Question
Using lens maker's formula obtain the equation of lens.
✓
Answer
→For refraction at two spherical surfaces of lens,
$\begin{aligned}& \frac{n_1}{ OB }+\frac{n_1}{ DI }=\left(n_2-n_1\right)\left(\frac{1}{ BC _1}+\frac{1}{ DC _2}\right) \\\therefore \quad & n_1\left(\frac{1}{ OB }+\frac{1}{ DI }\right)=\left(n_2-n_1\right)\left(\frac{1}{ BC _1}+\frac{1}{ DC _2}\right) \\\therefore \quad & \frac{1}{ OB }+\frac{1}{ DI }=\left(n_{21}-1\right)\left(\frac{1}{ BC _1}+\frac{1}{ DC _2}\right)\end{aligned}$
→using sign convention,
$OB =-u, DI =v, BC _1= R _1 \text { and } DC _2=- R _2.$
Substituting these values in above equation,
$\therefore-\frac{1}{u}+\frac{1}{v}=\left(n_{21}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$ ...(1)
→lens maker's formula,
$\therefore \quad \frac{1}{f}=\left(n_{21}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$ ...(2)
→Comparing equation (1) and (2),
$\therefore \frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
→This equation is known as lens equation.
→This equation is true for both convex and concave lens.
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