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Chapter 3: Problem 45

Electromagnetic "waves" strike a single slit of I \mum width. Determine the angular full width (angle from first minimum on one side of the center to first minimum on the other) in degrees of the central diffraction maximum if the waves are (a) visible light of wavelength \(500 \mathrm{nm}\) and (b) X-rays of wavelength \(0.05 \mathrm{nm}\). (c) Which more clearly demonstrates a wave nature?

Short Answer

Expert verified
The full angular width of the central diffraction maximum is approximately 57.3 degrees for visible light and 0.00573 degrees for X-rays. Hence, the wave nature of light is more clearly demonstrated by the visible light.

Step by step solution

01

Calculation for Visible Light

First, convert the given slit width and light wavelength to same units for a proper calculation. Given that the slit width is 1 渭m = \(1 \times 10^{-6}\) m and wavelength is 500 nm = \(500 \times 10^{-9}\) m. Now, use the formula \( \theta = \lambda / w \) to determine the angular position: \( \theta = \frac{500 \times 10^{-9}}{1 \times 10^{-6}} = 0.5 \) radians. To convert radians to degrees, multiply by \((180 / \pi)\). Hence, the full width is \( \theta_{full} = 2 \times \theta \times \frac{180}{\pi} \) degrees.
02

Calculation for X-rays

Given that the X-ray wavelength is 0.05 nm = \(0.05 \times 10^{-9}\) m, use the same formula and process as in step 1: \( \theta = \frac{0.05 \times 10^{-9}}{1 \times 10^{-6}} = 0.00005 \) radians. Hence, the full width is \( \theta_{full} = 2 \times \theta \times \frac{180}{\pi} \) degrees.
03

Comparison and Analysis

Now compare the results of steps 1 and 2. Visible light gives a wider diffraction pattern than X-rays, which means the wave nature (diffraction) is more evident with visible light in this case. This is in line with the understanding that wave phenomena such as diffraction and interference are more noticeable with longer wavelengths.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Waves
Electromagnetic waves are a type of energy that travel through space at the speed of light. These waves do not need a medium to propagate, which means they can travel through the vacuum of space. They have both electric and magnetic field components. The relationship between these fields is what defines them as electromagnetic waves.
  • Range: Electromagnetic waves cover a broad range of wavelengths and frequencies, forming the electromagnetic spectrum. This spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
  • Properties: All electromagnetic waves travel at the speed of light, which is approximately 299,792,458 meters per second in a vacuum. The speed can vary depending on the medium they pass through.
  • Applications: These waves are essential in various technologies and natural processes, including communication (radio waves), medical imaging (X-rays), and daily human vision (visible light).
Understanding electromagnetic waves is crucial for comprehending phenomena like diffraction.
Single Slit
When electromagnetic waves pass through a single slit, they spread out instead of traveling in a straight line. This phenomenon is called diffraction. The single slit setup demonstrates the wave nature of light and is an essential aspect of wave-based optics.
  • How it Works: When waves pass through a slit that is similar in size to their wavelength, they interact with the edges of the slit, causing them to spread out.
  • Central Maximum: The spread creates a pattern of light and dark areas on a screen where the light waves interfere constructively and destructively, respectively. The brightest part is the central maximum.
  • Applications: Single slit experiments help in measuring wavelengths and studying wave behaviors.
The single slit setup is fundamental in understanding broader concepts of wave interference.
Visible Light
Visible light is the part of the electromagnetic spectrum that can be detected by the human eye. It encompasses the range of wavelengths from approximately 400 nm (nanometers) to 700 nm. This type of light plays a critical role in human life and various technologies.
  • Spectrum: The visible spectrum consists of colors ranging from violet (shorter wavelengths) to red (longer wavelengths).
  • Significance: This light is responsible for our perception of color. In diffraction experiments, visible light is commonly used due to its relative ease of manipulation and detection.
  • Characteristics: Visible light waves exhibit properties like reflection, refraction, and diffraction, which are typical of electromagnetic waves.
When visible light passes through a single slit, diffraction patterns can be observed, showcasing its wave nature.
X-rays
X-rays are another form of electromagnetic waves with significantly shorter wavelengths compared to visible light. Typically, they range from 0.01 nm to 10 nm in wavelength. X-rays have unique properties that make them indispensable in certain fields, especially in medicine.
  • Higher Energy: Due to their shorter wavelengths, X-rays have more energy compared to visible light. This property makes them highly penetrative.
  • Application in Medical Imaging: X-rays can penetrate the human body, making them ideal for imaging internal structures like bones.
  • Challenges in Diffraction: Their short wavelength results in less visible diffraction patterns in single slit experiments, making it harder to observe their wave nature directly.
Understanding X-rays is crucial in fields like healthcare and material sciences, where their penetration ability is highly valued.
Angular Width
Angular width, sometimes referred to as angular spread, is an important concept in wave phenomena like diffraction. It measures the extent of spreading of a wavefront as it passes through a slit or around an object.
  • Calculation: The angular width can be determined using the formula \[ \theta = \frac{\lambda}{w},\] where \( \lambda \) is the wavelength and \( w \) is the width of the slit.
  • Measurement: Usually measured in radians, the angular width helps quantify the diffraction patterns observed in experiments. It is then converted to degrees for practical applications.
  • Influence of Wavelength: Longer wavelengths, like those of visible light, result in larger angular widths and thus more apparent diffraction.
Angular width is a quantitative reflection of the diffraction effect, providing valuable insights into wave behaviors.
Wave Nature of Light
One of the most fascinating aspects of light is its dual nature, exhibiting properties of both waves and particles. The wave nature of light is best illustrated through phenomena such as diffraction and interference.
  • Historical Discovery: Experiments like the famous double-slit experiment by Thomas Young provided strong evidence for the wave nature of light.
  • Wave Phenomena: Diffraction, the bending of light around obstacles, is a clear indication of light behaving as a wave. This behavior underlies many technological advancements, from cameras to telescopes.
  • Educational Significance: Understanding that light behaves as a wave helps explain many complex concepts in physics and everyday optical phenomena, such as rainbows and the diffraction of beams in optics.
Recognizing the wave nature of light is essential for foundational and advanced studies in the field of optics and wave mechanics.

