Two thin convex lenses L_{1} and L_{2} of focal lengths f_{1} and f_{2}, respectively, are placed coaxially in contact. An object is placed at a point beyond the focus of lens L_{1}. 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.

#### Solution

Consider two thin lens L_{1} and L_{2} of focal length f_{1} and f_{2} 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 L_{1} such that OP = u (object distance). Lens L_{1} alone forms the image at I_{1} where P I_{1} = v_{1} (image distance). The image I_{1} would serve as a virtual object for lens L_{2} 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 I_{1} formed by the lens L_{1}, we have

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

For the image formation by the second lens, L_{2}

`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)`