To solve this problem, let's break it down step by step.
Given Data:
Focal length of plano-convex lens = f
Refractive index (R.I) of plano-convex lens = 1.5
The lens is placed on a plane glass plate.
The gap between the lens and the glass plate is filled with a liquid.
The effective focal length of the system = 2f.
Step 1: Formula for the focal length of a plano-convex lens
For a thin plano-convex lens, the focal length is given by the lens maker’s formula:
(1/f) = (n - 1) (1/R)
where:
n = refractive index of the material of the lens,
R = radius of curvature of the convex surface (the plane surface has infinite radius of curvature, so it does not contribute).
Substituting the given values:
(1/f) = (1.5 - 1) (1/R)
(1/f) = (0.5) (1/R)
So, R = 0.5f.
Step 2: Finding the effective focal length of the combination
When the plano-convex lens is placed on the glass plate with a liquid in between, the system behaves like a plano-concave lens formed by the liquid layer (since it has a curved interface opposite to the convex surface of the lens).
The focal length of a thin plano-concave lens is given by:
(1/f') = (1 - n_liquid) (1/R)
where:
n_liquid = refractive index of the liquid,
R = 0.5f.
So,
(1/f') = (1 - n_liquid) (1/0.5f)
(1/f') = (2(1 - n_liquid))/f.
Step 3: Applying the formula for the combination of two lenses
The focal lengths of two lenses in contact are combined using:
1/F_effective = 1/f + 1/f'
Substituting values:
1/(2f) = 1/f + (2(1 - n_liquid))/f
Multiplying throughout by f:
1/2 = 1 + 2(1 - n_liquid)
1/2 - 1 = 2(1 - n_liquid)
-1/2 = 2(1 - n_liquid)
Dividing by 2:
-1/4 = 1 - n_liquid
n_liquid = 1 + 1/4
n_liquid = 1.25.
Final Answer:
The refractive index of the liquid is 1.25, so the correct option is (c) 1.25.
Sign Conventions for Plano-Convex and Plano-Concave Lenses:
Plano-Convex Lens:
The convex surface has a positive radius of curvature (R > 0) if measured from the lens to the center of curvature.
The plane surface has an infinite radius of curvature.
The focal length is positive (acts as a converging lens).
Plano-Concave Lens:
The concave surface has a negative radius of curvature (R < 0).
The plane surface has an infinite radius of curvature.
The focal length is negative (acts as a diverging lens).
These conventions follow the Cartesian sign convention:
Light is assumed to travel from left to right.
Distances measured in the direction of incident light are positive.
Distances measured opposite to the direction of light are negative.
Let me know if you need further clarification!
