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University Physics Volume 3

3.2 Mathematics of Interference

University Physics Volume 33.2 Mathematics of Interference
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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Determine the angles for bright and dark fringes for double slit interference
  • Calculate the positions of bright fringes on a screen

Figure 3.7(a) shows how to determine the path length difference ΔlΔl for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle θθ between the path and a line from the slits to the screen [part (b)] is nearly the same for each path. In other words, r1r1 and r2r2 are essentially parallel. The lengths of r1r1 and r2r2 differ by ΔlΔl, as indicated by the two dashed lines in the figure. Simple trigonometry shows

Δl=dsinθΔl=dsinθ
(3.3)

where d is the distance between the slits. Combining this result with Equation 3.1, we obtain constructive interference for a double slit when the path length difference is an integral multiple of the wavelength, or

dsinθ=mλ,form=0,±1,±2,±3,…(constructive interference).dsinθ=mλ,form=0,±1,±2,±3,…(constructive interference).
(3.4)

Similarly, to obtain destructive interference for a double slit, the path length difference must be a half-integral multiple of the wavelength, or

dsinθ=(m+12)λ,form=0,±1,±2,±3,…(destructive interference)dsinθ=(m+12)λ,form=0,±1,±2,±3,…(destructive interference)
(3.5)

where λλ is the wavelength of the light, d is the distance between slits, and θθ is the angle from the original direction of the beam as discussed above. We call m the order of the interference. For example, m=4m=4 is fourth-order interference.

Left picture is a schematic drawing that shows waves r1 and r2 passing through the two slits S1 and S2. The waves meet in a common point P on a screen. Distance between points S1 and S2 is d; distance between the screen with the two slits and the screen with point P is D. Point P is higher than the mid-point between S1 and S2 by the distance y. Imaginary line drawn from the point P to the mid-point between slits form an angle Theta with the x axis. Right picture is a schematic drawing that two slits separated by the distance d. Waves pass through the slits and travel to the screen P. Angle theta is formed by the travelling wave and x axis.
Figure 3.7 (a) To reach P, the light waves from S1S1 and S2S2 must travel different distances. (b) The path difference between the two rays is ΔlΔl.

The equations for double-slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes (Figure 3.8). The closer the slits are, the more the bright fringes spread apart. We can see this by examining the equation

dsinθ=mλ,form=0,±1,±2,±3dsinθ=mλ,form=0,±1,±2,±3. For fixed λλ and m, the smaller d is, the larger θθ must be, since sinθ=mλ/dsinθ=mλ/d. This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance d apart) is small. Small d gives large θθ, hence, a large effect.

Referring back to part (a) of the figure, θθ is typically small enough that sinθtanθym/Dsinθtanθym/D, where ymym is the distance from the central maximum to the mth bright fringe and D is the distance between the slit and the screen. Equation 3.4 may then be written as

dymD=mλdymD=mλ

or

ym=mλDd.ym=mλDd.
(3.6)
Left picture shows a double slit located a distance D from a screen, with the distance between the slits given as d. Right picture is a photograph of the fringe pattern that shows the bright lines at the positions where the waves interfere constructively.
Figure 3.8 The interference pattern for a double slit has an intensity that falls off with angle. The image shows multiple bright and dark lines, or fringes, formed by light passing through a double slit.

Example 3.1

Finding a Wavelength from an Interference Pattern Suppose you pass light from a He-Ne laser through two slits separated by 0.0100 mm and find that the third bright line on a screen is formed at an angle of 10.95°10.95° relative to the incident beam. What is the wavelength of the light?

Strategy The phenomenon is two-slit interference as illustrated in Figure 3.8 and the third bright line is due to third-order constructive interference, which means that m=3m=3. We are given d=0.0100mmd=0.0100mm and θ=10.95°θ=10.95°. The wavelength can thus be found using the equation dsinθ=mλdsinθ=mλ for constructive interference.

Solution Solving dsinθ=mλdsinθ=mλ for the wavelength λλ gives

λ=dsinθm.λ=dsinθm.

Substituting known values yields

λ=(0.0100mm)(sin10.95°)3=6.33×10−4mm=633nm.λ=(0.0100mm)(sin10.95°)3=6.33×10−4mm=633nm.

Significance To three digits, this is the wavelength of light emitted by the common He-Ne laser. Not by coincidence, this red color is similar to that emitted by neon lights. More important, however, is the fact that interference patterns can be used to measure wavelength. Young did this for visible wavelengths. This analytical techinque is still widely used to measure electromagnetic spectra. For a given order, the angle for constructive interference increases with λλ, so that spectra (measurements of intensity versus wavelength) can be obtained.

Example 3.2

Calculating the Highest Order Possible Interference patterns do not have an infinite number of lines, since there is a limit to how big m can be. What is the highest-order constructive interference possible with the system described in the preceding example?

Strategy The equation dsinθ=mλdsinθ=mλ (for m=0,±1,±2,±3m=0,±1,±2,±3) describes constructive interference from two slits. For fixed values of dandλdandλ, the larger m is, the larger sinθsinθ is. However, the maximum value that sinθsinθ can have is 1, for an angle of 90°90°. (Larger angles imply that light goes backward and does not reach the screen at all.) Let us find what value of m corresponds to this maximum diffraction angle.

Solution Solving the equation dsinθ=mλdsinθ=mλ for m gives

m=dsinθλ.m=dsinθλ.

Taking sinθ=1sinθ=1 and substituting the values of dandλdandλ from the preceding example gives

m=(0.0100mm)(1)633nm15.8.m=(0.0100mm)(1)633nm15.8.

Therefore, the largest integer m can be is 15, or m=15m=15.

Significance The number of fringes depends on the wavelength and slit separation. The number of fringes is very large for large slit separations. However, recall (see The Propagation of Light and the introduction for this chapter) that wave interference is only prominent when the wave interacts with objects that are not large compared to the wavelength. Therefore, if the slit separation and the sizes of the slits become much greater than the wavelength, the intensity pattern of light on the screen changes, so there are simply two bright lines cast by the slits, as expected, when light behaves like rays. We also note that the fringes get fainter farther away from the center. Consequently, not all 15 fringes may be observable.

Check Your Understanding 3.1

In the system used in the preceding examples, at what angles are the first and the second bright fringes formed?

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