Question

Two converging lenses are in contact. If the focal lengths are each \(5 \mathrm{~cm}\), what is the equivalent focal length of the combination? 1\. \(0.1 \mathrm{~cm}\) 2\. \(2.5 \mathrm{~cm}\) 3\. \(5.0 \mathrm{~cm}\) 4\. \(10.0 \mathrm{~cm}\)

Short answer

\(f_{eq} = 2.5 \mathrm{~cm}\)

Step-by-step solution

Lens formula

The lens maker's formula relates the focal length of a lens to its optical power, which operates according to the principle of superposition when multiple lenses are in contact. The equation for lens formula is: \( \frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} \) Where \(f_{eq}\) is the equivalent focal length, and \(f_1\) and \(f_2\) are the individual focal lengths of the lenses in contact.

Applying the lens formula

For this problem, the focal lengths of both lenses are given as 5 cm. We can thus substitute this into our lens formula: \( \frac{1}{f_{eq}} = \frac{1}{5} + \frac{1}{5} \)

Solve for equivalent focal length

Now, we can add the two fractions on the right side of the equation and solve for the equivalent focal length, \(f_{eq}\): \( \frac{1}{f_{eq}} = \frac{2}{5} \) To find the value of \(f_{eq}\), we can take the reciprocal of both sides: \( f_{eq} = \frac{5}{2} \)

Finding the answer

This gives us an equivalent focal length of \(f_{eq} = 2.5 \mathrm{~cm}\). Therefore, the correct answer is: 2. 2.5 cm

Key concepts

Lens Formula

The lens formula is a critical tool in optics. It helps us understand and calculate how multiple lenses affect light. The formula is written as:

• \( \frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} \)

Here, \(f_{eq}\) is the equivalent focal length of the lens system, while \(f_1\) and \(f_2\) are the focal lengths of individual lenses in contact.
When lenses are placed side by side, their individual effects on light combine to form a single optical system. As each lens bends the light, they work together under the principle of superposition. This means their powers add up, and the total effect can be calculated using the lens formula.
In practice, this means that you don't have to measure the created system's focal length directly. You can determine it mathematically, saving time and improving accuracy.

Focal Length

Focal length is a fundamental property that defines how a lens focuses light. It is the distance from the lens to the point where parallel light rays converge after passing through the lens.
For a converging lens, this spot is known as the focal point. The focal length, typically measured in centimeters or meters, determines the lens's power to converge or diverge light.
A shorter focal length indicates a stronger ability to focus light closely, while a longer focal length will focus light over a longer distance. This concept is central when working with lenses in cameras, glasses, and telescopes.
Different lenses can have various focal lengths based on their shape and material. Knowing a lens's focal length allows you to predict how it will behave in an optical system, helping with design and application.

Converging Lenses

Converging lenses, also known as convex lenses, are thicker at the middle compared to their edges. These lenses bend light rays toward each other, causing them to converge at the focal point. This behavior is crucial in many optical devices.
Converging lenses are used in everything from eyeglasses to improve vision, cameras to capture images, to scientific instruments like microscopes and telescopes.
Their ability to focus light makes them versatile tools in producing clear and magnified images. In a practical sense, by combining two converging lenses with equal focal lengths, as in this exercise, we observe an increase in optical strength and a reduced focal length for the system.
Understanding how these lenses work allows scientists and engineers to create precise systems for various applications, enhancing how we view and capture the world around us.