Two identical symmetric double convex lenses of focal length f are cut into two equal parts L₁, L₂ by AB plane and L₃, L₄ by XY plane as shown in figure respectively. The ratio of focal lengths of lenses L₁ and L₃ is

Solution

Physics sir · 4 steps

Ask what each cut does to the curved surfaces, not to the size of the lens.

1. Lens maker formula. For a symmetric double convex lens both faces have radius R, so the focal length depends only on these two curved surfaces.

   1/f = (μ - 1)(1/R + 1/R) = 2(μ - 1)/R

2. Cut along AB. AB is the principal axis, so it splits the lens into an upper half and a lower half. Each half still has both curved faces of radius R, so it has the same focal length: f₁ = f. Only the aperture is smaller, which reduces the brightness of the image.
3. Cut along XY. XY is the plane of the lens, so each piece is plano-convex: one face of radius R and one flat face (R = ∞). Then 1/f₃ = (μ - 1)/R = 1/2f, so f₃ = 2f.
4. So. f₁ : f₃ = f : 2f = 1 : 2, which is option (a).

The idea to keep. Cutting a lens along its axis keeps the focal length and only dims the image. Cutting it along its plane removes one curved face, so each half has twice the focal length.

Answer from the NTA final answer key. Solution by PrepWiser.