Lens Combination Focal Length Calculation

This question asks us to find the focal length of a system made by combining two lenses. We are given the individual powers of these lenses and need to determine the resulting focal length of the lens combination.

Understanding Lens Power and Focal Length

In optics, the power of a lens is a measure of its ability to converge or diverge light. It is defined as the reciprocal of the focal length.

• The formula relating power (P) and focal length (f) is: \(P = \frac{1}{f}\)
• Here, the power (P) must be in Diopters (D) and the focal length (f) must be in meters (m).

Combining Lens Powers

When multiple lenses are placed in contact with each other, their powers add up algebraically to give the total power of the combination.

Given the powers of the two lenses:

• Lens 1 power: \(P_1 = +2 \cdot 5 \ D\)
• Lens 2 power: \(P_2 = -2 \cdot 0 \ D\)

The total power (\(P_{total}\)) of the combination is calculated as:

Step-by-Step Calculation

1. Calculate Total Power: Substitute the given powers into the formula: \(P_{total} = (+2 \cdot 5 \ D) + (-2 \cdot 0 \ D)\) \(P_{total} = 2 \cdot 5 - 2 \cdot 0 \ D\) \(P_{total} = 0 \cdot 5 \ D\)
2. Calculate Total Focal Length: Now, use the total power (\(P_{total}\)) to find the total focal length (\(f_{total}\)) using the formula \(f = \frac{1}{P}\): \(f_{total} = \frac{1}{P_{total}}\) \(f_{total} = \frac{1}{0 \cdot 5 \ D}\) \(f_{total} = 2 \cdot 0 \ m\)

Result Analysis

The calculation shows that the combined focal length of the lens system is \(2 \cdot 0 \ m\). This positive value indicates that the combination behaves like a converging lens.

Comparing this result with the given options:

• Option 1: \(+0 \cdot 5 \ m\)
• Option 2: \(-0 \cdot 5 \ m\)
• Option 3: \(+2 \cdot 0 \ m\)
• Option 4: \(-2 \cdot 0 \ m\)

Our calculated focal length, \(2 \cdot 0 \ m\), matches Option 3.