Two systems are formed from a converging lens and a diverging lens, as shown in

Question

Two systems are formed from a converging lens and a diverging lens, as shown in parts a and b of the drawing. (The point labeled "Fconverging" is the focal point of the converging lens.) An object is placed inside the focal point of lens 1 at a distance of 5.70 cm to the left of lens 1. The focal lengths of the converging and diverging lenses are 15.0 and -20.0 cm respectively. The distance between the lenses is 50.0 cm. Determine the final image distance for each system, measured with respect to lens 2.

Explanation / Answer

What are the two systems?
I assume that 1) the converging lens is in front, and 2) the diverging lens is in front.

let s2' = distance of image from second lens
s1 = distance of object from first lens (to the left, if the first lens is on the left side of the second lens)
d = distance between lenses
f1 = focus length of first lens
f2 = focus length of second lens

for the distance of the image from the second lens use
1/s2' = 1/f2 - 1/(d - f1s1/(s1 - s1))

1) converging lens is in front:
1/s2' = 1/(- 0.2) - 1/(0.5 - 0.15*0.0570/(0.0570 - 1.5))
--> s2' = - 2.0700 m

The image is 207 cm to the left of the second lens (in front of the second lens)

2) diverging lens is in front:
1/s2' = 1/0.15 - 1/(0.5 - (-0.2*0.0570/(0.0570 - (-0.2)))
---> s2' = 6.710 m
The image is 6.710 m =671.0 cm behind the second lens

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