26. Capacitors & Dielectrics

Which of the following correctly expresses the capacitance of an air-filled parallel plate capacitor with plate area $A$ and separation $d$?

(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?

Which of the following will increase the capacitance $C$ of a parallel-plate capacitor?

If you have two identical capacitors, each with capacitance $C$, and an external potential source, what is the equivalent capacitance when the capacitors are connected in series?

A cylindrical capacitor (Example 24–2) has R_a = 3.5 mm and R_b = 0.50 mm. The two conductors have a potential difference of 625 V, with the inner conductor at the higher potential. Calculate the electric field at the surface of the inner conductor.

When two capacitors are connected in series, which of the following quantities must be the same for both capacitors?

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.

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. 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?

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.

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.

A switch that connects a battery to a 10 μF capacitor is closed. Several seconds later you find that the capacitor plates are charged to ±30 μC. What is the emf of the battery?

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 the voltage now across each capacitor.

Two 3.0-cm-diameter aluminum electrodes are spaced 0.50 mm apart. The electrodes are connected to a 100 V battery. What is the magnitude of the charge on each electrode?

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 R_a and R_b, 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 (K_liq) and its vapor (K_V), 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, R_a = 5.0 mm, and R_b = 4.5 mm. For liquid nitrogen (K_liq = 1.4, K_V = 1.0), what values of C (in pF) will correspond to the tank being completely full and completely empty?

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. What is now the net capacitance in terms of A, d, and ℓ?