Question

derive the lens formula

Answer 1

lens formula can be derived by above method.

Answer 2

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,

$\frac{A′B′}{AB} = \frac{OB′}{OB}$ .... (1)

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

$\frac{A′B′}{OC} = \frac{FB′}{OF}$

But, OC=AB

Hence,

$\frac{A′B′}{AB} = \frac{FB′}{OF}$.... (2)

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

$\frac{OB′}{OB} =\frac{FB′}{OF} = \frac{OB′′−OF}{OF}$

Substituting the sign convention, we get

OB= -u, OB’= v and OF=f

$\frac{v}{−u} = \frac{v−f}{f}$

vf = −uv+uf or uv = uf−vf

Dividing both the sides by uvf, we get

$\frac{uv}{uvf} = \frac{uf}{uvf} − \frac{vf}{uvf}$

⇒$\large\boxed{\frac{1}{f}\: {=} \:\frac{1}{v}\: − \:\frac{1}{u}}$

The above equation is known as the Lens formula.

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