Problem 41 - College Physics

Question

The commonly used CD (Compact Disc) consists of many closely spaced tracks that can be used as reflecting gratings. The industry standard for the track- to-track distance is \(1.6 \mu \mathrm{m} .\) If a He-Ne laser with a wavelength of \(632.8 \mathrm{nm}\) is incident normally onto a \(\mathrm{CD}\), calculate the angles for all the visible maxima.

Step-by-Step Solution

Answer

The visible angles are \(0^{\circ}, 23.3^{\circ},\) and \(52.1^{\circ}\).

Step 1: Understand the Problem

The problem is about finding the angles where you can see diffraction maxima when a laser hits a CD. This is like using the CD as a diffraction grating. The track-to-track distance is the distance between lines on the grating, given as \(d = 1.6 \mu m\). The laser has a wavelength \(\lambda = 632.8 nm\).

Step 2: Identify the Diffraction Condition

For a diffraction grating, the angles of maxima \( \theta \) are given by the equation for diffraction maxima: \[ d \sin(\theta) = m\lambda \] where \( d \) is the distance between tracks, \( \lambda \) is the wavelength of the laser, and \( m \) is the order of the maximum and can be 0, 1, 2, ...

Step 3: Convert Units Consistently

Make sure both distance \(d\) and wavelength \(\lambda\) are in the same units. Convert \(d = 1.6 \mu \mathrm{m}\) to nanometers: \[ 1.6 \mu m = 1600 nm \]This way both \(d\) and \(\lambda\) are in nanometers.

Step 4: Solve for \(m = 0\) (Zero Order)

For the zero order maximum: \[ 1600 \sin(\theta) = 0 \times 632.8 \]Thus, \( \sin(\theta) = 0 \), which means \( \theta = 0^{\circ} \).

Step 5: Solve for Higher Orders

Next, solve for higher orders (\(m = 1, 2, ...\)). - **First order (\(m = 1\))**:\[ 1600 \sin(\theta) = 1 \times 632.8 \]\[ \sin(\theta) = \frac{632.8}{1600} \approx 0.3955 \]Calculate \( \theta \) using \( \sin^{-1}(0.3955) \), which gives \( \theta \approx 23.3^{\circ} \).- **Second order (\(m = 2\))**:\[ 1600 \sin(\theta) = 2 \times 632.8 \]\[ \sin(\theta) = \frac{1265.6}{1600} \approx 0.791 \]Calculate \( \theta \) using \( \sin^{-1}(0.791) \), which gives \( \theta \approx 52.1^{\circ} \).- **Third order (\(m = 3\))**:\[ 1600 \sin(\theta) = 3 \times 632.8 \]\[ \sin(\theta) = \frac{1898.4}{1600} \]This results in \( \sin(\theta) > 1 \), meaning \(m = 3\) does not give a valid angle.

Step 6: Conclusion - List of Visible Maxima

The visible diffraction angles for the He-Ne laser incident on the CD are approximately \(0^{\circ}, 23.3^{\circ},\) and \(52.1^{\circ}\) for \(m=0, 1,\) and \(2\) respectively.

Key Concepts

Diffraction Maxima

Diffraction maxima occur when waves, such as light waves from a laser, pass through narrow slits or around barriers and interfere constructively. This results in a pattern of bright and dark regions which are known as diffraction patterns. For a diffraction grating, these bright spots indicate where the waves amplify each other the most, creating distinct angles known as diffraction maxima.

These maxima can be calculated using the formula \(d \sin(\theta) = m\lambda\), where:

• \(d\) is the distance between grating lines or the track-to-track distance.
• \(\lambda\) is the wavelength of the incident light.
• \(m\) is the order of maximum and represents the constructive interference order (\(m=0, 1, 2, ...\)).

Using this equation, you can find out at what angles \(\theta\) light will be most intense. In a practical scenario, this could be seen when shining a laser on a CD and observing the angles where bright spots appear.

Wavelength Conversion

Wavelength is the distance between successive peaks of a wave, such as a light wave, and is often measured in nanometers (nm) or micrometers (μm). When solving problems related to diffraction, it's crucial to ensure that all measurements are in the same unit.

For instance, in the provided problem, the track-to-track spacing \(d\) is given as \(1.6 \mu \mathrm{m}\), which needs to be converted into nanometers to match the wavelength \(\lambda = 632.8 \mathrm{nm}\) of the He-Ne laser. To convert micrometers to nanometers, remember that \(1\ \mu m = 1000\ nm\).

Thus, performing the conversion:

• \(d = 1.6 \mu m = 1.6 \times 1000 = 1600\ nm\).

This consistency in units ensures accurate calculations of the diffraction maxima angles.

He-Ne Laser

The Helium-Neon (He-Ne) laser is widely used as it emits a coherent and monochromatic red light with a wavelength of approximately 632.8 nm. It is often employed in educational and professional settings for experiments involving optics, such as diffraction, due to its stable and easily observable light characteristics.

The stability and coherence of the He-Ne laser make it suitable for precise measurements and demonstrations. In the context of diffraction, such a laser helps highlight the resulting patterns and angles clearly, as seen in this exercise where it is used incidentally on a CD acting as a grating.

Understanding the characteristics of lasers, especially their wavelength, is essential in analyzing and predicting the behavior of light as it interacts with different materials, including diffraction gratings like CDs.

Order of Maxima

The order of maxima \(m\) is a critical aspect in diffraction patterns. It indicates the sequence of bright spots formed as a result of constructive interference of light waves.

• Zero Order \(m = 0\): For zero order, all incoming waves are in the same direction, producing a maximum directly along the incident light's path (\(\theta = 0\ degree\)).
• First Order \(m = 1\): This represents the first set of bright spots away from the center, calculated from the equation \(d \sin(\theta) = m\lambda\).
• Second Order \(m = 2\): These are the next set of bright spots further out from the central maximum.

As \(m\) increases, the angle \(\theta\) at which these maxima appear also increases, until at some point, \(\sin(\theta)\) exceeds 1, making higher order maxima not possible or visible. In this particular problem, a third order \(m = 3\) is calculated but found unfeasible (\(\sin(\theta) \> 1\)).

Understanding these orders is crucial for analyzing the patterns produced in practical experiments and for interpreting the related equations effectively.