Chapter 24

The plates of a parallel-plate capacitor are 2.50 \(\mathrm{mm}\) apart, and each carries a charge of magnitude 80.0 \(\mathrm{nC}\) . The plates are in vacuum. The electric field between the plates has a magnitude of \(4.00 \times 10^{6} \mathrm{V} / \mathrm{m}\) . (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the capacitance?

The plates of a parallel-plate capacitor are 3.28 \(\mathrm{mm}\) apart, and each has an area of 12.2 \(\mathrm{cm}^{2} .\) Each plate carries a charge of magnitude \(4.35 \times 10^{-8} \mathrm{C}\) . The plates are in vacuum. (a) What is the capacitance? (b) What is the potential difference between the plates? (c) What is the magnitude of the electric field between the plates?

A parallel-plate air capacitor of capacitance 245 pF has a charge of magnitude 0.148\(\mu \mathrm{C}\) on each plate. The plates are 0.328 \(\mathrm{mm}\) apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electric- field magnitude between the plates? (d) What is the surface charge density on each plate?

Capacitance of an Oscilloscope. Oscilloscopes have parallel metal plates inside them to deflect the electron beam. These plates are called the deffecting plates. Typically, they are squares 3.0 \(\mathrm{cm}\) on a side and separated by \(5.0 \mathrm{mm},\) with vacuum in between. What is the capacitance of these deflecting plates and hence of the oscilloscope? (Note: This capacitance can sometimes have an effect on the circuit you are trying to study and must be taken into consideration in your calculations.)

A \(10.0-\mu \mathrm{F}\) parallel-plate capacitor with circular plates is connected to a 12.0 -V battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the \(12.0-\mathrm{V}\) battery after the radius of each plate was doubled without changing their separation?

A 10.0 -\muF parallel-plate capacitor is connected to a 12.0 -V battery. After the capacitor is fully charged, the battery is disconnected without loss of any of the charge on the plates. (a) A voltmeter is connected across the two plates without discharging them. What does it read? (b) What would the voltmeter read if (i) the plate separation were doubled; (ii) the radius of each plate were doubled but their separation was unchanged?

A \(5.00-\) pF, parallel-plate, air-filled capacitor with circular plates is to be used in a circuit in which it will be subjected to potentials of up to \(1.00 \times 10^{2} \mathrm{V}\) . The electric field between the plates is to be no greater than \(1.00 \times 10^{4} \mathrm{N} / \mathrm{C}\) . As a budding electrical engineer for Live-Wire Electronics, your tasks are to (a) design the capacitor by finding what its physical dimensions and separation must be; ( b) find the maximum charge these plates can hold.

A parallel-plate air capacitor is to store charge of magnitude 240.0 \(\mathrm{pC}\) on each plate when the potential difference between the plates is 42.0 \(\mathrm{V}\) (a) If the area of each plate is \(6.80 \mathrm{cm}^{2},\) what is the separation between the plates? (b) If the separation between the two plates is double the value calculated in part (a), what potential difference is required for the capacitor to store charge of magnitude 240.0 pC on each plate?

A cylindrical capacitor consists of a solid inner conducting core with radius \(0.250 \mathrm{cm},\) surrounded by an outer hollow conducting tube. The two conductors are separated by air, and the length of the cylinder is 12.0 \(\mathrm{cm}\) . The capacitance is 36.7 pF. (a) Calculate the inner radius of the hollow tube. (b) When the capacitor is charged to \(125 \mathrm{V},\) what is the charge per unit length \(\lambda\) on the capacitor?

A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is 10.0 \(\mathrm{pC}\) . The inner cylinder has radius \(0.50 \mathrm{mm},\) the outer one has radius \(5.00 \mathrm{mm},\) and the length of each cylinder is 18.0 \(\mathrm{cm} .\) (a) What is the capacitance? (b) What applied potential difference is necessary to produce these charges on the cylinders?

A cylindrical capacitor has an inner conductor of radius 1.5 \(\mathrm{mm}\) and an outer conductor of radius 3.5 \(\mathrm{mm}\) . The two conductors are separated by vacuum, and the entire capacitor is 2.8 \(\mathrm{m}\) long. (a) What is the capacitance per unit length? (b) The potential of the inner conductor is 350 \(\mathrm{mV}\) higher than that of the outer conductor. Find the charge (magnitude and sign) on both conductors.

A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has radius 15.0 \(\mathrm{cm}\) and the capacitance is 116 \(\mathrm{pF}\) (a) What is the radius of the outer sphere? (b) If the potential difference between the two spheres is \(220 \mathrm{V},\) what is the magnitude of charge on each sphere?

