Question

One surface of a lens is convex and the other is concave. If the radii of curvatures are \(R\) and \(r\) respectively, the lens will be convex if

• A. \(R>r\)
• B. \(R<r\)
• C. \(R = \frac{1}{r}\)
• D. \(R = r\)

Hint:

For a meniscus lens, the lens behaves as convex when the convex surface has smaller radius of curvature than the concave surface, because smaller radius means greater curvature.

Answer 1

The Correct Option is B

Step 1: Understand the nature of the lens.
The lens has one convex surface and one concave surface. Such a lens is called a meniscus lens.
A meniscus lens can behave either as a convex lens or as a concave lens depending on which surface has greater curvature.

Step 2: Curvature and radius relation.
Curvature of a spherical surface is inversely proportional to its radius of curvature.
\[ \text{Curvature} \propto \frac{1}{\text{Radius of curvature}} \]
So, smaller radius means greater curvature.

Step 3: Condition for lens to behave as convex lens.
For the lens to behave as a convex lens, the converging effect of the convex surface must be greater than the diverging effect of the concave surface.
This means the convex surface must have greater curvature than the concave surface.

Step 4: Compare the radii of curvature.
Radius of curvature of convex surface is \(R\), and radius of curvature of concave surface is \(r\).
For convex surface to have greater curvature:
\[ \frac{1}{R}>\frac{1}{r} \]

Step 5: Simplify the inequality.
Since \(R\) and \(r\) are positive magnitudes of radii of curvature, we can compare them directly.
\[ \frac{1}{R}>\frac{1}{r} \] \[ R<r \]

Step 6: Physical interpretation.
If \(R<r\), the convex surface is more strongly curved than the concave surface. Therefore, the net effect of the lens is converging.
Hence, the lens behaves as a convex lens.

Step 7: Final conclusion.
\[ \boxed{R<r} \] Hence, correct answer is option (B).