Lens Maker’s Formula

Consider a thin lens whose optical centre is C and is made of material of refractive index μ. Let f be the focal length of the lens and R1 and R2 be the radii of curvature of its two surfaces. Lens Maker’s formula gives a relation between the focal length, radii of curvature and the refractive index of the lens.
​Then, the thin lens behaves as a prism of small angle. We know, the deviation (δ) produced by a prism of small angle (say A) is independent of the angle of incidence and is given by, \[δ=A(μ-1)\text{ __(1)}\] Let a ray of light OP parallel to the principal axis is incident on the lens at a height h above the principal axis. Then, the ray gets refracted and the refracted ray cuts the principal axis at its focus F.

Lens Maker's Formula Figure (1)

​If deviation produced is δ, then, \[tanδ=\frac{PC}{CF}\] \[∴tanδ=\frac{h}{f}\] Since the lens is thin, δ will be very small and hence, \[tanδ ≈ δ\] \[∴δ=\frac{h}{f}\text{ __(2)}\] From (1) and (2), \[\frac{h}{f}=A(μ-1)\] \[∴\frac{1}{f}=\frac{A}{h}(μ-1)\text{ __(3)}\] Let C1 and C2 be the centres of curvature of the two surfaces of the lens. Then, XC2 and YC1 are drawn which are normal to the lens at the point X and Y respectively.

Lens Maker's Formula Figure (2)

​In quad. TXPY, \[∠XTY+∠XPY=180 \text{ [∵∠PXT+∠PYT=180]}\] \[∴A+∠XPY=180\text{ __(4)}\] also, \[∠XPY+∠YPC_2=180\text{ __(5) [Being straight angle}]\] From (4) and (5), \[∠YPC_2=A\] Similarly, \[∠XPC_1=A\] Now, \[A=α+β\text{ __(6)}\] and, \[tanα=\frac{PC}{C_1C}=\frac{h}{R_1}\] \[tanβ=\frac{PC}{CC_2}=\frac{h}{R_2}\] Since the angle of α and β are very small, α ≈ tanα and β≈tanβ. \[∴α=\frac{h}{R_1} \text{ and, } β=\frac{h}{R_2}\] From (6), \[A=\frac{h}{R_1}+\frac{h}{R_2}\] \[\frac{A}{h}=\frac{1}{R_1}+\frac{1}{R_2}\] From (3) and (7), \[\frac{1}{f}=(μ-1)\left(\frac{1}{R_1}+\frac{1}{R_2}\right)\] This is the Lens Maker’s formula.

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