Direct Energy, 2018a

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11 CALCULUS OF VARIATIONS 253 The Lagrangian represents the dierence between the forms of energy. L = E potential energy − E kinetic (11.32) ( L t, x, dx ) = 1 dt 2 Kx2 − 1 ( ) 2 dx 2 m (11.33) dt Both the Hamiltonian and Lagrangian have units ofjoules. The generalized potential is ∂L = Kx (11.34) ∂x in units ofnewtons. Note that Kx = − −→ F spring . The generalized momentum is M = ∂L ∂ ( ) = −m dx (11.35) dx dt dt in units of kg·m s which is the units ofmomentum. At t =0, the restraint is removed. The mass follows the path x(t). If we know the Lagrangian, we can nd the path by trial and error. To nd the path in this way, guess a path x(t) that the mass follows and calculate the action. ˆ t2 ( ) ∣ 2 1 dx S = ∣ t 1 2 m − 1 ∣∣∣∣ dt 2 Kx2 dt (11.36) Repeatedly guess another path, and calculate the action. The path with the least action ofall possible paths is the path that the mass follows. This path has the smallest dierence between the potential energy and the kinetic energy integrated over time. We can think ofmany possible, but not physical, paths x(t) that the mass can follow. Figure 11.2 illustrates two nonphysical paths as well as the physical path derived below. Paths are considered over the time interval 0
254 11.5 Mass Spring Example Path 1: x 1 (t) =2t − 1 , Action=0.355, Not physical Path 2: x 2 (t) =2t 2 − 1 , Action=0.364, Not physical Path 3: x 3 (t) =− cos(πt) , Action=0, Physical Figure 11.2: action. Possible paths taken by the mass and their corresponding
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Direct Energy Conversion by Andrea
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CONTENTS i Contents Contents i 1 In
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CONTENTS iii 6.3.2 Energy Levels in
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CONTENTS v 9.6 Fuel Cells . . . . .
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Acknowledgements I would like to th
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2 1.2 Preview of Topics to connect
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4 1.2 Preview of Topics Math throug
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6 1.2 Preview of Topics Process Spo
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8 1.2 Preview of Topics drodynamic
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10 1.4 Measures of Power and Energy
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12 1.5 Properties of Materials 1.5
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14 1.5 Properties of Materials most
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16 1.6 Electromagnetic Waves −→
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18 1.6 Electromagnetic Waves Relate
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20 1.6 Electromagnetic Waves A unif
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22 1.7 Problems
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24 2.2 Capacitors 2.2 Capacitors 2.
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26 2.2 Capacitors 2.2.3 Permittivit
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28 2.2 Capacitors â z â y â x No
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30 2.2 Capacitors Figure 2.3: Natur
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32 2.3 Piezoelectric Devices Piezoe
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34 2.3 Piezoelectric Devices Lattic
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36 2.3 Piezoelectric Devices Herman
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38 2.3 Piezoelectric Devices Figure
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40 2.3 Piezoelectric Devices c a β
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42 2.3 Piezoelectric Devices In cer
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44 2.3 Piezoelectric Devices made m
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46 2.3 Piezoelectric Devices tions.
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48 2.4 Problems 2.5. A piezoelectri
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50 2.4 Problems 2.13. Consider a pi
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52 2.4 Problems 2.16. The gure belo
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54 3.2 Pyroelectricity Material Che
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56 3.3 Electro-Optics KH 2 PO 4 [25
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58 3.3 Electro-Optics The rst two t
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60 3.3 Electro-Optics used as an el
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62 3.4 Notation Quagmire Notation i
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64 3.5 Problems 3.5 Problems 3.1. F
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66 3.5 Problems −→ DinC/m 2 unp
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68 4.1 Introduction Center-fed half
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70 4.2 Electromagnetic Radiation In
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72 4.2 Electromagnetic Radiation ar
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74 4.3 Antenna Components and Denit
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76 4.4 Antenna Characteristics ante
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78 4.4 Antenna Characteristics Impe
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80 4.4 Antenna Characteristics Azim
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82 4.4 Antenna Characteristics 4.4.
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84 4.4 Antenna Characteristics 5 Li
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86 4.5 Problems Figure 4.6: A snow
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88 4.5 Problems 4.5. Match the foll
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90 4.5 Problems 4.8. Radiation patt
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92 5.2 Physics of the Hall Eect The
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94 5.2 Physics of the Hall Eect The
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96 5.3 Magnetohydrodynamics This am
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98 5.5 Applications of Hall Eect De
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100 5.6 Problems 5.6 Problems 5.1.
