Question

A double-convex lens is to be made of glass with an index of refraction of $\mathbf{1}\mathbf{.}\mathbf{5}$.One surface is to have twice the radius of curvature of the other and the focal length is to be $\mathbf{60}\mathbf{}\mathbf{mm}$. What is the (a) smaller and (b) larger radius?

Short answer

1. Smaller radius of the lens is $45\mathrm{mm}$
2. Larger radius of the lens is $90\mathrm{mm}$

Step-by-step solution

Listing the given quantities

Refractive index of lens, $\mathrm{n}=1.5$

Focal length,$\mathrm{f}=60.0\mathrm{mm}$

Radius of one surface is twice the other, i.e.,${\mathrm{r}}_2=-2{\mathrm{r}}_1$

Understanding the concepts of lens maker equation

By using the Lens maker’s formula, we can find the radius of both curvatures of the lens.

Formula:

Lens maker’s formula

$\frac{\mathbf{1}}{\mathbf{f}}\mathbf{=}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\frac{\mathbf{1}}{{\mathbf{r}}_{\mathbf{1}}}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{r}}_{\mathbf{2}}}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{34}\mathbf{-}\mathbf{10}$

(a) Calculations of the smaller radius of the lens

Lens maker’s formula is given by equation,

$\frac{1}{\mathrm{f}}=\left(\mathrm{n}-1\right)\left(\frac{1}{{\mathrm{r}}_1}-\frac{1}{{\mathrm{r}}_2}\right)$

Since one surface has twice the radius of the other and since one surface is convex to the incoming light while the other is concave, we can write,

${\text{r}}_2=-2r_1$

So, we can get,

$$\begin{gathered} \frac{1}{\mathrm{f}\left(\mathrm{n}-1\right)}=\left(\frac{1}{{\mathrm{r}}_1}+\frac{1}{2{\mathrm{r}}_1}\right) \\ =\frac{3}{2{\mathrm{r}}_1} \\ {\mathrm{r}}_1=\frac{3}{2}\mathrm{f}\left(\mathrm{n}-1\right) \\ =1.5\left(60.0\right)\left(1.5-1\right) \\ =45\mathrm{mm} \end{gathered}$$

Smaller radius of the lens is $45\mathrm{mm}$.

(b) Calculations of the larger radius of the lens

As per the condition given in the problem, r²will be double of r₁

So, r₂ = 90 mm

Larger radius of the lens is 90 mm