Electromagnetic Waves and the Nature of Light

Classification of Waves according to the Direction of Vibration

• Transverse Waves

  • Definition: A transverse wave is a traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation.

  • Examples: Waves on a rope, light waves.

• Longitudinal Waves

  • Definition: A longitudinal wave is a traveling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation.

  • Example: Sound waves in air.

Classification of Waves according to the Nature of the Wave

• Mechanical Waves

  • Requirement: Require a material medium to propagate.

  • Vacuum Interaction: Cannot travel in a vacuum.

  • Examples: Sound waves, water waves.

• Electromagnetic Waves

  • Requirement: Do not require a material medium.

  • Vacuum Interaction: Can travel through a vacuum.

  • Composition: Consist of oscillating electric and magnetic fields.

  • Examples: Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.

Sinusoidal Waves

• Definition and Generation

  • A sinusoidal wave is represented by a curve identical to the function $\sin(\theta)$ plotted against $\theta$.

  • Generation: Can be established on a rope by shaking one end in simple harmonic motion.

• Visualizing Motion

  • Snapshots: A wave at $t=0$ can be compared to a snapshot at a later time $t$.

  • Element Motion: If focusing on a single element (e.g., at $x=0$), each element moves up and down along the y-axis in simple harmonic motion.

• Key Parameters and Definitions

  • Wavelength ( $\lambda$ ): The distance from one crest to the next, or the minimum distance between any two identical points (troughs, crests) on adjacent waves.

  • Period ( $T$ ): The time interval required for two identical points (such as crests) of adjacent waves to pass by a fixed point.

  • Frequency ( $f$ ): The number of crests (or troughs) that pass a given point in a unit time interval.

  • Relationship between Frequency and Period:         $f = \frac{1}{T}$

  • Wave Speed ( $v$ ): The wave travels one wavelength in one period.         $v = \frac{\lambda}{T} = \lambda f$

  • Amplitude ( $A$ ): The maximum displacement from equilibrium of an element of the medium.

  • Angular Wave Number ( $k$ ):         $k = \frac{2\pi}{\lambda}$

  • Angular Frequency ( $\omega$ ):         $\omega = 2\pi f = \frac{2\pi}{T}$

Plane Electromagnetic Waves

• Definition and Characteristics

  • Electromagnetic (EM) waves consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation.

  • Direction: If a wave travels in the $x$ direction, the electric field $\mathbf{E}$ is in the $y$ direction and the magnetic field $\mathbf{B}$ is in the $z$ direction.

  • Spatial Dependence: Magnitudes $E$ and $B$ depend only on $x$ and $t$, not on $y$ or $z$ coordinates.

• Mathematical Representation

  • Sinusoidal Variation:         $E = E_{max} \sin(kx - \omega t)$         $B = B_{max} \sin(kx - \omega t)$

  • Ratio of Fields: The ratio of the angular frequency to the wave number equals the speed of light $c$.         $\frac{\omega}{k} = c$

  • Field Relationship: At every instant, the ratio of the magnitude of the electric field to the magnetic field equals the speed of light.         $\frac{E}{B} = \frac{E_{max}}{B_{max}} = c$

• Properties of Electromagnetic Waves

  1. Travel through empty space at the speed of light $c$.

  2. Electric and magnetic components are perpendicular to each other and to the direction of propagation.

  3. They are transverse waves.

  4. The magnitudes are related by $\frac{E}{B} = c$.

• Quick Quiz: Phase Difference

  • Question: What is the phase difference between the sinusoidal oscillations of the electric and magnetic fields?

  • Options: (a) 180 (b) 90 (c) 0 (d) Impossible to determine.

  • Answer: (c) 0.

The Electromagnetic Spectrum

• General Features

  • Produced by accelerating charges.

  • All forms of EM radiation travel at speed $c$ in a vacuum.

• Spectral Classifications

  1. Radio Waves

     • Wavelengths: More than $10^{4}\,m$ to about $0.1\,m$.

     • Frequencies: $3\,kHz$ to $3\,GHz$.

     • Source: Charges accelerating through conducting wires (e.g., LC oscillators).

     • Use: Radio and television communication.

     • Nature: Non-ionizing, very low photon energy.

  2. Microwaves

     • Wavelengths: $0.3\,m$ to $10^{-4}\,m$.

     • Frequencies: $3\,GHz$ to $300\,GHz$.

     • Source: Electronic devices.

     • Use: Radar systems, studying atomic/molecular properties, microwave ovens.

     • Energy: Higher photon energy than radio waves ($E = hf$) but still non-ionizing.

  3. Infrared (IR) Waves

     • Wavelengths: $10^{-3}\,m$ to $7 \times 10^{-7}\,m$.

