Derive an expression for the focal length of the combination of two thin lenses when they are SEPERATED  by a small distance?

Consider two lenses L1 and L2 separated by a small distance 'd' apart, as shown below. A ray of light AB initially parallel to the principal axis hits the lens L1 and deviates and then hits lens L2.

here

f1 is the focal length of L1 and

f2 is the focal length of L2

and

δ1 is the deviation produced by L1

δ2 is the deviation produced by L2

so,

form simple geometry

δ1 = h1 / F1

and

δ2 = h2 / F2

now, 

total deviation of the ligth ray will be

δ = δ1  +δ2

or

δ =  (h1 / F1) +  (h2 / F2)

also

at lens 2 and triangle BCD

h2 = h1 - CD = h1 - BD.tanδ1

or

h2 = h1 - d.tanδ1

now as δ1 is very small, tanδ1 ~ δ1

thus,

h2 = h1 - d.δ1

so, from earlier relation

h2 = h1 - d.(h1/f1)

thus,

δ = δ1  +δ2 = (h1/f1) +  [(h1 - d.(h1/f1)) / f2]

or

δ =  (h1/f1) + (h1/f2) - (dh1 / f1.f2)

now,

for the combination of the two lenses let F be the combined focal length.

So, the total deviation will be given as 

δ =  h1 / F

or

 (h1/f1) + (h1/f2) - (d.h1 / f1.f2) = h1 / F

thus,

1/F = 1/f1 + 1/f2 - (d / f1.f2)

so, the combined focal length will be

F = f1.f2 / (f1 + f2 - d)

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Combination of Thin Lenses in Contact

Consider two lenses A and B of focal length f1and f2placed in contact with each other. An object is placed at a point O beyond the focus of the first lens A. The first lens produces an image at I1(real image), which serves as a virtual object for the second lens B, producing the final image atI.

Since the lenses are thin, we assume the optical centres (P) of the lenses to be co-incident.

For the image formed by the first lens A, we obtain

For the image formed by the second lens B, we obtain

Adding equations (i) and (ii), we obtain

If the two lens system is regarded as equivalent to a single lens of focal length f, we have

From equations (iii) and (iv), we obtain

For several thin lenses of focal length f1f2f3, …, the effective focal length

In terms of power, equation (vi) can be written as

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Is there one ?Even i will b looking forward dor an answer from someone.

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