Question

Derive lens formula for the $or\frac{OB\text{'}}{OB}=\frac{OB\text{'}-OF}{OF}(becauseFB\text{'}=OB\text{'}-OF$convex lens.

Answer 1

Step1: Lens formula.

The relationship between the distance of an image (v), the distance of an object (u), and the focal length (f) of the lens is given by the formula known as the Lens formula. The lens formula is applicable for convex as well as concave lenses. The formula is as given below

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

Step 2: Derivation of lens formula.

Consider a convex lens with an optical centre 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,

$\frac{A\text{'}B\text{'}}{AB}=\frac{OB\text{'}}{OB}$ ……..(1)

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

$\frac{A\text{'}B\text{'}}{OC}=\frac{FB\text{'}}{OF}$

But,

$OC=AB$

Hence,

$\frac{A\text{'}B\text{'}}{AB}=\frac{FB\text{'}}{OF}$ …………(2)

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

$\frac{OB\text{'}}{OB}=\frac{FB\text{'}}{OF}$

$or\frac{OB\text{'}}{OB}=\frac{OB\text{'}-OF}{OF}(becauseFB\text{'}=OB\text{'}-OF$

Using the sign convention, we get

$OB=-u,OB’=vandOF=f$

Therefore,

$$\begin{gathered} \frac{v}{-u}=\frac{v-f}{f} \\ \Rightarrow vf=-uv+uf \end{gathered}$$

Dividing both the sides by uvf, we get

$$\begin{gathered} \frac{vf}{uvf}=\frac{-uv}{uvf}+\frac{uf}{uvf} \\ \Rightarrow \frac{1}{u}=\frac{-1}{f}+\frac{1}{v} \\ \Rightarrow \frac{1}{v}-\frac{1}{u}=\frac{1}{f} \end{gathered}$$

Which is the required formula.