Two convex lenses with power 2 diopter are kept in contact with each other. The focal length of the combined lens system is
0.25 m
Calculating Combined Focal Length of Lenses
This question asks us to find the focal length of a system formed by two convex lenses placed in contact. We are given the power of each individual lens.
Understanding Lens Power and Focal Length
- Lens Power (P): This is a measure of a lens's ability to converge or diverge light. It is defined as the reciprocal of the focal length (f) when the focal length is measured in meters. The unit of lens power is the diopter (D).
- The relationship is given by the formula:
\(P = \frac{1}{f}\)(where \(f\) is in meters and \(P\) is in diopters). - A convex lens has positive power and a positive focal length.
Combining Lenses in Contact
When two or more thin lenses are placed in contact with each other, the total power of the combination is simply the algebraic sum of the individual powers of the lenses.
If we have two lenses with powers \(P_1\) and \(P_2\) in contact, the total power \(P_{total}\) is:
\(P_{total} = P_1 + P_2\)
Step-by-Step Calculation
We are given two convex lenses, each with a power of 2 diopters.
- Power of the first lens, \(P_1 = +2\) D (convex lens has positive power).
- Power of the second lens, \(P_2 = +2\) D (convex lens has positive power).
Now, we find the total power of the combined lens system:
\(P_{total} = P_1 + P_2\)
\(P_{total} = +2 \, D + (+2) \, D\)
\(P_{total} = 4 \, D\)
The total power of the combined lens system is 4 diopters.
Next, we need to find the focal length of this combined system. We use the relationship between power and focal length:
\(f_{total} = \frac{1}{P_{total}}\)
Substituting the value of \(P_{total}\):
\(f_{total} = \frac{1}{4 \, D}\)
\(f_{total} = 0.25 \, m\)
The focal length of the combined lens system is 0.25 meters.
Summary of Calculation
| Parameter | Value | Unit |
|---|---|---|
| Power of Lens 1 (\(P_1\)) | 2 | Diopters (D) |
| Power of Lens 2 (\(P_2\)) | 2 | Diopters (D) |
| Total Power (\(P_{total} = P_1 + P_2\)) | \(2 + 2 = 4\) |
Diopters (D) |
| Combined Focal Length (\(f_{total} = 1/P_{total}\)) | \(1/4 = 0.25\) |
Meters (m) |
Final Answer Determination
Based on our calculation, the focal length of the combined lens system is 0.25 m. We compare this value with the given options.
- Option 1: 0.10 m
- Option 2: 2 m
- Option 3: 4 m
- Option 4: 0.25 m
Our calculated value matches Option 4.
Revision Table: Lens Combination Concepts
| Concept | Formula/Rule | Notes |
|---|---|---|
| Lens Power | \(P = 1/f\) (f in meters) |
Unit is Diopter (D). Convex lens: P > 0, f > 0. Concave lens: P < 0, f < 0. |
| Lenses in Contact | \(P_{total} = P_1 + P_2 + \dots\) |
Powers add up algebraically. |
| Combined Focal Length | \(f_{total} = 1/P_{total}\) |
Derived from the total power. |
Additional Information: Uses of Lens Combinations
Combining lenses is a fundamental technique used in many optical instruments to achieve desired magnification and image quality. Some examples include:
- Microscopes: Use multiple lenses (objective and eyepiece) to produce highly magnified images of small objects.
- Telescopes: Use combinations of lenses or mirrors to view distant objects.
- Cameras: Lens systems are designed to focus light onto a sensor or film.
- Eyeglasses: Sometimes multiple lens elements are combined to correct vision problems and reduce aberrations.
The total power and focal length of the combined system determine its overall optical properties, such as magnification and where images are formed.
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