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Question

Derive Lens Maker's formula for a convex lens.

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Solution




The formula for refraction of light from the spherical surface is


n2vn1u=n2n1R where OB=u,OI=+v,BC=+R

n2 is refractive index of medium 2 i.e. spherical glass
and n1 is refractive index of medium 1.

Figure (a) shows the geometry of image formation by a double convex lens. the image formation can be seen in two steps, the first refracting surface forms an image I1 which acts as a virtual object for the second surface. applying the equation above written for surface 1

n1OB+n2BI1=n2n1BC2

similarly, form the second interface

n2DI1+n1DI=n2n1DC2 (refractive index of right side medium is n1 and for left side medium is n2, and also DI1is negative.)

now for thin lens BI1=DI1. adding both the equations we get
n1OB+n1DI=(n2n1)(1BC1+1DC2)

and assuming the object at infinity means OB will be very large
we get n1DI=(n2n1)(1BC1+1DC2)

and dividing both sides with n1
1DI=(n211)(1BC1+1DC2)

replacing DI=f,BC1=+R1,DC2=R2

Finally, we get 1f=(n211)(1R1+1R2).

This equation is known as lens makers formula.

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