Question

2) Determine the 'effective focal length of the combination of two lenses of focal lengths 30 cm
and –20 cm, if they are placed 8.0 cm apart with their principal axes coincident. Does the answer
depend on which side a beam parallel light is incident? Is the notion of effective focal length of
this system useful at all?
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Answer 1

Given info : focal length of 1st lens, f₁ = +30 cm

focal length of 2nd lens, f₂ = -20 cm

distance between lens, d = 8cm

Ask : find effective focal length.

Does the answer depend on which side a beam parallel light is incident ? Is the notion of effective focal length of this system useful at all ?

solution : case 1 : let a parallel beam of light incident on convex lens (1st lens).assume 2nd lens is absent.

now, object distance, u₁ = ∞ [ light comes from infinite are parallel so u₁ = ∞ ]

now, using formula, 1/v - 1/u = 1/f

1/v₁ - 1/∞ = 1/30

⇒v₁ = 30 cm , so the image would act as virtual object for the 2nd lens.

object distance for 2nd lens, u₂ = +(30 - 8) = +22 cm

using, 1/v - 1/u = 1/f, again,

1/v₂ - 1/+22 = 1/-20

1/v₂ = 1/22 - 1/20 = -2/440 = -1/220

⇒v₂ = -220 cm

Therefore the parallel beam would appear to diverge 220 - 4 = 216cm from the centre of two lens.

case 2 : now let's a parallel beam of light incident on concave lens at first (2nd lens).

object distance, u₂ = -∞

[ definitely we have to choose negative as sign because we have chosen object distance of 1st lens is +∞ ]

now, 1/v₂ - 1/-∞ = 1/-20

⇒v₂ = -20cm so the image would act as real for the 1st lens,

so object distance, u₁ = -(20 + 8) = -28 cm

now, 1/v₁ - 1/-28 = 1/30

⇒1/v₁ = 1/30 - 1/28 = -1/420

⇒v₁ = -420 cm

Therefore the parallel beam of light would appear to diverge 420 - 4 = 416 cm from the centre of lens.

from two cases, it can be clearly observed that the answer depends on the side of the system of lens where the parallel beam is incident. so the notion of effective focal length of this system never seems to be useful at all.