Problem 88 - Physics for Scientists and Engineers with Modern Physics

Question

A converging lens with focal length of \(13.0 \mathrm{~cm}\) is placed in contact with a diverging lens with a focal length of \(-20.0 \mathrm{~cm} .\) What is the focal length of the combination, and is the combination converging or diverging?

Step-by-Step Solution

Answer

The focal length of the combination is approximately \(37.2\) cm, and it is converging.

Step 1: Understanding Lens in Combination

When two lenses are placed in contact, the focal length of the combination can be found using the formula for lenses in contact: \( \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \), where \( f \) is the focal length of the combined lenses, \( f_1 \) is the focal length of the first lens, and \( f_2 \) is the focal length of the second lens.

Step 2: Plugging in Known Values

Substitute the given values into the formula: \( \frac{1}{f_1} = \frac{1}{13.0} \) cm and \( \frac{1}{f_2} = \frac{1}{-20.0} \) cm. The calculation becomes \( \frac{1}{f} = \frac{1}{13} + \frac{1}{-20} \).

Step 3: Calculating the Reciprocal of the Focal Lengths

Calculate \( \frac{1}{13} \approx 0.0769 \) and \( \frac{1}{-20} = -0.05 \). Now, combine these by adding the values: \( \frac{1}{f} \approx 0.0769 - 0.05 \).

Step 4: Calculate the Effective Focal Length

Perform the subtraction: \( 0.0769 - 0.05 = 0.0269 \). Therefore, \( \frac{1}{f} = 0.0269 \). To find \( f \), take the reciprocal: \( f = \frac{1}{0.0269} \approx 37.2 \) cm.

Step 5: Determine the Nature of the Combined Lens System

The focal length of the combination is positive, \( f \approx 37.2 \) cm, indicating that the combined lens system is converging.

Key Concepts

Focal Length

The focal length of a lens is a key concept in optics. It refers to the distance from the lens where light rays converge or diverge. If light rays parallel to the principal axis enter the lens, they will converge or appear to diverge from this focal point. The focal length determines how strong or weak a lens is in terms of focusing light.

In the context of lenses in combination, each lens contributes to the overall focal length of the system. When combining lenses, we use the formula \( \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \) to find the effective focal length \( f \) of two lenses placed in contact.

- Positive focal length means a converging lens, bringing light rays to a focus.
- Negative focal length indicates a diverging lens, causing light rays to spread out.

This formula allows us to calculate how lenses will work together, giving a single focal length for both.

Converging Lens

A converging lens, often known as a convex lens, has a positive focal length. This means it bends incoming parallel light rays towards each other, or converges them, to a focal point. Converging lenses are commonly used in a variety of optical devices such as cameras, glasses, and microscopes.

Key characteristics of a converging lens include:

• Convex shape, thicker at the center than at the edges.
• Focuses light to a point on the opposite side of the lens.
• Positive focal length values, which indicates a propensity to converge rays.

When combined with other lenses, a converging lens can alter the focal distance of the system. It can help offset the effects of a diverging lens and achieve a specific focal length for practical applications.

Diverging Lens

A diverging lens, also known as a concave lens, has a negative focal length. It spreads out light rays that are initially parallel to the principal axis, making them diverge from a focal point behind the lens. This type of lens is used to correct vision issues such as myopia (short-sightedness).

Characteristics of a diverging lens include:

• Concave shape, thinner at the center than at the edges.
• Spreads parallel light rays outward away from a central point.
• Negative focal length, reflecting its divergent properties.

Diverging lenses can be used in combination with converging lenses to manipulate light in desired ways. By changing the combined focal length of a lens system, they enable more versatile optical solutions by compensating for the convergence of another lens.