Problem 26 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A diverging lens is placed next to a converging lens of focal length \(f_{C},\) as in Fig. \(15 .\) If \(f\) represents the focal length of the combination, show that the focal length of the diverging lens, \(f_{\mathrm{D}},\) is given by \(\frac{1}{f_{\mathrm{D}}}=\frac{1}{f_{\mathrm{T}}}-\frac{1}{f_{\mathrm{C}}}\)

Step-by-Step Solution

Answer

The focal length of the diverging lens is given by \( \frac{1}{f_{D}} = \frac{1}{f_{T}} - \frac{1}{f_{C}} \).

Step 1: Understanding the Lens Combination

In this problem, we have two lenses: a diverging lens and a converging lens placed together. We are tasked with finding the focal length of the diverging lens given the focal length of the converging lens (\( f_{C} \)). The effect of the combined lenses can be condensed to a single equivalent focal length \( f \).

Step 2: Formula for Combined Focal Length

The focal length \( f \) of a combination of lenses in contact can be expressed as:\[\frac{1}{f} = \frac{1}{f_{C}} + \frac{1}{f_{D}}\]where \( f_{D} \) is the focal length of the diverging lens, and \( f_{C} \) is the focal length of the converging lens.

Step 3: Substituting Effective Focal Length

We replace \( f \) with \( f_{T} \) in our problem, assuming it represents the effective focal length of the lens system:\[\frac{1}{f_{T}} = \frac{1}{f_{C}} + \frac{1}{f_{D}}\]

Step 4: Solving for Diverging Lens Focal Length

To find an expression for the focal length of the diverging lens, \( f_{D} \), we rearrange the equation:\[\frac{1}{f_{D}} = \frac{1}{f_{T}} - \frac{1}{f_{C}}\]This formula represents the reciprocal of the focal length for the diverging lens expressed in terms of the combined focal length, \( f_{T} \), and the focal length of the converging lens, \( f_{C} \).

Key Concepts

Diverging Lens

A diverging lens, often known as a concave lens, spreads out light rays that are initially coming together. This type of lens is thinner in the center than at the edges. Due to its shape, it causes parallel rays of light that pass through it to spread apart—hence diverging from a common point on the object's side of the lens.
In optics, diverging lenses are used in eyeglasses for people who are nearsighted, as they help disperse the light before it reaches the eye, correcting focal point issues. The focal point of a diverging lens is located on the same side as the light source because the light rays never actually converge. Instead, they create an illusion of convergence if we trace them back.

• The image formed by a diverging lens is always virtual.
• The image is smaller than the object.
• It appears upright, as opposed to inverted.

The focal length of a diverging lens is considered negative because of its divergent nature, which is an essential concept when dealing with lens systems in physics.

Converging Lens

Converging lenses, also referred to as convex lenses, are the opposite of diverging lenses. They are thicker at the center than at the edges, causing parallel light rays passing through it to bend towards each other at the focal point. This type of lens is capable of converging light rays to a single point on the other side of the lens, thereby forming images that are real and can be displayed on a screen.
Converging lenses are commonly used in magnifying glasses, cameras, and corrective lenses for farsightedness. Its power to converge light is mathematically represented by a positive focal length.

• The image may be real or virtual depending on the position of the object.
• The image can appear larger or smaller than the object depending on its orientation with respect to the lens.
• The image can be inverted or upright.

Understanding how a converging lens functions is crucial when discussing lens combinations, especially when combined with a diverging lens, as it influences the overall focal properties of the combination.

Focal Length

The focal length of a lens, symbolized as \( f \), is a fundamental concept in optics that refers to the distance from the lens where parallel rays of light converge or appear to diverge from a common point after passing through the lens.
Depending on the type of lens:

• Converging lens: The focal length is positive, as light rays are brought to a single focus point on the opposite side of the light's entry into the lens.
• Diverging lens: The focal length is considered negative because the light rays appear to diverge from a point on the same side as the light source.

The focal length plays a vital role in determining how lenses are used. It affects magnification and the type of image produced. In multi-lens systems, knowing each lens's individual focal length helps calculate the effective focal length of the system, which is critical in applications like photography, astronomy, and everyday eyewear.

Lens Combination

A lens combination occurs when two or more lenses are placed in proximity, affecting the path of light and the overall focal characteristics of the system. By combining lenses, we can tailor optical systems to achieve specific goals, like altering the effective focal length \( f_T \).
When combining a converging and a diverging lens, the focal lengths of each lens affect the final focal length of the lens combination. For such a system, the formula is:\[ \frac{1}{f_T} = \frac{1}{f_C} + \frac{1}{f_D} \]

• \( f_T \) is the equivalent focal length of the entire lens system.
• \( f_C \) and \( f_D \) are the focal lengths of the converging and diverging lenses, respectively.

By re-arranging this formula, we can solve for unknown focal lengths, allowing for the design and understanding of complex optical systems. This equation highlights the reciprocal relationship between the lenses, showcasing how each affects the overall system's behavior. Understanding lens combinations is pivotal for designing devices like telescopes and microscopes, where precise image formation is required.