Download Thermodynamics in static electric and magnetic fields

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
dB
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  
dB
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