Problem 82 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An achromatic lens is made of two very thin lenses, placed in contact, that have focal lengths \(f_{1}=-28 \mathrm{~cm}\) and \(f_{2}=+25 \mathrm{~cm} .\) (a) Is the combination converging or diverging? (b) What is the net focal length?

Step-by-Step Solution

Answer

The lens combination is converging with a net focal length of approximately 233.33 cm.

Step 1: Understand the Concept of Focal Length

A lens' focal length explains whether it converges (positive focal length) or diverges (negative focal length) light rays. In this problem, you must find out if the combined lenses are converging or diverging by calculating their net focal length.

Step 2: Apply the Lens Combination Formula

When two thin lenses are in contact, their net focal length \( f \) is given by the formula:\[\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}\]Substitute \( f_1 = -28 \) cm and \( f_2 = 25 \) cm into the formula to find the net focal length.

Step 3: Calculate the Inverse of Each Focal Length

Calculate the inverses of the given focal lengths:\[\frac{1}{f_1} = \frac{1}{-28} = -0.0357\]\[\frac{1}{f_2} = \frac{1}{25} = 0.04\]

Step 4: Sum the Inverses of the Focal Lengths

Add the inverses of the focal lengths to find \( \frac{1}{f} \):\[\frac{1}{f} = -0.0357 + 0.04 = 0.0043\]

Step 5: Calculate the Net Focal Length

Determine the net focal length \( f \) by taking the reciprocal of the result from Step 4:\[f = \frac{1}{0.0043} \approx 233.33 \text{ cm}\]

Step 6: Determine Converging or Diverging

Since the net focal length \( f \approx 233.33 \text{ cm} \) is positive, the combination of the lenses is a converging lens.

Key Concepts

Focal Length

The concept of focal length is crucial in understanding how lenses interact with light. Simply put, the focal length of a lens is the distance over which parallel rays of light either converge to a single point or appear to diverge from a single point.

• A positive focal length indicates a converging lens, which brings light rays together at a focal point.
• A negative focal length indicates a diverging lens, which spreads light rays apart.

In an achromatic lens, which is composed of two or more lenses with different focal lengths, understanding the individual and combined focal lengths is essential to predict the lens' behavior. The goal is to either cancel or control chromatic aberration (color distortion) and achieve a specific focal outcome. When dealing with problems involving focal length like this, you always need to check both the sign and the magnitude to determine if the combined system focuses or spreads light.

Converging Lens

A converging lens bends incoming light rays toward a common focal point. This is why it's often referred to as a "positive lens"—its primary task is to focus light.

• Converging lenses have a thicker center compared to the edges, which helps them focus light inwards.
• They are commonly found in applications like magnifying glasses, camera lenses, and corrective lenses for farsighted vision.

The achromatic lens system in the example combines two thin lenses with different focal lengths but results in a net positive focal length of about 233.33 cm. This means the combination is acting as a converging lens, which is typical for applications where you need clearer imagery without color fringing.

Lens Combination Formula

The lens combination formula is a fundamental tool for working with systems of lenses in contact. It helps in calculating the effective focal length of the combined lenses, which is essential to know the resulting optical effect.The formula is:\[\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}\]This equation lets you determine the reciprocal of the net focal length, where \( f_1 \) and \( f_2 \) are the individual focal lengths of each lens in the system.Understanding each step:

• Step 1: Calculate the reciprocal of each focal length, for example, \( \frac{1}{f_1} = -0.0357 \) and \( \frac{1}{f_2} = 0.04 \).
• Step 2: Sum these reciprocals to find the reciprocal of the net focal length \( \frac{1}{f} = 0.0043 \).
• Step 3: Take the reciprocal of this result to find the net focal length \( f \), which in this case is approximately 233.33 cm.

By providing an accurate way to determine whether a lens system is converging or diverging, this formula is invaluable for designing lenses in everything from eyeglass prescriptions to complex optical instruments.