# Thermodynamic relations for dielectrics in an electric field

Thermodynamic relations for dielectrics in an electric field Section 10 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 Thermodynamic Potentials

In vacuum, they are all the same, since P = S = 0, so we just used U 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 Heat function or Enthalpy H is used in situations of constant pressure

e.g. chemistry in a test tube Helmholtz Free Energy F is used in situations of constant temperature, e.g. sample in helium bath 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 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). 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 doesnt describe the thermodynamic state of an inhomogeneous body as a

whole. F = F[intrinsic variable (TBD), extrinsic variable (TBD), T] 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 doesnt penetrate.

Conductors thermodynamic state is irrelevant. Situation is the same as for vacuum U = F = H = G. 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. 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 Direction of normal here is out of the dielectric and into the conductor.

Surface charge on conductor is extraneous charge on the dielectric Work done to increase charge by de is dR = f de ~same f

Gauss Volume outside conductor =volume of dielectric, including any vacuum The varied field must satisfy the field equations Work done on dielectric due to an increase of

the charge on the conductor Volume outside conductor =volume of dielectric, including any vacuum First Law of Thermodynamics expresses 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) Then dR = dU|S 1 law for dielectric

st No PdV term, since V is not a good variable. It does not characterize the thermodynamic state of body, which becomes inhomogeneous in an E-field. As long as there are no temperature gradients, T does characterized the thermodynamic state of the dielectric and is a good variable.

Helmholtz free energy has T as an independent variable. Legendre transform Are all extrinsic quantities proportional to the volume of material

Define new intrinsic quantities per unit volume Integral over volume removed New one First law

Energy per unit volume is a function of mass density, too. Chemical potential referred to unit mass For gas we use mdN, where

m = chemical potential referred to one particle Free energy per unit volume is found, as before for F, by Legendre transform Electric field

F is the more convenient potential: It is easier to hold T constant than S Change the independent variable from D to E by Legendre Transformation E T, r

We can also write the energy functions of dielectric in terms of e and f on the conductors, instead of E and D in the dielectric. For several conductors Potential on ath conductor

Charge on ath conductor The extrinsic internal energy of the whole dielectric with E as the independent variable can now be written in terms of potentials on conductors as the variables

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 Variation of free energy at constant T = work done on the body

Extra charge brought to the ath conductor from infinity Potential of ath conductor (potential energy per unit charge) Variation of free energy, with E as variable (at constant T) can be written instead in terms of potentials on conductors as variables.

Similarly for And 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. In general case, E(D) is some arbitrary function of D. But for linear isotropic dielectrics, the relationship is simple

integrate = internal energy per unit volume of dielectric inte gra te

Free energy per unit volume of dielectric The term is the change in U for constant S and r due to the field and it is the change in F for constant T and r due to the field.

But both U and F energy functions need to be written in terms of D, which is their proper variable. For and , E is the independent variable, so

Difference is in sign, just as in section 5 for field energy of conductors in vacuum. For conductors in dielectric, this only holds if the dielectric is linear. Total free energy = integral over space of free energy per unit volume If dielectric fills all space outside conductors For given changes on conductors ea

Dielectric reduces the fa by factor 1/e Field energy also reduce by factor 1/e For given potentials on conductors fa maintained by battery Charges on conductors increased by factor e Field energy also increased by factor e

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