A symmetric thin biconvex lens is cut into four equal parts by two planes AB and CD as shown in the figure. If the power of the original lens is 4D, then the power of a part of the divided lens is:
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When a lens is divided symmetrically, the power of each part is halved. This is because the focal length increases as the area decreases, and power is inversely proportional to focal length.
When a biconvex lens is divided symmetrically, the power of each part is proportional to its area. Since the lens is cut into four equal parts, the area of each part is one-fourth of the original lens. The power of the lens is inversely proportional to its focal length, and thus, the power of each part is the same as the original lens, but the focal length is reduced.
Since the focal length of each part is doubled, the power of each part is half of the original power:
\[
P_{\text{part}} = \frac{P_{\text{original}}}{2} = \frac{4D}{2} = 2D.
\]
Thus, the power of each part is \( \boxed{2D} \).
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Approach Solution -2
Step 1: Formula for the power of a lens. The power \( P \) of a lens is related to its focal length \( f \) by: \[ P = \frac{1}{f} \quad (\text{in meters, with } f \text{ in meters}). \]
Step 2: Effect of cutting a lens. When a lens is cut along its principal axis (i.e., through its optical center), its curvature remains unchanged. Thus, its focal length \( f \) and power \( P \) remain the same.
Step 3: Case 1 — Cut along the principal axis (plane AB). Each half lens has the same focal length, so: \[ P_{\text{each}} = P_{\text{original}} = 4\,\text{D}. \] However, each half transmits less light, so the intensity changes, but the focal length does not.
Step 4: Case 2 — Cut perpendicular to the principal axis (plane CD). Now, the aperture (area) is halved again, but the focal length remains unchanged. Hence, power is still the same.
Step 5: Combining both cuts. The lens is divided into four equal parts — each smaller part behaves as a separate lens with same focal length as the original but a smaller aperture.
Therefore, each of the four parts has the same focal length \( f \), and the same power \( P \).
But if each part is used separately, the effective aperture reduces, and hence the amount of bending (effective power) is halved in one direction (due to half height or half width of curvature). So effectively: \[ P_{\text{part}} = \frac{P_{\text{original}}}{2} = \frac{4}{2} = 2\,\text{D}. \]
