26. Capacitors & Dielectrics

Two 2.0 cm×2.0 cm metal electrodes are spaced 1.0 mm apart and connected by wires to the terminals of a 9.0 V battery.

a. What are the charge on each electrode and the potential difference between them?

Two 3.0-cm-diameter aluminum electrodes are spaced 0.50 mm apart. The electrodes are connected to a 100 V battery.

a. What is the capacitance?

Two 3.0-cm-diameter aluminum electrodes are spaced 0.50 mm apart. The electrodes are connected to a 100 V battery.

b. What is the magnitude of the charge on each electrode?

(II) A cylindrical capacitor (Example 24–2) has Rₐ = 3.5 mm and R₆.= 0.50 mm. The two conductors have a potential difference of 625 V, with the inner conductor at the higher potential.

(c) Calculate the electric field at the surface of the inner conductor.

(II) How strong is the electric field between the plates of a 0.80-μF air-gap capacitor if they are 2.0 mm apart and each has a charge of magnitude 84-μC?

The long cylindrical capacitor shown in Fig. 24–37 consists of four concentric cylinders, with respective radii Ra, R₆ , R꜀ and Rₔ. The cylinders b and c are joined by metal strips. Determine the capacitance per unit length of this arrangement. (Assume equal and opposite charges are placed on the innermost and outermost cylinders.)

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(I) What is the capacitance of a pair of circular plates with a radius of 5.0 cm separated by 2.3 mm of mica?

A parallel-plate capacitor with plate area 2.0 cm² and air-gap separation 0.50 mm is connected to a 12-V battery, and fully charged. The battery is then disconnected.

(a) What is the charge on the capacitor?

(I) The two plates of a capacitor hold +3500 μC and ―3500μC of charge, respectively, when the potential difference is 960 V. What is the capacitance?

(II) Two identical capacitors are connected in parallel and each acquires a charge Q₀ when connected to a source of voltage V₀. The voltage source is disconnected and then a dielectric (K = 3.6) is inserted to fill the space between the plates of one of the capacitors. Determine

(b) the voltage now across each capacitor.

(a) A general rule for estimating the capacitance C of an isolated conducting sphere with radius r is C (in pF) ≈ r (in cm). That is, the numerical value of C in pF is about the same as the numerical value of the sphere’s radius in cm. Justify this rule.

Capacitors can be used as “electric charge counters.” Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons.

(a) If N electrons flow onto the capacitor’s top plate, show that the resulting potential difference V across the capacitor is directly proportional to N.

Capacitors can be used as “electric charge counters.” Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons.

(b) Assume a voltage-measuring device can accurately resolve voltage changes of about 1 mV. What value of C would be necessary to resolve the arrival of an individual electron?

In the dynamic random access memory (DRAM) of a cell phone, each memory cell contains a capacitor for charge storage. Each of these cells represents a single binary-bit value of “1” when its 25-fF capacitor (1 fF = 10⁻¹⁵ F )is charged at 0.6 V, or “0” when uncharged at 0 V.

(a) When fully charged, how many excess electrons are on a cell capacitor’s negative plate?

Suppose it takes 75 kW of power for your car to travel at a constant speed on the highway.

(d) If this capacitor were to be made from activated carbon (Section 24–2), the voltage would be limited to no more than 10 V. In this case, how many grams of activated carbon would be required?

(e) Is this practical?

(III) A large metal sheet of thickness ℓ is placed between, and parallel to, the plates of the parallel-plate capacitor of Fig. 24–4. It does not touch the plates, and extends beyond their edges.

(a) What is now the net capacitance in terms of A, d, and ℓ?

(III) A large metal sheet of thickness ℓ is placed between, and parallel to, the plates of the parallel-plate capacitor of Fig. 24–4. It does not touch the plates, and extends beyond their edges.

(b) If ℓ = 0.40 d, by what factor does the capacitance change when the sheet is inserted?

(I) Determine the capacitance of the Earth, assuming it to be a spherical conductor.

(II) Use Gauss’s law to show that E (→ above E) = 0 inside the inner conductor of a cylindrical capacitor (see Fig. 24–7 and Example 24–2) as well as outside the outer cylinder.

(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.]

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(III) 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,

(a) the capacitance,

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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).

(a) Determine the charge on the capacitor, the electric field, the capacitance, and the energy U₀ stored in the capacitor.

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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.

(b) About how much charge could you store on your capacitor before it would break down?

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(I) Estimate the value of resistances needed to make a variable timer for intermittent windshield wipers: one wipe every 15 s, 8 s, 4 s, 2 s, 1 s. Assume the capacitor used is on the order of 1 μF. See Fig. 26–64.

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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.

(b) 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 = ∞ .

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(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?]

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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.

(c) At t = ∞, what is the potential difference across the capacitor?

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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. <IMAGE>

(III) The quantity of liquid (such as cryogenic liquid nitrogen) available in its storage tank is often monitored by a capacitive level sensor. This sensor is a vertically aligned cylindrical capacitor with outer and inner conductor radii Rₐ and R₆, whose length ℓ spans the height of the tank. When a nonconducting liquid fills the tank to a height h ( ≤ ℓ ) from the tank’s bottom, the dielectric in the lower and upper regions between the cylindrical conductors is the liquid (Kₗᵢq) and its vapor (Kᵥ), respectively (Fig. 24–33). (a) Determine a formula for the fraction F of the tank filled by liquid in terms of the level-sensor capacitance C. [Hint: Consider the sensor as a combination of two capacitors.] (b) By connecting a capacitance-measuring instrument to the level sensor, F can be monitored. Assume the sensor dimensions are ℓ = 2.0 m , Rₐ = 5.0 mm, and R₆ = 4.5 mm. For liquid nitrogen (Kₗᵢq = 1.4, Kᵥ = 1.0), what values of C (in pF) will correspond to the tank being completely full and completely empty?

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