- #1
johnmadsen88
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Today, during class, our professor went through a simple example about Piezoelectricity. We have a cylinder with a trapped ideal gas and a piston which is part of a capacitor together with the bottom of the cylinder. The voltage across the capacitor is [tex]\Phi[/tex], and the charge is [tex]Q[/tex]. The distance between the piston and the bottom of the cylinder is [tex]l[/tex]. A potential for such a system could be
with [tex]k_B[/tex] and [tex]N[/tex] denoting the Boltzman constant and the number of molecules, respectively. [tex]\epsilon[/tex] is the dielectric constant. He then said that one can (I assume, by Legendre transformation) express this using another potential [tex]\mathcal{B}[/tex] which is relevant for variables [tex](T,Q,l)[/tex]:
I was wondering if someone could give this a more physical twist. In other words, I'd like it if someone explained where these contributions come from and what characterizes them. If I have been too unclear, please tell me, so I can reword my writing. Thanks.
PS: I cannot currently see my TeX code properly when previewing (either my computer or the site is having problems), but I hope I haven't made errors.
[tex]\mathcal{A}(T,l,\Phi )=-\frac{\epsilon A}{2l}\Phi ^2-Nk_BT\ln{Al}+\frac{3}{2}Nk_BT-\frac{3}{2}Nk_BT\ln{T}[/tex]
with [tex]k_B[/tex] and [tex]N[/tex] denoting the Boltzman constant and the number of molecules, respectively. [tex]\epsilon[/tex] is the dielectric constant. He then said that one can (I assume, by Legendre transformation) express this using another potential [tex]\mathcal{B}[/tex] which is relevant for variables [tex](T,Q,l)[/tex]:
[tex]\mathcal{B}(T,Q,l)=-\frac{1}{2\epsilon A}Q^2-Nk_BT\ln{Al}+\frac{3}{2}Nk_BT-\frac{3}{2}Nk_BT\ln{T}[/tex]
I was wondering if someone could give this a more physical twist. In other words, I'd like it if someone explained where these contributions come from and what characterizes them. If I have been too unclear, please tell me, so I can reword my writing. Thanks.
PS: I cannot currently see my TeX code properly when previewing (either my computer or the site is having problems), but I hope I haven't made errors.