Chapter 25, Problem 46P

An achromatic lens is made of two very thin lenses, placed in contact, that have focal lengths f₁ = -27.8 cm and f₂= +25.3 cm. (a) Is the combination converging or diverging? (b) What is the net focal length?

To determine

Part (a)To determine:

The net focal length of the combination of the two thin lenses.

Answer to Problem 46P

Solution:

The net focal length of the combination of the two thin lenses in contact is found to be +281 cm.

Explanation of Solution

Given:

The focal length of the thin lenses $f_1=-27.8\text{ cm;}{f}_2=25.3\text{ cm}$

Formula used:

The net focal length of the combination of lenses is given by

$F=\frac{f_1{f}_2}{f_1+f_2}$

Calculation:

Use the given values of $f_1\text{ and }{f}_2$ in the expression to calculate the value of the net focal length.

$\begin{array}{c}F=\frac{f_1{f}_2}{f_1+f_2}\\ =\frac{\left(-27.8\text{ cm}\right)\left(25.3\text{ cm}\right)}{\left(-27.8\text{ cm}\right)+\left(25.3\text{ cm}\right)}\\ =+281\text{ cm}\end{array}$

To determine

Part (b)To determine:

If the given combination of lenses is converging or a diverging.

Answer to Problem 46P

Solution:s:

The given combination of the lenses is a converging lens.:

Explanation of Solution

Given:

The focal length of the thin lenses $f_1=-27.8\text{ cm;}{f}_2=25.3\text{ cm}$

The focal length of a converging lens is positive and that of a diverging lens is negative. When two thin lenses are kept in contact, the combination may behave as a converging lens if the equivalent focal length of the combination is positive. The combination may behave as a diverging lens if the equivalent focal length is negative.

From the previous section, it can be seen that the equivalent focal length of the combination of the given lenses has a value +281 cm. Since the focal length is positive, the combination functions as a converging lens.