Consider the situation shown in figure. ADBE is a thin lens. An object 0 is placed on its principal axis. The two spherical surfaces of the lens have their centres at C
1
and C
2
. The optical centre is at P and the principal axis cuts the two spherical surfaces at D and E.
As shown in the figure, after the first refraction image is formed at O
1
and after second refraction the final image is formed at I.
As
the lens is thin, the points D,PandE are all close to each other and we may take the origin at P for both these refractions.
The general equation for refraction at a spherical surface is
v
μ
2
−
u
μ
1
=
R
μ
2
−μ
1
..........(i)
For the first refraction, the object is at O, the
image is at O
1
and the centre of curvature is at C
1
. If
u, v, and R, denote their x-coordinates,
v
1
μ
2
−
u
μ
1
=
R
1
μ
2
−μ
1
.......(ii)
For the second refraction at AEB, the incident rays GH and DE diverge from O
1
. Thus, O
1
is the object for
this refraction and its x-coordinate is v
1
. The image is
formed at I and the centre of curvature is at C
2
. Their
x-coordinates are v and R respectively. The light goes
from the medium μ
2
to medium μ
1
.
Hence
v
μ
1
−
v
2
μ
2
=
R
2
μ
1
−μ
2
.......(iii)
Adding (ii) and (iii),
v
1
−
u
1
=(
μ
1
μ
2
−1)×(
R
1
1
−
R
2
1
)
If the object O is taken far away from the lens, the image is formed close to the focus. Thus, for u= ∞ , v=f
Hence,
f
1
=(
μ
1
μ
2
−1)×(
R
1
1
−
R
2
1
)
This is lens makers formula.
