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| Lens Maker's Formula |
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| It is a relation that connects focal length of a lens to radii of curvature of the two surfaces of the lens and refractive index of the material of the lens. |
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| The following assumptions are made for the derivation: |
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 The lens is thin, so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens. |
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The aperture of the lens is small. |
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Point object is considered. |
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Incident and refracted rays make small angles. |
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| Consider a convex lens (or concave lens) of absolute refractive index m2 to be placed in a rarer medium of absolute refractive index m1. |
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| Considering the refraction of a point object on the surface XP1Y, the image is formed at I1 who is at a distance of V1. |
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| CI1= P1I1 = V1 (as the lens is thin) |
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| CC1 = P1C1 = R1 |
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| CO = P1O = u |
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| It follows from the refraction due to convex spherical surface XP1Y |
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| The refracted ray from A suffers a second refraction on the surface XP2Y and emerges along BI. Therefore I is the final real image of O. |
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| Here the object distance is |
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| (Note P1P2 is very small) |
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| (Final image distance) |
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| Let R2 be radius of curvature of second surface of the lens. |
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| \ It follows from refraction due to concave spherical surface from denser to rarer medium that |
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| Adding (1) & (2) |
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| Note: The lens maker's formula can be derived for a concave lens in the same way. The ray diagram is as follows: |
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| Note: The lens maker's formula indicates that a convex lens can behave like a diverging one if m1 > m2 i.e., if the lens is placed in a medium whose m is greater than the m of lens. Similarly a concave lens can be made convergent. |
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