(a) Determine the 'effective focal length' of the combination of the two lenses in Exercise 9.10, if they are placed 8.0cm apart with their principal axes coincident. Does the answer depend on which side of the combination a beam of parallel light is incident? Is the notion of effective focal length of this system useful at all? (b) An object 1.5 cm in size is placed on the side of the convex lens in the arrangement (a) above. The distance between the object and the convex lens is 40cm. Determine the magnification produced by the two-lens system, and the size of the image.
The effective focal length is -300 cm (or -3.0 m). The effective focal length is a single value, so it doesn't depend on which side the parallel beam is incident. However, the front focal length (-420 cm) and back focal length (-220 cm) are different, which means the position where parallel rays converge (or appear to diverge from) depends on which side they enter. The notion of effective focal length is not very useful for this asymmetric system because the front and back focal lengths are different, so the system behaves differently depending on which side light enters.
The total magnification is 15/23 0.652. The image size is 45/46 cm 0.98 cm.
Explanation
For part (a), the effective focal length is calculated using the formula 1/f_eff = 1/f₁ + 1/f₂ - d/(f₁ × f₂), where f₁ = 30 cm (convex lens), f₂ = -20 cm (concave lens), and d = 8 cm. The front and back focal lengths are calculated using f_F = f_eff × (1 - d/f₂) and f_B = f_eff × (1 - d/f₁). For part (b), the magnification is calculated by finding the image formed by each lens sequentially.