Direct Energy, 2018a

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11 CALCULUS OF VARIATIONS 267 11.8. A pendulum converts kinetic energy to and from gravitational potential energy. As shown in the gure, a ball of mass m is hung by a string 1 m long. The pendulum is mounted on a base that is 3 m high. As shown in the gure, θ(t) is the angle of the pendulum. The kinetic energy of the ball is given by E kinetic = 1m ( ) dθ 2, 2 dt and the gravitational potential energy is given by E p.e. = mg (3 − cos θ). The quantity g is the gravitational constant, g =9.8 m s . 2 1 θ 3 3 − cos(θ) (a) Find L, the Lagrangian of the system. (b) Find H, the Hamiltonian of the system. (c) Set up the Euler-Lagrange equation, and use it to nd the equation of motion for θ(t), the angle of the pendulum as a function of time. (d) Show that energy is conserved in this system by showing that dH =0. dt The equation of motion found in part c is nonlinear, so don't try to solve it. Interestingly, it does have a closed form solution [164, Ch. 6]. (This problem is a modied version of an example in reference [163].) 11.9. As shown in the gure, an object of charge Q 1 and mass m moves near a stationary object with charge Q 2 . Assume the mass and the charges are constants, and assume the objects are surrounded by free space. The kinetic energy of the moving object is converted to or from energy stored in the electric eld between the objects. The kinetic energy of the moving object is given by 1m ( ) dx 2. 2 dt The energy of the electric eld is given by Q 1Q 2 where ɛ 4πɛ 0 x o is the permittivity of free space. The distance between the objects is given by x(t). (a) Find the Lagrangian of the system. (b) Find the generalized momentum.
268 11.8 Problems (c) Find the generalized potential. (d) Find the equation of motion for the path x(t) of the system. (Don't try to solve this nonlinear equation.) (e) Find the total energy of the system. (f) Show that energy is conserved in this system. x(t) Q 1 Q 2 moving stationary
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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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202 9.2 Measures of the Ability of
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212 9.3 Charge Flow in Batteries an
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214 9.3 Charge Flow in Batteries an
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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
Page 251 and 252: 240 10.5 Resistive Sensors capacito
Page 253 and 254: 242 10.6 Electrouidics However, ene
Page 255 and 256: 244 10.6 Electrouidics
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
Page 263 and 264: 252 11.5 Mass Spring Example not in
Page 265 and 266: 254 11.5 Mass Spring Example Path 1
Page 267 and 268: 256 11.5 Mass Spring Example We can
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
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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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Page 319 and 320: 308 13.5 From Thomas Fermi Theory t
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Page 323 and 324: 312 14.1 Introduction damental equa
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318 14.3 Continuous Symmetries and
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320 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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Direct Energy, 2018a

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