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Step-by-Step Solution
Step 1: Identify the Known Quantities
• Diameter of the plano-convex lens, D = 6 cm ⇒ Radius of the lens’ surface, D/2 = 3 cm
• Thickness at the center, t = 3 mm = 0.3 cm
• Speed of light in the lens material, v = 2 × 108 m/s
• Speed of light in vacuum, c = 3 × 108 m/s
Step 2: Determine the Refractive Index of the Lens Material
The refractive index n is given by the ratio of the speed of light in vacuum to that in the material:
$ n = \frac{c}{v} = \frac{3 \times 10^8}{2 \times 10^8} = 1.5 $
Step 3: Relate Thickness to the Radius of Curvature
For a plano-convex lens, one surface is flat (R = ∞), and the other surface has radius of curvature R. The center thickness t (also called the sag) is related to the radius of curvature R and the lens’ radius D/2 by:
$ t = R - \sqrt{R^2 - \left(\frac{D}{2}\right)^2} $
Substituting t = 0.3 cm and D/2 = 3 cm:
$ 0.3 = R - \sqrt{R^2 - 3^2} $
Let x = $ \sqrt{R^2 - 9} $. Then:
$ 0.3 = R - x \quad \Rightarrow \quad x = R - 0.3. $
Also,
$ x^2 = R^2 - 9 \quad \Rightarrow \quad (R - 0.3)^2 = R^2 - 9. $
Expanding and simplifying:
$ R^2 - 0.6R + 0.09 = R^2 - 9 \quad \Rightarrow \quad -0.6R + 0.09 = -9 \quad \Rightarrow \quad -0.6R = -9.09 \quad \Rightarrow \quad R = \frac{9.09}{0.6} \approx 15.15\text{ cm}. $
Step 4: Use the Lens Formula for a Plano-Convex Lens
For a plano-convex lens with radius of curvature R for the curved surface and the other surface being flat (R = ∞), the focal length f is given by:
$ f = \frac{R}{(n - 1)}. $
Substituting R = 15.15 cm and n = 1.5:
$ f = \frac{15.15}{(1.5 - 1)} = \frac{15.15}{0.5} \approx 30.3\text{ cm}. $
Rounded appropriately, this is about 30 cm.
Step 5: Conclude the Focal Length
Hence, the focal length of the given plano-convex lens is approximately 30 cm.
