JEE Physics 2026 Question with Solution

Given: A convex lens has refractive index $\mu_{\text{lens}}=1.5$ and focal length in air $f_{\text{air}}=18 \, \text{cm}$. The refractive index of water is $\mu_{\text{water}}=\dfrac{4}{3}$.

Find: The value of $\alpha$ if the difference in focal lengths is $\alpha f$.

Concept: For a thin lens immersed in a medium, use the refractive index of the lens relative to the medium in the lens maker formula.

$$\frac{1}{f_m}=\left(\frac{\mu_{\text{lens}}}{\mu_{\text{medium}}}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$$

In air,

$$\frac{1}{f_{\text{air}}}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$$

Using $f_{\text{air}}=18 \, \text{cm}$ and $\mu=1.5$,

$$\left(\frac{1}{R_1}-\frac{1}{R_2}\right)=\frac{1}{18(1.5-1)}=\frac{1}{9}$$

Now for water, the relative refractive index is

$$\mu_{\text{rel}}=\frac{\mu_{\text{lens}}}{\mu_{\text{water}}}=\frac{1.5}{4/3}=\frac{9}{8}$$

So,

$$\frac{1}{f_{\text{water}}}=\left(\frac{9}{8}-1\right)\frac{1}{9}=\frac{1}{8}\cdot\frac{1}{9}=\frac{1}{72}$$

Hence,

$$f_{\text{water}}=72 \, \text{cm}$$

The difference in focal lengths is

$$\Delta f=f_{\text{water}}-f_{\text{air}}=72-18=54 \, \text{cm}$$

Given that

$$\Delta f=\alpha f$$

therefore,

$$\alpha=\frac{54}{18}=3$$

Therefore, the value of $\alpha$ is $3$.

Given: The lens has refractive index $1.5$, focal length in air $18 \, \text{cm}$, and water has refractive index $\dfrac{4}{3}$.

Find: The value multiplying $f$ in the focal length difference.

The key idea is that when the lens is placed in water, its power reduces because the contrast between lens material and surrounding medium decreases.

In air, lens power is proportional to

$$(1.5-1)=0.5$$

In water, the effective refractive index factor becomes

$$\frac{1.5}{4/3}-1=\frac{9}{8}-1=\frac{1}{8}$$

Thus the power in water is

$$\frac{1/8}{1/2}=\frac{1}{4}$$

Since focal length is inversely proportional to power,

$$f_{\text{water}}=4f_{\text{air}}=4\times 18=72 \, \text{cm}$$

Hence the change in focal length is

$$72-18=54 \, \text{cm}$$

Now,

$$54=\alpha \times 18$$

so

$$\alpha=3$$

Therefore, the correct numerical answer is $3$.