Let us assume a system consist of two lenses with focal length f₁ and f₂ respectively. How do you find the focal length of the system experimentally? When
i) two lenses are touching each other
ii) they are separated by a distance 'd' with common principle axis. (AS3)

Let us take two convex lenses made up of the same material so that their refractive index (n) is same.

i) The first case given is two lenses are touching. So let us define the lens as lens A & lens B and their focal length be f₁ & f₂ respectively.

Now place a point object in front of the two lenses at a distance of u. Rays coming from the object after refraction from lens A forms an image I1 at a distance v1. Now this I1 will act as an object for lens B and after refraction from the lenses will form an image I at distance v.

Using lens formula:

Lens A: ….(1)

Lens B: 1/v-1/v1 = 1/f2 …(2)

Adding equation (1) & (2) we get,

Where F = effective focal length of lens

ii) Lenses are separated by a distance 'a' with common principle axis.

Let O be the point object placed at a distance u1. Rays from it after refraction from lens 1 alone will converge at I1 at a distance v1. But lens 2 will converge it before I1 creating the actual image I t distance v.

Object distance for lens 1 = u₁ = OC₁

where C₁ is center of curvature of lens 1

Image distance for lens 1 = v₁ = I₁C₁

Object distance for lens 2 = I₁C₂ = u₂

Image distance for lens 2 = IC₂ = v₂

where C₂ is center of curvature of lens 2

Now from the figure: I₁C₁ = I₁C₁ + a

using lens formula:

Lens 1: = ….(1)

Lens 2:

….(2)

…(3)

Solving [1],[2], & [3]