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Question
Arrive at lens equation from lens maker’s formula.
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Solution
From refraction through a double convex lens, the relation between the object distance u, image distance v1 and radius of curvature R1 as
`mu_2/"v"_1 - mu_1/"u" = (mu_2 - mu_1)/"R"_1` ....(1)
The relation between the object distance image distance v1 and radius of curvature R2 can be
`mu_1/"v" - mu_2/"v"_1 = (mu_1 - mu_2)/"R"_2` ....(2)
Adding equation (1) and (2)
`mu_1/"v" - mu_1/"u" = (mu_2 - mu_1) [1/"R"_1 - 1/"R"_2]`
`1/"v" - 1/"u" = ((mu_2 - mu_1)/mu_1) [1/"R"_1 - 1/"R"_2]`
If the object is placed at infinity (µ = ∞), the image will be formed at the focus. i.e.v = ƒ
`1/"f" = ((mu_2 - mu_1)/mu_1) [1/"R"_1 - 1/"R"_2]`
This is len’s maker’s formula. When the lens is placed in air µ1 = 1 and µ2 = µ.
Equation (4) becomes,
`1/"f" = (mu - 1) [1/"R"_1 - 1/"R"_2]`
From equation (3) and (4), we have `1/"v" - 1/"u" = 1/"f"`
This is the len’s equation.
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