BIO Electric Eels. Electric eels and electric fish generate large potential differences that are used to stun enemies and prey. These potentials are produced by cells that each can generate 0.10 \(\mathrm{V}\) . We can plausibly model such cells as charged capacitors. (a) How should these cells be connected (in series or in parallel) to produce a total potential of more than 0.10 \(\mathrm{V} ?\) (b) Using the connection in part (a), how many cells must be connected together to produce the \(500-\mathrm{V}\) surge of the electric eel?

A \(5.80-\mu \mathrm{F}\), parallel-parallel-plate, air capacitor has a plate separation of 5.00 \(\mathrm{mm}\) and is charged to a potential difference of 400 \(\mathrm{V}\) . Calculate the energy density in the region between the plates, in units of \(\mathrm{J} / \mathrm{m}^{3} .\)

An air capacitor is made from two flat parallel plates 1.50 \(\mathrm{mm}\) apart. The magnitude of charge on each plate is 0.0180\(\mu \mathrm{C}\) when the potential difference is 200 \(\mathrm{V}\) . (a) What is the capacitance? (b) What is the area of each plate? (c) What maximum voltage can be applied without dielectric breakdown? (Dielectric breakdown for air occurs at an electric-field strength of \(3.0 \times 10^{6} \mathrm{V} / \mathrm{m} .\) ) (d) When the charge is \(0.0180 \mu \mathrm{C},\) what total energy is stored?

A parallel-plate vacuum capacitor with plate area \(A\) and separation \(x\) has charges \(+Q\) and \(-Q\) on its plates. The capacitor is disconnected from the source of charge, so the charge on each plate remains fixed. (a) What is the total energy stored in the capacitor? (b) The plates are pulled apart an additional distance \(d x\) . What is the change in the stored energy? (c) If \(F\) is the force with which the plates attract each other, then the change in the stored energy must equal the work \(d W=F d x\) done in pulling the plates apart. Find an expression for \(F .(\mathrm{d})\) Explain why \(F\) is not equal to \(Q E,\) where \(E\) is the electric field between the plates.

A parallel-plate vacuum capacitor has 8.38 \(\mathrm{J}\) of energy stored in it. The separation between the plates is 2.30 \(\mathrm{mm}\) . If the separation is decreased to 1.15 \(\mathrm{mm}\) , what is the energy stored (a) if the capacitor is disconnected from the potential source so the charge on the plates remains constant, and (b) if the capacitor remains connected to the potential source so the potential difference between the plates remains constant?

You have two identical capacitors and an external potential source. (a) Compare the total energy stored in the capacitors when they are connected to the applied potential in series and in parallel. (b) Compare the maximum amount of charge stored in each case. (c) Energy storage in a capacitor can be limited by the maximum electric field between the plates. What is the ratio of the electric field for the series and parallel combinations?

A \(3.350-\mathrm{m}\) -long cylindrical capacitor consists of a solid conducting core with a radius of 1.20 \(\mathrm{mm}\) and an outer hollow conducting tube with an inner radius of 2.00 \(\mathrm{mm}\) . The two conductors are separated by air and charged to a potential difference of 6.00 \(\mathrm{V}\) . Calculate (a) the charge per length for the capacitor; (b) the total charge on the capacitor; (c) the capacitance; (d) the energy stored in the capacitor when fully charged.

A cylindrical air capacitor of length 15.0 \(\mathrm{m}\) stores \(3.20 \times 10^{-9} \mathrm{J}\) of energy when the potential difference between the two conductors is 4.00 \(\mathrm{V}\) . (a) Calculate the magnitude of the charge on each conductor. (b) Calculate the ratio of the radii of the inner and outer conductors.

A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius \(12.5 \mathrm{cm},\) and the outer sphere has radius 14.8 \(\mathrm{cm}\) . A potential difference of 120 \(\mathrm{V}\) is applied to the capacitor. (a) What is the energy density at \(r=12.6 \mathrm{cm},\) just outside the inner sphere? (b) What is the energy density at \(r=14.7 \mathrm{cm},\) just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?

A 12.5 -\muF capacitor is connected to a power supply that keeps a constant potential difference of 24.0 \(\mathrm{V}\) across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the insertion? Did it increase or decrease?

A parallel-plate capacitor has capacitance \(C_{0}=5.00 \mathrm{pF}\) when there is air between the plates. The separation between the plates is 1.50 \(\mathrm{mm}\) (a) What is the maximum magnitude of charge \(Q\) that can be placed on each plate if the electric field in the region between the plates is not to exceed \(3.00 \times 10^{4} \mathrm{V} / \mathrm{m}\) (b) A dielectric with \(K=2.70\) is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed \(3.00 \times 10^{4} \mathrm{V} / \mathrm{m}\) ?

