Animated Solution for Physics - Optics: A thin glass (refractive index $1.5$) lens has optical power of $-5\text{ D}$ in air. Its optical power in a liquid medium with refractive index $1.6$ will be

When we think of the optical power of a lens, we often default to the simple formula $P=\frac{1}{f}$. While this is perfectly fine for a lens in air or vacuum, it hides the true physical mechanism of why a lens bends light.

The ability of a lens to converge or diverge light depends entirely on the contrast between the refractive index of the lens material and the refractive index of the surrounding medium. A more robust and generalized definition of optical power for a lens in any medium is given by:

This elegant equation separates the optical environment $({\mu }_{\text{lens}}-{\mu }_{\text{medium}})$ from the physical geometry of the lens $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$.

Let's apply this generalized formula to our given scenario. We have a glass lens with a refractive index ${\mu }_g=1.5$ placed in air (${\mu }_a=1$). We are told its power in air is $P_a=-5\text{ D}$. The negative sign immediately tells us that this is a diverging (concave) lens.

Substituting these values into our power equation:

Solving for the geometric factor, we get:

This geometric factor is a permanent physical property of the lens. Whether you take this lens to space, submerge it in water, or bury it in honey, this factor remains exactly $-10{\text{ m}}^{-1}$.

Now comes the exciting part. We take this exact same lens and submerge it in a liquid medium with a refractive index ${\mu }_m=1.6$. Notice a critical detail here: the surrounding medium is now optically denser than the glass itself!

Let's calculate the new optical power $P_m$ using our generalized formula. We plug in the new medium's refractive index and our previously calculated geometric factor:

The math reveals a beautiful physical truth. The double negative has flipped the sign of the power! The lens, which was diverging in air, is now converging in the liquid. Because light travels slower in the liquid than in the glass, the bending of light at the interfaces reverses direction.

This phenomenon is a classic trap in competitive exams. A lens does not have an absolute nature; its behavior is entirely relative to its environment.

Consider a fascinating thought experiment: what if we submerged the lens in a liquid with a refractive index of exactly $1.5$? The term $({\mu }_{\text{lens}}-{\mu }_{\text{medium}})$ would become zero, resulting in an optical power of $0\text{ D}$. The lens would lose all ability to bend light and would effectively become invisible in the liquid!