Question

Two lenses of powers \(-1.75 \, \text{D}\) and \(+2.25 \, \text{D}\) are placed in contact. Find the focal length of the combination.

• A. 100 cm
• B. 50 cm
• C. 200 cm
• D. 150 cm

Hint:

For lenses in contact, total power \(P = P_1 + P_2\) and focal length \(f = 1/P\) (in meters). Convert to desired units as needed.

Answer 1

The Correct Option is C

Step 1: Find each lens's own focal length.
Using $f = \dfrac{100}{P}$ in cm,
\[ f_1 = \frac{100}{-1.75} = -57.14\ \text{cm}, \qquad f_2 = \frac{100}{2.25} = 44.44\ \text{cm} \]
Step 2: Combine the two lenses with the combination rule.
\[ \frac{1}{f} = \frac{1}{f_1}+\frac{1}{f_2} = -\frac{1.75}{100}+\frac{2.25}{100} = \frac{0.50}{100} \]
This matches simply adding the powers, $P_1+P_2 = 0.50\ \text{D}$, a quick cross-check.
Step 3: Invert to get the focal length.
\[ f = \frac{100}{0.50} = 200\ \text{cm} \]
\[ \boxed{200\ \text{cm}} \]