Chapter 24, Problem 6E

(a)

To determine

The radius of curvature of the other surface and draw a sketch of the lens.

Answer to Problem 6E

The radius of curvature of the other surface is $-0.3\text{m}$.

Explanation of Solution

Write the expression for the Lens Makers formula.

  $\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Here $f$ is the focal length of the lens, $\mu$ is the refractive index of the lens, $R_1$ is the radius of the convex surface and $R_2$ is the radius of the other surface.

Rearrange the above expression in terms of $R_2$.

  $R_2=\left(\frac{R_{1}f\left(\mu -1\right)}{f\left(\mu -1\right)-R_1}\right)$        (I)

Conclusion:

Substitute $0.1\text{m}$ for $R_1$, $1.5$ for $\mu$  and $0.15\text{m}$ for $f$ in equation (I).

  $\begin{array}{c}{R}_2=\left(\frac{\left(0.1\text{m}\right)\left(0.15\text{m}\right)\left(1.5-1\right)}{\left(0.15\text{m}\right)\left(1.5-1\right)-\left(0.1\text{m}\right)}\right)\\ =\left(\frac{0.0075}{-0.025}\right)\text{m}\\ =-0.3\text{m}\end{array}$

Therefore, the other side is concave.

The sketch for the lens is drawn below.

Thus, the radius of curvature of the other surface is $-0.3\text{m}$.

(b)

To determine

The radius of curvature of the other surface and draw a sketch of the lens.

Answer to Problem 6E

The radius of curvature of the other surface is  $-0.1\text{m}$.

Explanation of Solution

Write the expression for the Lens Makers formula.

  $\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Here $f$ is the focal length of the lens, $\mu$ is the refractive index of the lens, $R_1$ is the radius of the convex surface and $R_2$ is the radius of the other surface.

Rearrange the above expression in terms of $R_2$.

  $R_2=\left(\frac{R_{1}f\left(\mu -1\right)}{f\left(\mu -1\right)-R_1}\right)$        (I)

Conclusion:

Substitute $0.1\text{m}$ for $R_1$, $1.5$ for $\mu$  and $0.1\text{m}$ for $f$ in equation (I).

  $\begin{array}{c}{R}_2=\left(\frac{\left(0.1\text{m}\right)\left(0.1\text{m}\right)\left(1.5-1\right)}{\left(0.1\text{m}\right)\left(1.5-1\right)-\left(0.1\text{m}\right)}\right)\\ =\left(\frac{0.005}{-0.05}\right)\text{m}\\ =-0.1\text{m}\end{array}$

Therefore, the other side is concave.

The sketch for the lens is drawn below.

Thus, the radius of curvature of the other surface is $-0.1\text{m}$.

(c)

To determine

The radius of curvature of the other surface and draw a sketch of the lens.

Answer to Problem 6E

The radius of curvature of the other surface is $0.043\text{m}$ .

Explanation of Solution

Write the expression for the Lens Makers formula.

  $\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Here $f$ is the focal length of the lens, $\mu$ is the refractive index of the lens, $R_1$ is the radius of the convex surface and $R_2$ is the radius of the other surface.

Rearrange the above expression in terms of $R_2$.

  $R_2=\left(\frac{R_{1}f\left(\mu -1\right)}{f\left(\mu -1\right)-R_1}\right)$        (I)

Conclusion:

Substitute $0.1\text{m}$ for $R_1$, $1.5$ for $\mu$  and $-0.15\text{m}$ for $f$ in equation (I).

  $\begin{array}{c}{R}_2=\left(\frac{\left(0.1\text{m}\right)\left(-0.15\text{m}\right)\left(1.5-1\right)}{\left(-0.15\text{m}\right)\left(1.5-1\right)-\left(0.1\text{m}\right)}\right)\\ =\left(\frac{-0.0075}{-0.175}\right)\text{m}\\ =0.043\text{m}\end{array}$

Therefore, the other side is convex.

The sketch for the lens is drawn below.

Thus, the radius of curvature of the other surface is $0.043\text{m}$ .