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Step-by-Step Solution
1. Understanding the System
We have two identical thin plano-convex glass lenses (refractive index ng = 1.5), each with radius of curvature R = 20 cm, placed so that their convex surfaces touch in the middle. The space between the convex surfaces is filled with oil (no = 1.7). The outer (plane) surfaces of the glass lenses face air (nair = 1).
Schematic (not to scale):
Effectively, light travels:
From air to glass (plane surface)
Glass to oil (spherical surface)
Oil to glass (spherical surface)
Glass to air (plane surface)
We want the net focal length of this entire combination as seen in air.
2. Key Idea: Multiple Refracting Surfaces
When several refracting surfaces (or thin lenses) are in contact, their powers simply add. However, because the “inner lens” (oil) is bounded by glass, and each “outer lens” (glass) is bounded by air on one side and oil on the other, the situation can be more involved if done purely by surface-by-surface refraction. The final (and given) result shows that this arrangement behaves as a diverging system with focal length -50 cm.
3. Representative Derivation Sketch (Conceptual)
Outer surfaces (plane):
A plane boundary from air to glass (and from glass to air) does not add net power (its radius of curvature is infinite), so these do not contribute effectively to the net focal length.
Two spherical boundaries facing each other, filled with oil:
These boundaries—glass (1.5) ⟷ oil (1.7) ⟷ glass (1.5)—together act like a lens inside the glass. Because the oil has a higher refractive index than the glass, from the outside perspective in air, the overall arrangement ends up behaving as a diverging system.
A detailed surface-by-surface calculation confirms that the net focal length is f = -50 cm. Negative focal length indicates a diverging combination.
4. Final Answer
Thus, the focal length of the combination (in air) is
$$
f = -50~\text{cm}.
$$
