Question

A convex lens of focal length \( f \) is cut into two equal parts perpendicular to the principal axis. The focal length of each part will be:

• A. \(f\)
• B. \(2f\)
• C. \(\frac f2\)
• D. \(\frac f4\)

Hint:

When a lens is cut along the principal axis, the focal length of each part is halved.

Answer 1

The Correct Option is C

To determine the new focal length when a convex lens is divided into two equal parts, one must analyze the impact of this division on the lens's optical characteristics. The focal length (\( f \)) of a lens is defined by the lens maker's formula:

\( \frac{1}{f} = (\mu-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \)

Here, \( \mu \) represents the refractive index, and \( R_1, R_2 \) are the radii of curvature of the lens surfaces. When a symmetrical and thin convex lens is physically bisected, its curvature and refractive index remain constant. However, the aperture area is reduced.

Each resulting lens piece retains its curvature properties, but its diameter, or aperture, is halved. This reduction in aperture affects the lens's ability to converge light rays. A larger aperture facilitates better convergence by minimizing diffraction effects at the edges. Dividing the lens decreases the aperture, thereby influencing the effective focal length of each segment.

When a lens is cut along its principal axis, each resulting piece possesses half the aperture area. This modification enhances the converging power, leading to a new focal length calculated as:

\( \text{New focal length} = \frac{f}{2} \)

This relationship arises because each partitioned section effectively functions as a smaller, complete lens with half the original aperture, focusing light to the same point but requiring a numerical adjustment based on geometric and optical considerations.

Consequently, the focal length of each half of the original lens is found to be \(\frac{f}{2}\).