6.1 Electromagnetic Waves

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1 6.1 Electromagnetic Waves electromagnetic radiation can be described as a harmonic wave in either time or distance waves are characterized by their period, frequency, wavelength and wave number Planck's Law gives the energy of electromagnetic quanta ν and λ are related by the speed of light, which in turn is determined by the refractive index electromagnetic radiation is characterized by both an electric field and magnetic field 6.1 : 1/11

2 Temporal Harmonic Wave An electromagnetic wave can be described in either. When amplitude is measured as the wave passes a fixed location the result is a temporal wave. amplitude (V/m) 1 t The wave is given by -1 time (fs) where t is the period. Usually the expression is written using frequency, where ν = 1/t. ν has units of. In some cases it is convenient to use angular frequency, ω = 2πν. ω also has units of (which may be called radians per second). ( πν ) ( ω ) E = E cos 2 t E = E cos t 6.1 : 2/11

3 Spatial Harmonic Wave When a "snapshot" of the wave is taken, the wave will extend over distance in the laboratory. The wave is given by amplitude (V/m) 1 λ where λ is the wavelength. In the visible λ has units of, in the infrared units of. -1 distance (nm) 6.1 : 3/11 ν = 1 λ Sometimes the expression is written using wave number, has units of reciprocal centimeters (cm -1 ). In physical optics it is convenient to use the k-vector, k = 2π/λ. The units of k are usually reciprocal meters (m -1 ). ν ( πν ) ( ) E = E cos 2 x E = E cos kx

4 Plane, Spherical and Cylindrical Waves a three dimensional plane wave is given by ( ) ( ) ( ) E = E cos k x + E cos k y + E cos k z x x y y z z a spherical wave is given by E = E cos( kr) r a cylindrical wave is given by E = E cos( kr) kr z r plane wave spherical wave 6.1 : 4/11

5 Energy per Quantum The energy per quantum is given by Planck's Law, hc E = hν = = hcν λ where h = J s. Note that E is proportional to ν and, but not λ! The concept of a quantum is only relevant when the energy delivered is above the thermodynamic noise given by 4kT, where k = J K -1. At room temperature this limit is given approximately by, 4kT 13 ν = = Hz h which is. For a bond energy of 4 kj mol -1, the bond energy per molecule is J. This is delivered by one quantum of 1 15 Hz (3 nm) radiation. All higher frequencies are considered. ν 6.1 : 5/11

6 Refractive Index For electromagnetic radiation, frequency and wavelength are related by the speed of light, c = m s -1. When electromagnetic radiation travels through material with polarizable electrons, it slows down. Because the quantum of energy remains fixed, the frequency remains fixed. This means that the wavelength decreases. The refractive index is given by the ratio of velocities,. 6.1 : 6/11 substance n 589 substance n 589 perfluorohexane 1.25 quartz 1.46 water 1.33 benzene 1.5 ethanol 1.36 glass 1.52 carbon disulfide 1.63

7 Dispersion The refractive index depends upon the wavelength, ordinarily from the violet toward the red. The change in value is called dispersion. Shown at the right are dispersion curves for three materials used to make prisms. Dispersion is greatest near an. Quartz is a good prism material for the UV, while flint glass is good for the visible and near IR. refractive index 1.75 flint glass crown glass 1.5 quartz wavelength (nm) 6.1 : 7/11

8 Propagation of Light Maxwell's equations show that a time-varying electric field (E in V m -1 ) generates a time varying magnetic field (H in A m -1 ), and that a time varying magnetic field generates a time varying electric field. Thus, electromagnetic radiation is. When E is aligned in the x-direction and the wave is traveling in the z-direction, the generated magnetic field is in the y-direction. ( π λ) ( π λ) E = E cos 2 z/ H = H cos 2 z/ x In a vacuum the ratio of E x and H y are given by the impedance of free space. E H x = 1 ε c = y Note that the two waves are in phase and that the direction of propagation is given by E H. The radiant flux density transmitted in the z-direction is given by E x H y in units of W m -2. When intensity is used the flux is given by photons s -1 m -2. y 6.1 : 8/11

9 Electromagnetic Spectrum frequency (Hz) gamma rays ultraviolet infrared radio x-rays visible microwaves wavelength (m) : 9/11

10 Spectral Regions and Phenomena (1) radio waves: below 1 GHz, natural units Hz P nuclear magnetic resonance, used to determine bonding in molecules P electron spin resonance, used to determine location of unpaired electrons microwaves: 1 GHz to 1 THz, natural units Hz or cm -1 P, used to determine bond angles and lengths of small molecules in the gas phase infrared (below red): 1, cm -1 to 5 cm -1, natural units cm -1 or μm P of molecules, used to determine functional groups existing within a molecule visible: 4 nm to 75 nm, natural units nm or cm -1 P valence shell electronic transitions, used to identify atoms in a sample and concentration P molecular electronic transitions involving, used to determine structure and concentration 6.1 : 1/11

11 Spectral Regions and Phenomena (2) ultraviolet (beyond violet): 1 nm to 38 nm, natural units nm or D P valence shell, used to identify atoms in a sample and their concentration P molecular electronic transitions involving single bonds, used to determine structure and concentration x-rays:.1 D to 1 D, natural units D P, used to identify atoms in solids and provide information about adjacent bonded atoms gamma rays: greater than.1 MeV, natural units electron volts (1 ev = J) P, used to identify isotopes 6.1 : 11/11

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