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Thermodynamics in static electric and magnetic fields 1st law reads: dU dQ dW -so far focus on PVT-systems where dW PdV originates from mechanical work Now: -additional work terms for matter in fields Source of D is density of 1 Dielectric Materials A + -q Ve dielectric material +q L D D d3r D d 2 r DA VGauss VGauss free charges. Here: charge q on capacitor plate with area A d3r q VGauss Ve L q -displacement field D given by the free charges on the capacitor plates: D A -electric field inside the capacitor: E -Reduction of q Wcap Ve dq With Energy content in capacitor reduced which means work Wcap>0 done by the capacitor (in accordance with our sign convention for PVT systems) (dq<0 and Ve>0 yields Wcap>0) Ve dq E L A dD Ve dq VEdD V=volume of the dielectric material Wcap V E dD -When no material is present: still work is done by changing the field energy in the capacitor D 0E Wempty cap V 0 E dE -Work done by the material exclusively: parameterized e.g., with time (slow changes!) dE (t ) dD(t ) Wsys Wcap Wempty cap V EdD V 0 EdE V E (t ) 0 dt dt dt dE (t ) dD(t ) Wsys V E (t ) 0 dt dt dt With D 0 E P Polarization=total dipole moment per volume Wsys V E (t ) d P (t ) dt dt dW VEd P With dU dQ dW dU dQ EVd P (where V=const. is assumed so With V P : Pe we define the total dipole moment of the dielectric material that PdV has not to be considered ) Comparing dU dQ E dPe with dU dQ P dV Correspondence (where work is done mechanically via volume change against P) E P and Pe V -Legendre transformations (providing potentials depending on useful natural variables) dU TdS E dPe making electric field E variable d ( U EPe ) TdS Pe dE dU TdS d ( E Pe ) Pe dE dH TdS Pe dE H=H(S,E) dH TdS Pe dE making T variable d ( H TS ) SdT Pe dE dH d (TS ) SdT Pe dE G=G(T,E) G S T E and G Pe E T dG SdT Pe dE 2 Magnetic Materials R I dB Faraday’s law: E(r )dr dt N: # of turns of the wire A: cross sectional where B B d 2 r B A B here voltage Vind induced in 1 winding Ampere’s law: Hdr I tot where Itot N I here area of the ring magn. flux lines -Reduction of the current I work done by the ring dWring N Vind I work done by the ring per time dt Hdr 2 R H N I dWring dt A 2 R dB dB H Vring H dt dt makes sure that reduction of B ( dB / dt 0 ) corresponds to work done by the ring dWring / dt 0 I 2 R H N E(r )dr dB dB A dt dt -Again, when no material is present: still work is done on the source by changing the field energy In general: B 0 H M No material M=0 where M is the magnetization = magnetic dipole moment per volume B 0 H dWmm dWring dH Vring H 0 dt dt dt dH dB V H ring 0 dM dt dt Vring 0 H dt dt rate at which work is done by the magnetic material dWmm dM Vring 0 H dt dt B 0 H M Wmm dW V 0 HdM -Legendre transformations (providing potentials depending on useful natural variables) dU TdS 0VHdM making magnetic field H variable dU TdS 0Vd ( HdM ) 0VMdH d ( U 0VMH ) TdS 0VMdH Henth =Henth(S,H) dH enth TdS 0VMdH dHenth TdS 0VMdH making T variable dH enth d (TS ) SdT 0VMdH dG SdT 0VMdH G S T H d ( H enth TS ) SdT 0VMdH G=G(T,H) and M 1 G 0V H T