BackA 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. 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. 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. 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. 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?
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?
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. If ℓ = 0.40 d, by what factor does the capacitance change when the sheet is inserted?
Determine the capacitance of the Earth, assuming it to be a spherical conductor.
(II) Use Gauss’s law to show that 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.]
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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(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. 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 = ∞.