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.