Thermodynamic relations for dielectrics in an electric field Section 10.
Thermodynamic relations for dielectrics in an electric field
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Thermodynamic relations for dielectrics in an electric field
Section 10
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Basic thermodynamics
• We always need at least 3 thermodynamic variables– One extrinsic, e.g. volume– One intrinsic, e.g. pressure– Temperature
• Because of the equation of state, only 2 of these are independent
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Thermodynamic Potentials
In vacuum, they are all the same, since P = S = 0, so we just used U
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Internal energy and Enthalpy
• U is used to express the 1st law (energy conservation) dU = TdS – PdV
= dQ + dR = Heat flowing in + work done on
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Heat function or Enthalpy
H is used in situations of constant pressuree.g. chemistry in a test tube
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Helmholtz Free Energy
• F is used in situations of constant temperature, e.g. sample in helium bath
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Gibbs Free Energy or Thermodynamic Potential
• G is used to describe phase transitions– Constant T and P
– G never increases– Equality holds for reversible processes– G is a minimum in equilibrium for constant T & P
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Irreversible processes at constant V and T
• dF is negative or zero.– F can only decrease– In equilibrium, F = minimum
• F is useful for study of condensed matter– Experimentally, it is very easy to control T, but it is
hard to control S• For gas F = F(V,T), and F seeks a minimum at constant V
& T, so gas sample needs to be confined in a bottle.• For solid, V never changes much (electrostriction).
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What thermodynamic variables to use for dielectric in an electric field?
• P cannot be defined because electric forces are generally not uniform or isotropic in the body.
• V is also not a good variable: it doesn’t describe the thermodynamic state of an inhomogeneous body as a whole.
• F = F[intrinsic variable (TBD), extrinsic variable (TBD), T]
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Why for conductors did we use only U?
• E = 0 inside the conductor.• The electric field does not change the
thermodynamic state of a conductor, since it doesn’t penetrate.
• Conductor’s thermodynamic state is irrelevant.
• Situation is the same as for vacuumU = F = H = G.
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Electric field penetrates a dielectric and changes its thermodynamic state
• What is the work done on a thermally insulated dielectric when the field in it changes?
• Field is due to charged conductors somewhere outside.
• A change in the field is due to a change in the charge on those conductors.
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Dielectric in an external field caused by some charged conductors
Simpler, but equivalent: A charged conductor surrounded by dielectric
Might be non-uniform and include regions of vacuum
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Electric induction exists in the dielectric
Conductor
Take Dn to be the component of D out of the dielectric and into the conductor.
Surface charge on conductor is extraneous charge on the dielectric
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Work done to increase charge by de is dR = f de
Volume outside conductor=volume of dielectric, including any vacuum
Gauss
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The varied field must satisfy the field equations
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Work done on dielectric due to an increase of the charge on the conductor
Volume outside conductor=volume of dielectric, including any vacuum
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First Law of Thermodynamics(conservation of energy)
• Change in internal energy = heat flowing in + work done on
• dU = dQ + dR = TdS + dR• For thermally insulated body, dQ = TdS = 0– Constant entropy
dR = dU|S
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1st law for dielectrics in an E-field
No PdV term, since V is not a good variable when body becomes inhomogeneous in an E-field.
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For uniform T, T is a good variable, and Helmholtz free energy is useful
Legendre transform
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Are all extrinsic quantities proportional to the volume of material
Define new intrinsic quantities per unit volume
Integral over volume removed
New one
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First law Energy per unit volume is a function of mass density, too.
Chemical potential referred to unit mass
For gas we had mdN, where m = chemical potential referred to one particle
Basis of thermodynamics of dielectrics
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Free energy
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F is the more convenient potential:It is easier to hold T constant than S
Electric field
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Define new potentials by Legendre Transformation
E T, r
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For conductor embedded in a dielectric
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For several conductors
Potential on ath conductor
Charge on ath conductor
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Extrinsic internal energy with E as a the independent variable
This is the same relation as (5.5) for conductors in vacuum, where mechanical energy in terms of ea was and in terms of fa was
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Variation of free energy at constant T = work done on the body
Potential of ath conductor(potential energy per unit charge)
Extra charge brought to the ath conductor from infinity
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Variation of free energy, with E as variable, at constant T
Similarly for And
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For T and ea constant, a body will undergo irreversible processes until is minimized. Then equilibrium is established.
For T and fa constant, a body will undergo irreversible processes until is minimized. Then equilibrium is established.
For S and ea constant, a body will undergo irreversible processes until is minimized. Then equilibrium is established.
For S and fa constant, a body will undergo irreversible processes until is minimized. Then equilibrium is established.
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Linear isotropic dielectrics
integrate
= internal energy per unit volume of dielectric
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inte
grat
e
Free energy per unit volume of dielectric
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The term
is the change in U for constant S and r due to the fieldand
it is the change in F for constant T and r due to the field.
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For and , E is the independent variable, so
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Difference is in sign, just as in section 5 for vacuum field energy. Result good only for linear dielectric
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Total free energy = integral over space of free energy per unit volume
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If dielectric fills all space outside conductorsFor given changes on conductors ea
Dielectric reduces the fa by factor 1/eField energy also reduce by factor 1/e
For given potentials on conductors fa maintained by batteryCharges on conductors increased by factor eField energy also increased by factor e