Two parallel plates have equal and opposite charges. When the space between the plates is evacuated, the electric field is \(E=3.20 \times 10^{5} \mathrm{V} / \mathrm{m}\) . When the space is filled with dielectric, the electric field is \(E=2.50 \times 10^{5} \mathrm{V} / \mathrm{m}\) . (a) What is the charge density on each surface of the dielectric? (b) What is the dielectric constant?

A budding electronics hobbyist wants to make a simple 1.0 -nf capacitor for tuning her crystal radio, using two sheets of aluminum foil as plates, with a few sheets of paper between them as a dielectric. The paper has a dielectric constant of \(3.0,\) and the thickness of one sheet of it is 0.20 \(\mathrm{mm}\) . (a) If the sheets of paper measure \(22 \times 28 \mathrm{cm}\) and she cuts the aluminum foil to the same dimensions, how many sheets of paper should she use between her plates to get the proper capacitance? (b) Suppose for convenience she wants to use a single sheet of posterboard, with the same dielectric constant but a thickness of \(12.0 \mathrm{mm},\) instead of the paper. What area of aluminum foil will she need for her plates to get her 1.0 nF of capacitance? (c) Suppose she goes high-tech and finds a sheet of Teflon of the same thickness as the posterboard to use as a dielectric. Will she need a larger or smaller area of Teflon than of poster board? Explain.

The dielectric to be used in a parallel-plate capacitor has a dielectric constant of 3.60 and a dielectric strength of \(1.60 \times\) \(10^{7} \mathrm{V} / \mathrm{m}\) . The capacitor is to have a capacitance of \(1.25 \times 10^{-9} \mathrm{F}\) and must be able to withstand a maximum potential difference of 5500 \(\mathrm{V} .\) What is the minimum area the plates of the capacitor may have?

B10 Potential in Human Cells. Some cell walls in the human body have a layer of negative charge on the inside surface and a layer of positive charge of equal magnitude on the outside surface. Suppose that the charge density on either surface is \(\pm 0.50 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2},\) the cell wall is 5.0 \(\mathrm{nm}\) thick, and the cell-wall material is air. (a) Find the magnitude of \(\vec{\boldsymbol{E}}\) in the wall-between the two layers of charge. (b) Find the potential difference between the inside and the outside of the cell. Which is at the higher potential? (c) Atypical cell in the human body has a volume of\(10^{-16} \mathrm{m}^{3} .\) Estimate the total electric-field energy stored in the wall of a cell of this size. (Hint: Assume that the cell is spherical, and calculate the volume of the cell wall.) (d) In reality, the cell wall is made up, not of air, but of tissue with a dielectric constant of \(5.4 .\) Repeat parts (a) and (b) in this case.

When a 360 -nF air capacitor \(\left(1 \mathrm{nF}=10^{-9} \mathrm{F}\right)\) is connected to a power supply, the energy stored in the capacitor is \(1.85 \times 10^{-5} \mathrm{J}\) . While the capacitor is kept connected to the power supply, a slab of dielectric is inserted that completely fills the space between the plates. This increases the stored energy by \(2.32 \times 10^{-5} \mathrm{J}\) . (a) What is the potential difference between the capacitor plates? (b) What is the dielectric constant of the slab?

A parallel-plate capacitor has capacitance \(C=12.5 \mathrm{pF}\) when the volume between the plates is filled with air. The plates are circular, with radius 3.00 \(\mathrm{cm}\) . The capacitor is connected to a battery, and a charge of magnitude 25.0 pC goes onto each plate. With the capacitor still connected to the battery, a slab of dielectric is inserted between the plates, completely filling the space between the plates. After the dielectric has been inserted, the charge on each plate has magnitude 45.0 \(\mathrm{pC}\) (a) What is the dielectric constant \(K\) of the dielectric? (b) What is the potential difference between the plates before and after the dielectric has been inserted? (c) What is the electric field at a point midway between the plates before and after the dielectric has been inserted?

A parallel-plate capacitor has plates with area 0.0225 \(\mathrm{m}^{2}\) separated by 1.00 \(\mathrm{mm}\) of Teflon. (a) Calculate the charge on the plates when they are charged to a potential difference of 12.0 \(\mathrm{V}\) . (b) Use Gauss's law (Eq. 24.23 ) to calculate the electric field inside the Teflon. (c) Use Gauss's law to calculate the electric field if the voltage source is disconnected and the Teflon is removed.

Electronic flash units for cameras contain a capacitor for storing the energy used to produce the flash. In one such unit, the flash lasts for \(\frac{1}{677}\) s with an average light power output of \(2.70 \times 10^{5} \mathrm{W}\) (a) If the conversion of electrical energy to light is 95\(\%\) efficient (the rest of the energy goes to thermal energy), how much energy must be stored in the capacitor for one flash? (b) The capacitor has a potential difference between its plates of 125 \(\mathrm{V}\) when the stored energy equals the value calculated in part (a). What is the capacitance?

