Question

Derive the lens formula with the ray diagram :-

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Answer 1

Answer:

Consider a convex lens with an optical center O. Let F be the principle focus and f be the focal length. An object AB is held perpendicular to the principal axis at a distance beyond the focal length of the lens. A real, inverted magnified image A’B’ is formed as shown in the figure.

From the given figure, we notice that △ABO and △A’B’O are similar.

Therefore,

A′B′AB=OB′OB (1)

Similarly, △A’B’F and △OCF are similar, hence

A′B′OC=FB′OF

But, OC=AB

Hence,

A′B′AB=FB′OF (2)

Equating eq (1) and (2), we get

OB′OB=FB′OF=OB′−OFOF

Substituting the sign convention, we get

OB=-u, OB’=v and OF=f

v−u=v−ff

vf=−uv+uf or uv=uf−vf

Dividing both the sides by uvf, we get

uv/uvf=uf/uvf−vf/uvf

⇒1f=1v−1u

The above equation is known as the Lens formula.