Cylindrical Capacitor in Dielectric Bath

The electrostatic field energy U_e of a capacitor can be expressed as a function of a parameter x (e.g., the plate separation) and the fixed plate charge (no charging battery present) or as a function of and of the electromotive force V_b of a battery to which the plates are connected.

a) Show that the generalized force F_ex corresponding to the parameter x is given by

b) Verify these formulae for the case of a parallel plate capacitor.

c) A cylindrical capacitor is lowered vertically into a reservoir of liquid dielectric of mass density ρ. If a voltage V is applied between the inner cylinder (radius a) and the outer shell (radius b), the liquid

Figure 1.1

rises to a height h between them (see Figure 1.1). Show that

SOLUTION

a) For the first case (fixed charge), the generalized force can be calculated as usual by considering the change in potential energy of the capacitor (field source) written in terms of the charge of the capacitor (which is a closed system). So, in this case,

(1)

If the capacitor is connected to the battery, it is no longer a closed system, and we have to consider the energy of the battery also. The battery must do some work to keep the potential of the plate constant. This work ∆W is

The energy change of the total system

Therefore, the force

(2)

b) The energy of the capacitor

where A is the area of the plates. From (1)

In the case of constant voltage

c) The capacitance of a cylindrical capacitor may be found by calculating the potential outside a uniformly charged cylinder. Gauss’s theorem gives

(3)

where λ is the linear charge density of the cylinder, and a is the radius of the cylinder. The potential of the outer cylinder of the capacitor in the problem is V. So

(4)

For a cylinder of length H, the capacitance may be found from (4) by substituting λ = Q/H:

(5)

(6)

 Figure 1.2

For the capacitor of length H filled with dielectric up to a height (see Figure 1.2)

Here, we use (2) to obtain

(7)

The liquid is drawn into the capacitor. The weight of the liquid in the capacitor

(8)

where π(b² – a²) h is the volume of the liquid drawn up between the cylinders and ρ is the mass density of the liquid. Equating (7) and (8), we get