     • Frequencies: $300\,GHz$ to $400\,THz$.

     • Source: Molecules and room-temperature objects.

     • Interaction: Readily absorbed by material; absorbed IR energy increases internal energy (vibrational/translational motion), resulting in temperature increase.

     • Use: Physical therapy, IR photography, vibrational spectroscopy.

  4. Visible Light

     • Definition: Part of the spectrum detectable by the human eye.

     • Source: Rearrangement of electrons in atoms and molecules.

     • Range: Red ($\lambda = 7 \times 10^{-7}\,m$) to Violet ($\lambda = 4 \times 10^{-7}\,m$).

     • Eye Sensitivity: Maximum sensitivity at $\lambda \approx 5.5 \times 10^{-7}\,m$ (yellow-green region).

  5. Ultraviolet (UV) Waves

     • Wavelengths: $4 \times 10^{-7}\,m$ to $6 \times 10^{-10}\,m$.

     • Source: The Sun is a major source.

     • Effects: Sunburn, formation of cataracts (clouding of the eye lens).

     • Protection: Sunscreens absorb UV; ozone ($O_3$) in the stratosphere absorbs most solar UV, converting it to IR radiation that warms the stratosphere.

  6. X-rays

     • Wavelengths: $10^{-8}\,m$ to $10^{-12}\,m$.

     • Source: Stopping high-energy electrons bombarding a metal target.

     • Use: Medical diagnostics, cancer treatment, studying crystal structures (wavelengths are comparable to atomic separation distances, $\approx 0.1\,nm$).

     • Safety: Can damage or destroy living tissue; must avoid overexposure.

  7. Gamma Rays

     • Wavelengths: $10^{-10}\,m$ to less than $10^{-14}\,m$.

     • Source: Radioactive nuclei (e.g., $^{60}Co$, $^{137}Cs$), nuclear reactions, and cosmic rays.

     • Nature: Highly penetrating, produces serious damage to living tissues.

     • Shielding: Requires heavy absorbing materials like thick lead layers.

Theoretical History of the Nature of Light

• Newton's Corpuscular Theory (17th Century)

  • Light composed of "corpuscles" (extremely small particles).

  • Properties: Travel in straight lines at high speeds, undergo elastic collisions with objects.

  • Success: Explained reflection and refraction.

  • Flaw: Assumed speed of light is higher in denser media (e.g., water) than air, which was later proven false.

• Huygens' Wave Theory (1678)

  • Light consists of waves.

  • Huygens' Principle: A wave front is a sphere centered on the source; every point on the front acts as a secondary source of disturbance.

  • Support: Thomas Young (interfering beams, 1801) and Maxwell (EM waves, mid-19th century).

  • Success: Explained reflection, refraction, interference, diffraction, and polarization.

• De Broglie's Dual Nature Theory (1924)

  • Contradiction: Late 19th/early 20th-century experiments (Black body radiation, photoelectric effect) indicated particle behavior.

  • Einstein (1905): Explained photoelectric effect using "photons".

  • Synthesis: De Broglie proposed light has a dual nature. It behaves as a wave under some conditions and a particle/photon under others.

Reflection

• Types of Reflection

  • Diffuse Reflection: Occurs on rough surfaces; incident parallel rays reflect in various directions.

  • Specular Reflection: Occurs on smooth, mirror-like surfaces; reflected rays remain parallel. A surface is considered "smooth" if surface variations are much smaller than the incident wavelength.

• Laws of Reflection

  • First Law: The incident ray, reflected ray, and normal to the surface all lie in the same plane perpendicular to the surface.

  • Second Law: The angle of incidence equals the angle of reflection.         $\theta_{1} = \theta_{1}^{\prime}$

Refraction

• General Principles

  • Occurs when light encounters a boundary between two transparent media.

  • Refraction is the bending of the ray as it enters the second medium.

  • The angle of refraction $\theta_{2}$ depends on medium properties and angle of incidence $\theta_{1}$.

• Laws of Refraction

  • First Law: Incident ray, reflected ray, refracted ray, and normal all lie in the same plane.

  • Second Law (Snell's Law): The ratio of indices of refraction is relates to the sines of the angles.         $\frac{\sin(\theta_{1})}{\sin(\theta_{2})} = \frac{v_{1}}{v_{2}} = \text{constant}$         $n_{1} \sin(\theta_{1}) = n_{2} \sin(\theta_{2})$

• Bending Behavior

  • High speed to lower speed (v_1 > v_2): \theta_2 < \theta_1 (Bends toward the normal).

  • Slow speed to higher speed (v_1 < v_2): \theta_2 > \theta_1 (Bends away from the normal).

• Index of Refraction ($n$)

  • Definition: Ratio of the speed of light in vacuum ($c$) to the speed in the material ($v$).         $n = \frac{c}{v}$

  • Dimensionless number; $n \ge 1$ because $v \le c$.

