Question

Consider a simple electrostatic capacitor constructed from two electrodes consisting of perfect conductors, where the electric charge Q on the positively-charged conductor is related to the potential difference V between the electrodes by Q=CV. Suppose the potential difference V of the capacitor is held fixed, by attaching its electrodes via very thin, perfectly conducting leads to the terminals of an ideal battery. Define a free energy G for the capacitor, such that dG=đW for work done by the external agent with the capacitor's voltage held at fixed potential difference V. Show that G may be regarded as a suitable Legendre transformation of the electrostatic energy U.

Answer 1

consider a simple electrostatic capacitor
consisting of two electrodes with charge seperation Q
potential difference between the plates = V

hence for capacitance C
Q = CV

now consider V is held fixed by connecting the electrodes to a battery
free energy for the capacitor = G

now dG = dW

now electrostatic energy of the capacitor is given by U
U = 0.5CV^2

consider f(C) = 0.5CV^2 ( as V is held constant)
then
f'(C) = 0.5V^2

now
legrande transformation of f will be f*
also
f*'(f'(C)) = C
f*'(0.5V^2) = C
f*(0.5V^2) = C*dV + k

also
f'(f*'(0.5V^2) = 0.5V^2
hence
f(f*(0.5V^2)) = 0.5V^2*dC + k'

hence
comparing
we have legrande transformation of the electrostatic energy as
f* = 0.5V^2*dC + CdV

now, dG = dW = dU = 0.5(dC)V^2 + 0.5C(dV^2) = 0.5V^2*dC + CdV
hence
dG = 0.5V^2*dC + CdV = f*
hence dG can be used as legrande transformation of the electrostatic energy U of the capacitor