Question

A convex lens of refractive index 1.5 is kept in a liquid medium having same refractive index. What is the focal length of the lens in this medium?

Hint:

A lens is only a "lens" because its refractive index is different from its surroundings. If \(n_{lens} = n_{medium}\), the lens becomes optically invisible and has no focusing power.

Answer 1

Step 1: Understanding the Concept:
A lens works by refracting light. Refraction, or the bending of light, occurs only when light passes from one medium to another with a different refractive index. The focal length of a lens is determined by its curvature and the difference in refractive index between the lens material and the surrounding medium.
Step 2: Key Formula or Approach:
The focal length (\(f\)) of a lens is given by the Lens Maker's Formula:
\[ \frac{1}{f} = \left(\frac{n_l}{n_m} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] where:
- \(n_l\) is the refractive index of the lens material.
- \(n_m\) is the refractive index of the surrounding medium.
- \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces.
Step 3: Detailed Explanation:
According to the problem statement:
- Refractive index of the lens, \(n_l = 1.5\).
- Refractive index of the medium, \(n_m = 1.5\).
Now, let's substitute these values into the Lens Maker's Formula:
\[ \frac{1}{f} = \left(\frac{1.5}{1.5} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] \[ \frac{1}{f} = (1 - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] \[ \frac{1}{f} = (0) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] \[ \frac{1}{f} = 0 \] If the reciprocal of the focal length is zero, the focal length itself must be infinite:
\[ f = \infty \] A lens with an infinite focal length has zero power (\(P = 1/f = 0\)). This means it does not bend light at all and acts as a transparent, parallel-sided plate of glass. The light rays will pass through it undeviated.
Step 4: Final Answer:
The focal length of the lens will be infinite, and it will lose its converging property, effectively becoming invisible in the liquid.