To determine the new focal length when a convex lens is cut into two equal parts, we need to consider the effect of dividing the lens on its optical properties. The focal length \( f \) of a lens is given by the lens maker's formula:
where \( \mu \) is the refractive index and \( R_1, R_2 \) are the radii of curvature of the lens surfaces. However, when we physically cut a convex lens (which is initially symmetrical and thin) into two equal halves, the curvature and refractive index remain unchanged, but the aperture area is reduced.
For the new lens piece, the lens maintains its curvature properties but the diameter (or the aperture of the lens) is halved. This effectively changes the lens's ability to converge rays. A larger aperture allows better convergence due to less diffraction at the edges. Dividing the lens reduces the aperture, influencing the effective focal length of each piece.
For a lens cut along the principal axis, you essentially have a lens with half the aperture area, which increases the converging power given by:
\( \text{New focal length} = \frac{f}{2} \)
This is because the portioned lens should act equivalently to the original smaller, complete lens with half the aperture area: focusing light to the same point but needing adjustment numerically due to geometry and optics context.
Therefore, the focal length of each half of the original lens is \(\frac{f}{2}\).
