells us exactly how the focal length $f$ is related to the refractive index $n$ and the geometry of the lens (radii of curvature $R_1$ and $R_2$):

For a given lens, the term in brackets is a constant (determined by the shape). So the focal length depends only on $(n - 1)$.

Let’s walk through the reasoning step by step:

1. Identify the relationship. From the formula, $f$ is inversely proportional to $(n - 1)$. A larger $n$ means a smaller $f$, and a smaller $n$ means a larger $f$.

2. Compare refractive indices. For typical crown glass or flint glass, $n_{\text{blue}} > n_{\text{red}}$. For example, in common borosilicate crown glass (BK7), $n_{\text{blue}} \approx 1.53$ and $n_{\text{red}} \approx 1.51$ (approximate values for illustration).

3. Apply to the formula. Since $n_{\text{blue}} > n_{\text{red}}$, we have $(n_{\text{blue}} - 1) > (n_{\text{red}} - 1)$. Therefore:

which implies $f_{\text{blue}} < f_{\text{red}}$. …

Method: Comparing Focal Lengths for Different Colours Using the Lens Maker's Formula

Use this method whenever a question asks how a lens's focal length differs between two colours (or wavelengths) of light.

Steps

Step 1: Write the lens maker's formula and identify what's fixed vs. variable

For a single given lens, the geometric term $\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ is fixed by its shape and does not change with colour — only the refractive index $n$ varies with wavelength.

Step 2: Recall the dispersion trend for ordinary glass

Refractive index increases as wavelength decreases: shorter-wavelength (blue) light experiences a higher $n$ than longer-wavelength (red) light in typical optical glass. …