  • For vacuum, $n = 1$.

  • Representative Indices at $20^{\circ}C$ (Solids/Liquids):

    • Cubic zirconia: 2.20

    • Diamond: 2.419

    • Fluorite: 1.434

    • Fused quartz: 1.458

    • Glass (Crown): 1.52

    • Glass (Flint): 1.66

    • Ice ($0^{\circ}C$): 1.309

    • Water: 1.333

    • Benzene: 1.501

    • Ethyl alcohol: 1.361

    • Air ($0^{\circ}C, 1\,atm$): 1.000293

• Frequency and Wavelength in Media

  • When light enters a new medium, its frequency ($f$) remains constant, but speed ($v$) and wavelength ($\lambda$) change.

  • Equation: $v = \lambda f$ implies:         $\lambda_{1} n_{1} = \lambda_{2} n_{2}$         $\lambda_{n} = \frac{\lambda}{n}$     where $\lambda$ is the wavelength in vacuum and $\lambda_{n}$ is the wavelength in the medium.

Total Internal Reflection

• Condition for Occurrence

  • Light must travel from a denser medium ($n_1$) to a less dense medium ($n_2$).

  • As $\theta_{1}$ increases, the refracted ray bends away from the normal until it reaches the critical angle.

• Critical Angle ( $\theta_{cr}$ )

  • Definition: The angle of incidence that corresponds to an angle of refraction of $90^{\circ}$.         $n_{1} \sin(\theta_{cr}) = n_{2} \sin(90^{\circ})$         \sin(\theta_{cr}) = \frac{n_{2}}{n_{1}} \text{ (where } n_{1} > n_{2})

• Total Internal Reflection Phenomenon

  • If \theta_{1} > \theta_{cr}, the ray is entirely reflected back into the denser medium; no light is refracted.

Examples and Numerical Problems

• EM Wave Calculation (Example 10/11)

  • Given: $f = 40.0\,MHz$, travels in free space.

  • (A) Determine $\lambda$ and $T$:         $T = \frac{1}{f} = \frac{1}{4.00 \times 10^{7}\,s^{-1}} = 2.50 \times 10^{-8}\,s$         $\lambda = \frac{c}{f} = \frac{3.00 \times 10^{8}\,m/s}{4.00 \times 10^{7}\,s^{-1}} = 7.50\,m$

  • (B) Given $E_{max} = 750\,N/C$ along y-axis. Find $B$:         $B_{max} = \frac{E_{max}}{c} = \frac{750}{3.00 \times 10^{8}} = 2.50 \times 10^{-6}\,T$         Direction: Because propagation is along $x$ and $E$ is along $y$, $B$ must be along $z$.

• Laser in Compact Disc (Example 35.5)

  • Given: $\lambda_{air} = 780\,nm$, plastic disc $n = 1.55$.

  • Speed in plastic: $v = \frac{c}{n} = \frac{3.00 \times 10^{8}}{1.55} = 1.94 \times 10^{8}\,m/s$.

  • Wavelength in plastic: $\lambda_{n} = \frac{\lambda}{n} = \frac{780\,nm}{1.55} = 503\,nm$.

• Refraction Example (Page 39)

  • Given: $\lambda = 550\,nm$ in air ($n_{1} = 1.00$), incident angle $\theta_{1} = 40.0^{\circ}$, refracted angle $\theta_{2} = 26.0^{\circ}$.

  • Find $n_{2}$:         $n_{2} = \frac{n_{1} \sin(\theta_{1})}{\sin(\theta_{2})} = \frac{1.00 \sin(40.0^{\circ})}{\sin(26.0^{\circ})} \approx 1.47$

• Eye Sensitivity (Example 39)

  • Given: $\lambda = 5.50 \times 10^{-7}\,m$.

  • Find frequency: $f = \frac{c}{\lambda} = \frac{3.00 \times 10^{8}}{5.50 \times 10^{-7}} \approx 5.45 \times 10^{14}\,Hz$.

Questions & Discussion

• Quick Quiz 35.3: Light passes from material $n=1.3$ to $n=1.2$. Compared to incident ray, refracted ray: (a) bends toward normal (b) undeflected (c) bends away from normal.

  • Answer: (c). Light speed increases in the lower refractive index medium.

• Quick Quiz 35.6: Five light rays enter a glass prism. How many undergo total internal reflection at the slanted surface?

  • Answer: (b) 2. The other three rays exit as both reflected and refracted parts.

• Tennis Ball Inquiry: Why are tennis balls often yellow-green?

  • Context: Human eye sensitivity peaks at roughly $5.5 \times 10^{-7}\,m$, which corresponds to yellow-green.