(III) Suppose one plate of a parallel-plate capacitor is tilted so it makes a small angle θ with the other plate, as shown in Fig. 24–29. Determine a formula for the capacitance C in terms of A, d, and θ, where A is the area of each plate and θ is small. Assume the plates are square. [Hint: Imagine the capacitor as many infinitesimal capacitors in parallel.]

A slab of width d and dielectric constant K is inserted a distance 𝓍 into the space between the square parallel plates (of side ℓ) of a capacitor as shown in Fig. 24–32. Determine, as a function of 𝓍, the capacitance.

A parallel-plate capacitor with plate area A = 2.0 m² and plate separation d = 3.0 mm is connected to a 45-V battery (Fig. 24–39a). Determine the charge on the capacitor, the electric field, the capacitance, and the energy U₀ stored in the capacitor.

Paper has a dielectric constant K = 3.7 and a dielectric strength of 15 x 10⁶ V/m. Suppose that a typical sheet of paper has a thickness of 0.11 mm. You make a “homemade” capacitor by placing a sheet of 21 cm x 14 cm paper between two aluminum foil sheets (Fig. 24–40) of the same size. About how much charge could you store on your capacitor before it would break down?

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(II) Consider the circuit shown in Fig. 26–67, where all resistors have the same resistance R. At t = 0, with the capacitor C uncharged, the switch is closed. At t = ∞, the currents can be determined by analyzing a simpler, equivalent circuit. Identify this simpler circuit and implement it in finding the values of I₁, I₂ and I₃ at t = ∞.

(II) Two different dielectrics fill the space between the plates of a parallel-plate capacitor as shown in Fig. 24–31. Determine a formula for the capacitance in terms of K₁, K₂, the area A of the plates, and the separation d₁ = d₂ = d/2. [Hint: Can you consider this capacitor as two capacitors in series or in parallel?]

(II) Consider the circuit shown in Fig. 26–67, where all resistors have the same resistance R. At t = 0, with the capacitor C uncharged, the switch is closed. At t = ∞, what is the potential difference across the capacitor?

The variable capacitance of an old radio tuner consists of four plates connected together placed alternately between four other plates, also connected together (Fig. 24–36). Each plate is separated from its neighbor by 1.6 mm of air. One set of plates can move so that the area of overlap of each plate varies from 2.0 cm² to 9.0 cm².

(a) Are these seven capacitors connected in series or in parallel?

(b) Determine the range of capacitance values.