Given Data:
1. A plano-convex lens is silvered on different sides and behaves like a concave mirror.
2. When silvered on the plane side, it behaves like a concave mirror with focal length f1=60f_1 = 60 cm.
3. When silvered on the convex side, it behaves like a concave mirror with focal length f2=20f_2 = 20 cm.
4. We need to determine the refractive index μ\mu of the lens.
Concept Used:
When a lens is silvered on one side, it behaves as a combination of a mirror and a lens. The effective focal length FF of such a system is given by:
1F=1fL+2fM\frac{1}{F} = \frac{1}{f_L} + \frac{2}{f_M}
where:
• fLf_L is the focal length of the lens,
• fMf_M is the focal length of the mirror.
Step 1: Case 1 - Silvered on the Plane Side
• The plane side acts like a mirror with infinite radius of curvature (fM=∞f_M = \infty).
• The effective focal length is given as F=60F = 60 cm.
Using the formula:
1F=1fL+2fM\frac{1}{F} = \frac{1}{f_L} + \frac{2}{f_M}
Since the plane side has an infinite radius of curvature, fM=∞f_M = \infty, so:
160=1fL+0\frac{1}{60} = \frac{1}{f_L} + 0 fL=60 cmf_L = 60 \text{ cm}
Step 2: Case 2 - Silvered on the Convex Side
• The convex side behaves like a concave mirror with focal length fM=R/2f_M = R/2.
• The effective focal length is given as F=20F = 20 cm.
Using the same formula:
120=1fL+2fM\frac{1}{20} = \frac{1}{f_L} + \frac{2}{f_M}
Substituting fL=60f_L = 60:
120=160+2fM\frac{1}{20} = \frac{1}{60} + \frac{2}{f_M} 120−160=2fM\frac{1}{20} - \frac{1}{60} = \frac{2}{f_M} 3−160=2fM\frac{3 - 1}{60} = \frac{2}{f_M} 260=2fM\frac{2}{60} = \frac{2}{f_M} fM=60 cmf_M = 60 \text{ cm}
Since fM=R/2f_M = R/2, we get:
R=2×60=120 cmR = 2 \times 60 = 120 \text{ cm}
Step 3: Using Lens Maker’s Formula
For a plano-convex lens, the focal length is given by:
1fL=(μ−1)1R\frac{1}{f_L} = (\mu - 1) \frac{1}{R}
Substituting fL=60f_L = 60 and R=120R = 120:
160=(μ−1)1120\frac{1}{60} = (\mu - 1) \frac{1}{120} μ−1=12060=2\mu - 1 = \frac{120}{60} = 2 μ=3.0\mu = 3.0
Final Answer:
The refractive index of the lens is 3.0\boxed{3.0}.
Thus, the correct option is A) 3.0.
