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?
Given:
Focal length of the convex lens, f1 = 30 cm
Focal length of the concave lens, f2 = −20 cm
Distance between the two lenses, d = 8.0 cm
When the parallel beam of light is incident on the convex lens first:
According to the lens formula, we have:
…(1)
Where,
u1 = Object distance
v1 = Image distance
f1 = Focal length of convex lens.
By putting the values in equation (1), we get,
By solving the above equation we get,
v1 = 30cm
This image will act as a virtual object for the concave lens.
Applying lens formula to the concave lens, we have:
…(2)
Where,
u2 = Object distance
u2 = image distance- distance between lenses
⇒ u2 = (30 − d) = 30 − 8 = 22 cm
v2 = image distance
f2 = focal length of concave lens
Now, putting the values in equation (2) we get,
⇒ 
Therefore v2 = −220cm
The parallel incident beam appears to diverge from a point that
is (220−d2) = (220−4) = 216cm from the centre of the combination of the two lenses.
When the parallel beam of light is incident, from the left, on the concave lens I)
According to the lens formula, we have:
…(3)
From equation (3) we get,
Where,
u2 = Object distance = −∞
v2 = Image distance
by putting the values in above equation we get,
Therefore v2 = −20cm
The image will act as a real object for the convex lens.
Applying lens formula to the convex lens, we have:
…(4)
Where,
u1 = Object distance
⇒ u1 = −(20 + d) = −(20 + 8) = −28 cm
v1 = Image distance
Therefore v2 = −420cm
Hence, the parallel incident beam appear to diverge from a point that is (420 − 4) = 416 cm from the left of the centre of the combination of the two lenses. The answer does depend on the side of the combination at which the parallel beam of light is incident. The notion of effective focal length does not seem to be useful for this combination.