Problem 106 - Physics for Scientists and Engineers with Modern Physics

Question

The proper functioning of certain optical devices (e.g., optical fibers and spectrometers) requires that the input light be a collection of diverging rays within a cone of halfangle \(\theta\) (Fig. \(33-50\) ). If the light initially exists as a collimated beam (i.e., parallel rays), show that a single lens of focal length \(f\) and diameter \(D\) can be used to create the required input light if \(D / f=2 \tan \theta .\) If \(\theta=3.5^{\circ}\) for a certain spectrometer, what focal length lens should be used if the lens diameter is \(5.0 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

A lens with a focal length of approximately 40.8 cm is required.

Step 1: Understanding the Problem

We need to show that a lens can create the required divergent light if the condition \( D / f = 2 \tan \theta \) is satisfied. We are also asked to find the focal length \( f \) when \( \theta = 3.5^{\circ} \) and \( D = 5.0 \text{ cm} \).

Step 2: Relationship Setup

Given that the diameter of the lens \( D \) and focal length \( f \) are related by \( D / f = 2 \tan \theta \), we can rearrange to express \( f \) as \( f = \frac{D}{2 \tan \theta} \).

Step 3: Calculate \( \tan \theta \)

First, convert \( \theta = 3.5^{\circ} \) into radians and then compute \( \tan(3.5^{\circ}) \). \( \theta = \frac{3.5 \pi}{180} \text{ radians} \). Compute \( \tan(3.5^{\circ}) \approx 0.0612 \).

Step 4: Substitute Values

Substitute \( D = 5 \text{ cm} \) and \( \tan \theta \approx 0.0612 \) into the equation \( f = \frac{D}{2 \tan \theta} \): \[ f = \frac{5}{2 \times 0.0612} \approx \frac{5}{0.1224} \approx 40.8 \text{ cm} \].

Step 5: Conclusion of Solution

Thus, the required focal length \( f \) of the lens to create the desired diverging rays is approximately \( 40.8 \text{ cm} \).

Key Concepts

Optical Fibers

Optical fibers are slender and flexible strands of glass or plastic that transmit light from one point to another. These fibers are exceptionally efficient as they can carry light over long distances with minimal loss. This is primarily due to the principle of total internal reflection within the fiber:

• The light is confined within the core of the fiber as it travels.
• It reflects off the core-cladding boundary due to differences in the refractive indices.

Optical fibers are commonly used in telecommunications to transmit data at high speeds, making our internet faster and more reliable. They are also used in medical imaging and lighting, such as endoscopies, where light needs to be transported to areas that are hard to access.
To function efficiently, optical fibers often require input light to diverge slightly, allowing them to gather more light beams within a specific acceptance angle. This ensures that the light remains trapped and efficiently travels through the fiber.

Spectrometers

Spectrometers are instruments that analyze the properties of light. They are vital in scientific research for identifying substances based on the light they emit or absorb.
Spectrometers separate and measure the intensity of the light spectrum:

• They split incoming light into its constituent colors using a prism or diffraction grating.
• Each color corresponds to a different wavelength of light.
• Sensors then measure the intensity of each wavelength, revealing information about the sample.

Applications of spectrometers range from identifying unknown materials to measuring pollutants in the environment or determining the composition of distant stars. In order to function accurately, spectrometers need divergent light entering the device so that the instruments inside can effectively separate and analyze each wavelength of the spectrum.

Diverging Rays

Diverging rays refer to a pattern of light in which the rays spread apart from a single point. This spread is crucial for various optical devices, including lenses, to function properly.

• Diverging rays enable a broad examination of objects through devices like spectrometers.
• In applications such as projectors or flashlights, diverging rays help spread light evenly over a surface.

Creating diverging rays usually requires a lens system that adjusts the initial collimated (parallel) light rays into those that fan out in a cone shape. This spread is defined by an angle, often called the half-angle, which is essential for devices relying on the ray spread for accurate readings or projections. Managing the divergence correctly ensures that optical instruments can capture as much information as possible from the incoming light.

Lens Focal Length

The focal length of a lens is a critical measurement that denotes the distance from the lens to its focal point, where rays of light meet. It determines how a lens bends light and is crucial in applications across various optical devices.

• A longer focal length creates less bending and is used for applications like telescopes to view distant objects.
• A shorter focal length creates more bending, ideal for microscopes to magnify tiny objects.

In the context of achieving specific divergences of light, the focal length plays a pivotal role. By adjusting the focal length, lenses can convert parallel rays into a diverging pattern required by devices like optical fibers and spectrometers. The equation \( f = \frac{D}{2 \tan \theta} \) from the original exercise highlights how the diameter of the lens \( D \), the focal length \( f \), and the angle \( \theta \) work together. The focal length helps determine the extent to which the light rays will diverge upon passing through the lens.