Combination of Lenses

Two thin convex lenses A and B having focal lengths f₁ and f₂, respectively have been kept in contact with each other. O is a point object placed on the common principal axis of the lenses.

Lens A forms the image of object O at I₁. This image serves as the virtual object for lens B and the final image is thus formed at I.

If u be the object distance and v₁ the image distance for the lens A, then using the lens formula,

1/v₁ – 1/u = 1/f₁

If v is the final image distance for the lens B,

1/v – 1/v₁ = 1/f₂

Adding these two equations,

1/v – 1/u = 1/f₁ + 1/f₂

If the combination of lenses is replaced by a single lens of focal length F such that it forms the image of O at the same position I, then this lens is said to be equivalent to both the lenses. It is also called the equivalent lens for the combination. For the equivalent lens,

1/v – 1/u = 1/F

1/F = 1/f₁ + 1/f₂

If P is power of the equivalent lens and P₁ and P₂ are respectively the powers of individual lenses, then

P = P₁ + P₂