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102 6.2 The Wave and Particle Natur
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104 6.3 Semiconductors and Energy L
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120 6.4 Crystallography Revisited L
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122 6.5 Pn Junctions E Indirect Sem
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124 6.5 Pn Junctions an excess of n
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126 6.5 Pn Junctions V x - + Energy
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128 6.6 Solar Cells to day and loca
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130 6.6 Solar Cells dot based mater
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132 6.7 Photodetectors Solar Panel
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134 6.7 Photodetectors 6.7.2 Measur
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6 PHOTOVOLTAICS 137 6.3. The gure i
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7 LAMPS, LEDS, AND LASERS 139 7 Lam
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7 LAMPS, LEDS, AND LASERS 141 Photo
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7 LAMPS, LEDS, AND LASERS 143 Absor
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7 LAMPS, LEDS, AND LASERS 145 We ca
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7 LAMPS, LEDS, AND LASERS 147 If we
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7 LAMPS, LEDS, AND LASERS 149 tube
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7 LAMPS, LEDS, AND LASERS 151 easil
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7 LAMPS, LEDS, AND LASERS 153 Power
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7 LAMPS, LEDS, AND LASERS 155 Two L
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7 LAMPS, LEDS, AND LASERS 157 but n
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7 LAMPS, LEDS, AND LASERS 159 of ec
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7 LAMPS, LEDS, AND LASERS 161 the o
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7 LAMPS, LEDS, AND LASERS 163 Ti:Sa
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7 LAMPS, LEDS, AND LASERS 165 the b
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7 LAMPS, LEDS, AND LASERS 167 Gas D
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7 LAMPS, LEDS, AND LASERS 169 categ
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7 LAMPS, LEDS, AND LASERS 171 7.8.
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8 THERMOELECTRICS 173 8 Thermoelect
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8 THERMOELECTRICS 175 Symbol Name V
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8 THERMOELECTRICS 177 per unit volu
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8 THERMOELECTRICS 179 kPa, and the
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8 THERMOELECTRICS 181 Seebeck Effec
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8 THERMOELECTRICS 183 gradient acro
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8 THERMOELECTRICS 185 thermocouples
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8 THERMOELECTRICS 187 The gure of m
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8 THERMOELECTRICS 189 or mass invol
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8 THERMOELECTRICS 191 temperatures?
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8 THERMOELECTRICS 193 Figure 8.4: L
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8 THERMOELECTRICS 195 8.9 Problems
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8 THERMOELECTRICS 197 8.8. A thermo
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8 THERMOELECTRICS 199 Figure 8.5 8.
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Page 227 and 228: 216 9.4 Measures of Batteries and F
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Page 241 and 242: 230 9.6 Fuel Cells Fuel cell compon
Page 243 and 244: 232 9.6 Fuel Cells 9.6.3 Practical
Page 245 and 246: 234 9.7 Problems 9.7 Problems 9.1.
Page 247 and 248: 236 9.7 Problems A E - + I B D H 2
Page 249 and 250: 238 10.3 Radiation Detectors high e
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Page 253 and 254: 242 10.6 Electrouidics However, ene
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Page 257 and 258: 246 11.2 Lagrangian and Hamiltonian
Page 259 and 260: 248 11.3 Principle of Least Action
Page 261 and 262: 250 11.4 Derivation of the Euler-La
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Page 269 and 270: 258 11.6 Capacitor Inductor Example
Page 271 and 272: 260 11.6 Capacitor Inductor Example
Page 273 and 274: 262 11.7 Schrödinger's Equation 11
Page 275 and 276: 264 11.8 Problems 11.4. Figure 11.2
Page 277 and 278: 266 11.8 Problems a famous result k
Page 279 and 280: 268 11.8 Problems (c) Find the gene
Page 281 and 282: 270 12.2 Electrical Energy Conversi
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Page 299 and 300: 288 12.6 Problems 12.6 Problems 12.
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Page 303 and 304: 292 13.1 Introduction • Describe
Page 305 and 306: 294 13.2 Preliminary Ideas Assuming
Page 307 and 308: 296 13.3 Derivation of the Lagrangi
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304 13.3 Derivation of the Lagrangi
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306 13.4 Deriving the Thomas Fermi
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308 13.5 From Thomas Fermi Theory t
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310 13.6 Problems (a) Write the Ham
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312 14.1 Introduction damental equa
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314 14.2 Types of Symmetries be wri
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316 14.3 Continuous Symmetries and
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322 14.4 Derivation of the Innitesi
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330 14.5 Invariants and we end up w
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332 14.5 Invariants in the limit ε
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334 14.5 Invariants Next use Eq. 14
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336 14.6 Summary 14.6 Summary In th
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338 14.7 Problems 14.6. The equatio
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340 14.7 Problems
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342 Appendices Symbol Quantity Unit
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350 Appendices Prex name Symbol Val
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352 Appendices for voltage. (As an
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354 Appendices Material or Device S
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356 REFERENCES [15] E. C. Jordan an
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358 REFERENCES [45] V. K. Tikhomiro
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360 REFERENCES [73] M. G. Thomas, H
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362 REFERENCES [100] A. Ishibashi,
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364 REFERENCES [125] D. Wright, Req
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366 REFERENCES [150] J. P. Owejan,
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368 REFERENCES [175] G. K. Woodgate
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Index Absorption, 139 Action, 247 A
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372 INDEX Nernst equation, 220 Neur
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About the Book Direct Energy Conver
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Direct Energy, 2018a

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