Lens formula: An equation which relates the object distance, image distance and focal length of a lens is called the lens formula. It is given as
1v−1u=1f
The figure shows the formation of a real, inverted and diminished image AB of the object AB placed beyond the centre of curvature at a distance u from the convex lens.
Let v be the image distance.
According to Cartesian sign convention,
Object distance (OB)=−u
Image distance (OB′)=+v
Focal length (OF1=OF2)=+f
From the geometry of the figure above, right angled △ABO and △A′B′O are similar.
∴ABA′B′=OBOB′=−uv....(1)
Similarly, from the geometry of the figure, right angled △ODF2 and △B′A′F2 are similar.
∴ODA′B′=OF2F2B′
∴ABA′B′=OF2F2B′(∵OD=AB)
∴ABA′B′=OF2OB′−OF2
∴ABA′B′=fv−f....(2)
From (1) and (2),
−uv=fv−f
⇒−u(v−f)=vf
⇒−uv+uf=vf
Dividing each term by uvf,
−uvuvf+ufuvf=vfuvf
∴−1f+1v=1u
∴1v−1v=1f
This equation is the 'Lens formula'.