26. Capacitors & Dielectrics

Two circular plates of radius 2cm are brought together so their separation is 5mm. What is the capacitance of these plates?

A 3 F capacitor is given a potential difference across its plates of 10 V. What is the charge built up on its plates? If the source of the potential difference across the plates is removed, but the plates maintain their charge, what is the new potential difference across the capacitor if the distance between the plates is doubled?

A parallel-plate air capacitor of capacitance 245 pF has a charge of magnitude 0.148 uC on each plate. The plates are 0.328 mm apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electricfield magnitude between the plates? (d) What is the surface charge density on each plate?

The plates of a parallel-plate capacitor are 2.50 mm apart, and each carries a charge of magnitude 80.0 nC. The plates are in vacuum. The electric field between the plates has a magnitude of 4.00x10^6 V/m. What is (a) the potential difference between the plates; (b) the area of each plate; (c) the capacitance?

(II) An isolated capacitor C₁ carries a charge Q₀. Its wires are then connected to those of a second capacitor C₂, previously uncharged. What charge will each carry now? What will be the potential difference across each?

(I) What is the capacitance per unit length (F/m) of a coaxial cable whose inner conductor has a 1.0-mm diameter and the outer cylindrical sheath has a 5.0-mm diameter? Assume the space between is filled with air.

(II) To get an idea how big a farad is, suppose you want to make a 1-F air-filled parallel-plate capacitor for a circuit you are building. To make it a reasonable size, suppose you limit the plate area to 1.0 cm². What would the gap have to be between the plates? Is this practically achievable?

(III) Small distances can be measured using a capacitor whose plate separation 𝓍 is variable. Consider an air-filled parallel-plate capacitor with fixed plate area A = 25 mm² separated by a variable distance 𝓍. Assume this capacitor is attached to a capacitance-measuring instrument which can measure capacitance C in the range 1.0 pF to 1000.0 pF with an accuracy of ∆C = 0.1 pF .

(a) If C is measured while 𝓍 is varied, over what range ( 𝓍ₘᵢₙ ≤ 𝓍 ≤ 𝓍ₘₐₓ) can the plate-separation distance (in μm) be determined by this setup?

(III) Small distances can be measured using a capacitor whose plate separation 𝓍 is variable. Consider an air-filled parallel-plate capacitor with fixed plate area A = 25 mm² separated by a variable distance 𝓍. Assume this capacitor is attached to a capacitance-measuring instrument which can measure capacitance C in the range 1.0 pF to 1000.0 pF with an accuracy of ∆C = 0.1 pF .

(b) Define ∆𝓍 to be the accuracy (magnitude) to which 𝓍 can be determined, and determine a formula for ∆𝓍.

(III) Small distances can be measured using a capacitor whose plate separation 𝓍 is variable. Consider an air-filled parallel-plate capacitor with fixed plate area A = 25 mm² separated by a variable distance 𝓍. Assume this capacitor is attached to a capacitance-measuring instrument which can measure capacitance C in the range 1.0 pF to 1000.0 pF with an accuracy of ∆C = 0.1 pF .

(c) Determine the percent accuracy to which 𝓍ₘᵢₙ and 𝓍ₘₐₓ can be measured.

A capacitor is made from two 1.1-cm-diameter coins separated by a 0.10-mm-thick piece of paper (K = 3.7) . A 12-V battery is connected to the capacitor. How much charge is on each coin?

A parallel-plate capacitor has square plates 12 cm on a side separated by 0.10 mm of plastic with a dielectric constant of K = 3.8. The plates are connected to a battery, causing them to become oppositely charged. Since the oppositely charged plates attract each other, they exert a pressure on the dielectric. If this pressure is 40.0 Pa, what is the battery voltage?

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.

(b) The plates are now pulled to a separation of 0.85 mm. What is the charge on the capacitor now?

A student wearing shoes with thin insulating soles is standing on a grounded metal floor when he puts his hand on the metal case of a high voltage supply. The voltage inside, 6.3 mm from his hand, is 25,000 V. The student’s hand and the voltage supply form a capacitor, considering the student as a conductor. Another capacitor is between the floor and the student (his feet). Using reasonable numbers for hand and foot areas, estimate the student’s voltage relative to the floor. Assume vinyl-soled shoes 1 cm thick.

In lightning storms, the potential difference between the Earth and the bottom of the thunderclouds can be as high as 35,000,000 V. The bottoms of thunderclouds are typically 1500 m above the Earth, and can have an area of 110 km². Modeling the Earth–cloud system as a huge capacitor, calculate

(a) the capacitance of the Earth–cloud system,