Two Questions from Electrostatics-- Capacitor Combinations

Here are two multiple choice questions on effective capacitance of capacitor networks. [The first question was found to be some what difficult for the students (from their response)]:

(1) An unusual type of capacitor is made using four identical plates P₁, P₂, P₃ and P_4, each of area A arranged in air with the same separation d as shown in the figure. A thin wire outside the system of plates connects the plates P₂ and P₄. Wires soldered to the plates P₁ and P₃ serve as terminals T₁ and T₂ of the system. What is the effective capacitance between the terminals T₁ and T₂?

(a) (2ε₀A)/3d

(b) (3ε₀A)/2d

(c) (3ε₀A)/d

(d) (ε₀A)/d

(e) (ε₀A)/2d

The arrangement contains 3 identlcal capacitors C₁, C₂ and C₃ each of capacitance (ε₀A)/d arranged as shown in the adjoining figure. Plate P₂ is common for C₁ and C₂. Likewise, plates P₃ is common for C₂ and C₃. The capacitors C₂ and C₃ are connected in parallel (to produce an effective value 2ε₀A/d) and this parallel combination is connected in series with the capacitor C₁ of value ε₀A/d . The effective capacitance C between the terminals T₁ and T₂ is therfore given by

C = [(2ε₀A/d) ×(ε₀A/d)] / [(2ε₀A/d) +(ε₀A/d)].

Therefore, C = (2ε₀A)/3d

(2) Four 2 μF capacitors and a 3 μF capacitor are connected as shown in the figure. The effective capacitance between the points A and B is
(a) 3 μF

(b) 2 μF

(c) 1.5 μF

(d) 1 μF

(e) 0.5 μF

This is a very simple question once you identify the circuit to be a balanced Wheatstone’s bridge. The 3 μF capacitor is connected between equipotential points and you can ignore it. Without the diagonal branch, there are four 2 μF capacitors only. The capacitors in the upper pair are in series and produce a value of 1 μF. The capacitors in the lower pair also produce a value of 1 μF. Since they are in parallel the effective capacitance between the points A and B is 2 μF.