Question

Two thin convex lenses L₁ and L₂ of focal lengths f₁ and f₂, respectively, are placed coaxially in contact. An object is placed at a point beyond the focus of lens L₁. Draw a ray diagram to show the image formation by the combination and hence derive the expression for the focal length of the combined system.

 Draw a ray diagram to show the image formation by a combination of two thin convex lenses in contact. Obtain the expression for the power of this combination in terms of the focal lengths of the lenses. 

Answer 1

Consider two thin lens L₁ and L₂ of focal length f₁ and f₂ held coaxially in contact with each other. Let P be the point where the optical centres of the lenses coincide (lenses being thin).

Let the object be placed at a point O beyond the focus of lens L₁ such that OP = u (object distance). Lens L₁ alone forms the image at I₁ where P I₁ = v₁ (image distance). The image I₁ would serve as a virtual object for lens L₂ which forms a final image I at distance PI = v. The ray diagram showing the image formation by the combination of these two thin convex lenses will be as shown below:

 

From the lens formula, for the image I₁ formed by the lens L₁, we have

`1/v_1-1/u=1/f_1 `

For the image formation by the second lens, L₂

`1/v-1/v_1=1/f_2 """...........2"`

Adding (1) and (2) we get:

`1/v_1-1/u+1/v-1/v_1=1/f_1+1/f_2`

`-1/u+1/v=1/f_1+1/f_2`

`1/v-1/u=1/f_1+1/f_2`

If the two lenses are considered a single lens of focal length f, which forms an image I at a distance v with an object distance being u, then we get

`1/v-1/u=1/f """ (where"1/f=1/f_1+1/f_2)`

`=>f=(f_1f_2)/(f_1+f_2)`

Hence, the focal length of the combined system is given by `f=(f_1f_2)/(f_1+f_2)`

In terms of power, equation can be written as

`P = 1/f_1 + 1/f_2   (as P = 1/f)`