Chapter 39, Problem 14SP

Two thin lenses having focal lengths of +200 cm and –400 cm are glued together so as to have a common axis. Determine the dioptric power of the combination. Discuss the physics of your answer. [Hint: Go back to Eq. (38.4), and then study the definition of dioptric power.]

To determine

The dioptric power of the combination of two lenses when the focal length of lensesis $+200\text{ cm}$ and $-400\text{ cm}$.

Answer to Problem 14SP

Solution:

$0.25\text{ D}$ and the combination behaves as a convex lens of focal length $4\text{ m}$.

Explanation of Solution

Given data:

The focal length of first lens is $+200\text{ cm}$.

The focal length of second lens is $-400\text{ cm}$.

Formula used:

Write the expression for the focal length of the combination of lenses in contact.

$\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}$

Here, $F$ is the focal length of the combination of lenses, $f_1$ is the focal length of the first lens, and $f_2$ is the focal length of the second lens.

Write the expression for power of the combined lens.

$P=\frac{1}{F}$

Here, $P$ is the power of the combined lens.

Explanation:

Recall the expression for the focal length of the combination of lenses in contact.

$\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}$

Substitute $+200\text{ cm}$ for $f_1$ and $-400\text{ cm}$ for $f_2$

$\begin{array}{c}\frac{1}{F}=\frac{1}{\left(+200\text{ cm}\right)\left(\frac{1\text{ m}}{100\text{ cm}}\right)}+\frac{1}{\left(-400\text{ cm}\right)\left(\frac{1\text{ m}}{100\text{ cm}}\right)}\\ =\left(\frac{1}{2\text{ m}}-\frac{1}{4\text{ m}}\right)\\ \text{=}\frac{1}{4}{\text{ m}}^{-1}\\ F=4\text{ m}\end{array}$

Recall the expression for the power of the combined lens.

$P=\frac{1}{F}$

Substitute $\text{4 m}$ for $F$

$\begin{array}{c}P=\frac{1}{4\text{ m}}\\ =0.25{\text{ m}}^{-1}\cdot \frac{1\text{ D}}{1{\text{ m}}^{-1}}\\ =0.25\text{ D}\end{array}$

When the lens of focal length $+200\text{ cm}$, that is a convex lens, and lens of focal length $-400\text{ cm}$, that is a concave lens, are kept in contact, the combination behaves as a convex lens of focal length $4\text{ m}$. The power of the combination of lenses is $0.25\text{ D}$.

Conclusion:

The dioptric power of combination of lenses is $0.25\text{ D}$ and the combination behaves as a convex lens of focal length $4\text{ m}$.