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Most popular questions from this chapter

According to the energy-lime uncertainty principle. the lifetime \(\Delta f\) of a state and the uncertainty \(\Delta E\) in its energy are inversely proportional. Hydrogen's \(656 \mathrm{~nm}\) red spectral line is the result of an electron making a transition "downward" from a quantum state whose lifetime is about \(10^{-8} s\) (a) What inherent uncertainty in the energy of the emitted photon docs this imply? (Note: Unfortunately. we might use the symbol \(\Delta E\) for the energy difference - i.e., the energy of the photon - but here if means the uncertain in that energy difference.) (b) To what range in wavelengths does this correspond? (As noted in Exercise \(2.57\). the uncertainty principle is one contributor to the broadening of spectral lines.) (c) Obtain a general formula relating \(\Delta \lambda\) to \(\Delta t\).

In the Compton effect, we choose the electron to be at the origin and the initial photon's direction of motion to be in the \(+x\) direction. (a) We may also choose the xy-plane so that it contains the velocities of the outgoing electron and photon. Why? (b) The incoming photon's wavelength \(\lambda\) is assumed to be known. The unknowns after the collision are the outgoing photon's wavelength and direction, \(\Lambda^{\prime}\) and \(\theta,\) and the speed and direction of the electron, \(u_{e}\) and \(\phi .\) With only three equations \(-t w_{0}\) components of momentum conservation and one of energy - we can't find all four. Equation \((3-8)\) gives \(\lambda^{\prime}\) in terms of \(\theta .\) Our lack of knowledge of \(\theta\) after the collision (without an experiment) is directly related to a lack of knowledge of something before the collision. What is it? (lmagine the two objects are hard spheres.) (c) Is it reasonable to suppose that we could know this? Explain.

You are conducting a photoelectric effect experiment by shining light of \(500 \mathrm{nm}\) wavelength a a piece of metal and determining the stopping potential. If, unbeknownst to you, your 500 nm light source actually contained a small amount of ultraviolet light, would it throw off your results by a small amount or by quite a bit? Explain.

A stationary muon \(\mu^{-}\) annihilates with a stationary antimuon \(\mu^{+}\) (same mass. \(1.88 \times 10^{\circ} 28 \mathrm{~kg}\), but opposite charge). The two disappear, replaced by electromagnetic radiation. (a) Why is it not possible for a single photon to result? (b) Suppose two photons result. Describe their possible directions of motion and wavelengths.

A bedrock topic in quantum mechanics is the uncertainty principle. It is discussed mostly for massive objects in Chapter \(4,\) but the idea also applies to light: Increasing certainty in knowledge of photon position implies increasing uncertainty in knowledge of its momentum, and vice versa. A single-slit pattern that is developed (like the double-slit pattern of Section 3.6 ) one photon at a time provides a good example. Depicted in the accompanying figure, the pattern shows that photons emerging from a narrow slit are spread allover; a photon's \(x\) -component of momentum can be any value over a broad range and is thus uncertain. On the other hand, the \(x\) -coordinate of position of an emerging photon covers a fairly small range, for \(w\) is small. Using the single-slit diffraction formula \(n \lambda=w \sin \theta,\) show that the range of likely values of \(p_{x}\), which is roughly \(p \sin \theta\), is inversely proportional to the range \(w\) of likely position values. Thus, an inherent wave nature implies that the precisions with which the particle properties of position and momentum can be known are inversely proportional.
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