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Ideal dielectric gas in a capacitor

  • Thread starter neworder1
  • Start date
  • #1
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Homework Statement



Ideal dielectric gas is in a container closed by a movable piston and in thermal contact with its surroundings, so is kept at constant tempertaure [tex]T_0[/tex] and pressure [tex]p_0[/tex]. Inside there is a capacitor with fixed voltage and total electric field [tex]E[/tex]. The gas has permittivity [tex]\epsilon(n,T) = 1 + n\alpha(T)[/tex], where [tex]n[/tex] is the density of gas in the capacitor and [tex]\alpha[/tex] is some function of temperature.

Find equilibrium value of [tex]n[/tex].

Homework Equations



Capacitor energy [tex]U = \frac{1}{2}\epsilon E^2 V_{cap}[/tex]

The Attempt at a Solution



With constant pressure and temperature, the quantity minimized at equilibrium is Gibbs free energy, so at equilibrium chemical potentials [tex]\mu_1[/tex] and [tex]\mu_2[/tex] of the gas inside and outside the capacitor must be equal. While we can find [tex]\mu_2[/tex] easily, since this is an ideal gas, I'm not sure about [tex]\mu_2[/tex].
 

Answers and Replies

  • #2
45
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For the gas in the capacitor you should add to the Gibbs free energy a term describing the electric field:
[tex]
dG' = dG + \frac{1}{4\pi}V_{\text{cap}}\vec{E}d\vec{D}
[/tex]
This additional term is proportional to dn and makes a contribution to the chemical potential.
 

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