Analyzing the Setup
The Film's Deception
Imagine you are looking at a long glass cylinder with a beautifully curved, convex end. To make things interesting, a thin transparent film is coated right over this curved surface. Our goal is to find where parallel rays of light will focus when they travel from the air into the glass, and conversely, when they travel from deep inside the glass back out into the air.
Before we dive into the heavy math, let's address the elephant in the room: the thin film. Does it actually do anything?
Since the film has a uniform thickness, both its outer and inner surfaces share the exact same radius of curvature, R. If we apply the lens maker's formula to this film:
ffilm1=(n1−1)(R1−R1)=0
The focal length ffilm turns out to be infinite! This means the film acts exactly like a flat, parallel glass slab. It might slightly shift the rays laterally, but it has absolutely zero converging or diverging power. We can safely ignore it for our focal length calculations.
Case 1
Air to Glass (The Convex Journey)
Now, let's track the rays traveling from the air (n0=1) into the glass cylinder (n2=1.5). The rays are parallel to the principal axis, which means our object is effectively at infinity (u=−∞).
When light refracts at a single spherical surface, the master equation governing its path is:
Here, the light hits a convex surface. According to our sign convention, the center of curvature lies in the direction of the incident light, making the radius positive (+R). Let's substitute our known values to find the image distance, which will be our first focal length, f1:
Since any number divided by infinity is zero, the equation simplifies beautifully:
Solving for f1, we get:
So, the magnitude of the first focal length is ∣f1∣=3R.
Case 2
Glass to Air (The Concave Return)
What happens if we reverse the scenario? Imagine rays traveling from deep inside the glass cylinder, parallel to the axis, heading towards the air.
We use the exact same master equation, but the roles have changed. The light originates in the glass (n2=1.5) and enters the air (n0=1). More importantly, from the perspective of these outgoing rays, the boundary surface is now concave. This means the center of curvature lies against the direction of the incident light, making the radius negative (−R).
Let's plug these new parameters into our equation to find the second focal length, f2:
Again, the infinity term vanishes, leaving us with:
Solving for f2, we find:
Thus, the magnitude of the second focal length is ∣f2∣=2R.
The Grand Conclusion
We've successfully navigated both scenarios! We found that ∣f1∣=3R and ∣f2∣=2R. This perfectly matches options (a) and (c).
This problem is a fantastic reminder that for a single refracting surface, the focal length is not a single fixed number—it fundamentally depends on the direction the light is traveling!
