Formula for a thin lens
The formula for a thin lens can be shown to be
Lens formula:
1/object distance + 1/image distance = 1/focal length
1/u + 1/v = 1/f
This applies to all types of lens as long as the correct sign convention is used when
substituting values for the distances.
(Reminder: we use the ‘real is positive, virtual is negative’ sign convention.)
Two proofs of the formula will be given here, one a geometrical proof and the other an
optical version.
(a) Geometrical proof of the lens formula
Consider a plano-convex lens, as shown in Figure 1.
d
d
h
d
f
Figure 1
F
O

h
u

I
v
Figure 2
If we consider the action of the lens to be like that of a small-angle prism, then all rays have
the same deviation. Therefore, in Figure 2,
Deviation (d) =  + and so for small angles tan d = tan  + tan 
Therefore: h/f = h/u + h/v and so 1/f = 1/u + 1/v and the formula is proved.
Example
An object is placed in front of a converging lens and gives a real image with magnification 5; when
the object is moved 6 cm along the axis of the lens a real image of magnification 2 is obtained.
What is the focal length of the lens?
Let the initial object and image distances be u and v respectively.
Therefore v/u = 5, v = 5u, and v’/u + 6 = 2 where v' is the new image distance. Also
1/u + 1/5u = 1/f and 1/[u+6] + 1/[2u + 12]
These equations give:
6f = 5u and 3f = 2u + 12, and so f = 20 cm.
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(b) Optical proof of the lens formula
We will only consider the case for a biconvex lens here. (see Figure 3).
R1
n1
O
R2
n2
I'
n1
I
v
u
v'
Figure 3
Consider the two spherical surfaces of the lens. For the first surface we have
n2/v' + n1/u = [n2 – n1]/R1
For the second surface we have
n2/-v' + n1/v = [n2 – n1]/R2
(note the negative sign denoting a virtual object for the second surface).
Combining these two equations gives:
n1/u + n1/v + = [n2 – n1][1/R1 + 1/R2 ] = n1/f
If n1 = 1 (i.e. the lens is in air) the formula becomes:
1/u + 1/v = 1/f
This formula could be used to calculate the refractive index (n2) of the glass of the lens.
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