The thickness at the centre
Question:

The thickness at the centre of a plano convex lens is $3 \mathrm{~mm}$ and the diameter is $6 \mathrm{~cm}$. If the speed of light in the material of the lens is $2 \times 10^{8} \mathrm{~ms}^{-1}$. The focal length of the lens is _______.

1. $0.30 \mathrm{~cm}$

2. $15 \mathrm{~cm}$

3. $1.5 \mathrm{~cm}$

4. $30 \mathrm{~cm}$

Correct Option: , 4

Solution:

$R^{2}=r^{2}+(R-t)^{2}$

$R^{2}=r^{2}+R^{2}+t^{2}-2 R t$

Neglecting $t^{2}$, we get

$R=\frac{r^{2}}{2 t}$

$\therefore \frac{1}{f}=(\mu-1)\left(\frac{1}{R}-\frac{1}{\infty}\right)=\frac{\mu-1}{R}$

$f=\frac{R}{\mu-1}=\frac{r^{2}}{2 t(\mu-1)}=\frac{\left(3 \times 10^{-2}\right)^{2}}{2 \times 3 \times 10^{-3} \times\left(\frac{3}{2}-1\right)}$

$=\frac{9 \times 10^{-4}}{6 \times 10^{-3} \times 1} \times 2$

$f=0.3 \mathrm{~m}=30 \mathrm{~cm}$