A parallel-plate air capacitor is made by using two plates 16 \(\mathrm{cm}\) square, spaced 3.7 \(\mathrm{mm}\) apart. It is connected to a \(12-\mathrm{V}\) battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of \(7.4 \mathrm{mm},\) what are the answers to parts (a)-(d)?

A capacitor is made from two hollow, coaxial copper cylinders, one inside the other. There is air in the space between the cylinders. The inner cylinder has net positive charge and the outer cylinder has net negative charge. The inner cylinder has radius \(2.50 \mathrm{mm},\) the outer cylinder has radius \(3.10 \mathrm{mm},\) and the length of each cylinder is 36.0 \(\mathrm{cm} .\) If the potential difference between the surfaces of the two cylinders is \(80.0 \mathrm{V},\) what is the magnitude of the electric field at a point between the two cylinders that is a distance of 2.80 \(\mathrm{mm}\) from their common axis and midway between the ends of the cylinders?

In one type of computer keyboard, each key holds a small metal plate that serves as one plate of a parallel-plate, air-filled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard, the area of each metal plate is 42.0 \(\mathrm{mm}^{2}\) , and the separation between the plates is 0.700 \(\mathrm{mm}\) before the key is depressed. (a) Calculate the capacitance before the key is depressed. (b) If the circulatry can detect a change in capacitance of 0.250 \(\mathrm{pF}\) , how far must the key be depressed before the circuitry detects its depression?

A 20.0 -\muF capacitor is charged to a potential difference of 800 V. The terminals of the charged capacitor are then connected to those of an uncharged \(10.0-\mu \mathrm{F}\) capacitor. Compute (a) the original charge of the system, (b) the final potential difference across each capacitor, (c) the final energy of the system, and (d) the decrease in energy when the capacitors are connected.

You are working on an electronics project requiring a variety of capacitors, but you have only a large supply of 100 -nF capacitors available. Show how you can connect these capacitors to produce each of the following equivalent capacitances: (a) 50 \(\mathrm{nF}\) ; (b) \(450 \mathrm{nF} ;\) (c) \(25 \mathrm{nF} ;\) (d) 75 \(\mathrm{nF}\) .

Three capacitors having capacitances of \(8.4,8.4,\) and 4.2\(\mu \mathrm{F}\) are connected in series across a \(36-\mathrm{V}\) potential difference. (a) What is the charge on the \(4.2-\mu F\) capacitor? (b) What is the total energy stored in all three capacitors? (c) The capacitors are disconnected from the potential difference without allowing them to discharge. They are then reconnected in parallel with each other, with the positively charged plates connected together. What is the voltage across each capacitor in the parallel combination? (d) What is the total energy now stored in the capacitors?

Capacitance of a Thundercloud. The charge center of a thundercloud, drifting 3.0 \(\mathrm{km}\) above the earth's surface, contains 20 \(\mathrm{C}\) of negative charge. Assuming the charge center has a radius of \(1.0 \mathrm{km},\) and modeling the charge center and the earth's surface as parallel plates, calculate: (a) the capacitance of the system; (b) the potential difference between charge center and ground; (c) the average strength of the electric field between cloud and ground; (d) the electrical energy stored in the system.

A parallel-plate capacitor with only air between the plates is charged by connecting it to a battery. The capacitor is then disconnected from the battery, without any of the charge leaving the plates. (a) A voltmeter reads 45.0 \(\mathrm{V}\) when placed across the capacitor. When a dielectric is inserted between the plates, completely filling the space, the voltmeter reads 11.5 \(\mathrm{V}\) . What is the dielectric constant of this material? (b) What will the voltmeter read if the dielectric is now pulled partway out so it fills only one-third of the space between the plates?

An air capacitor is made by using two flat plates, each with area \(A,\) separated by a distance \(d\) . Then a a metal slab having thickness a (less than \(d\) ) and the same shape and size as the plates is inserted between them, parallel to the plates and not touching either plate (Fig. P24.66). (a) What is the capacitance of this arrangement? (b) Express the capacitance as a multiple of the capacitance \(C_{0}\) when the metal slab is not present. (c) Discuss what happens to the capacitance in the limits \(a \rightarrow 0\) and \(a \rightarrow d\) .

Capacitance of the Earth. Consider a spherical capacitor with one conductor being a solid conducting sphere of radius \(R\) and the other conductor being at infinity. (a) Use Eq. (24.1) and what you know about the potential at the surface of a conducting sphere with charge \(Q\) to derive an expression for the capacitance of the charged sphere. (b) Use your result in part (a) to calculate the capacitance of the earth. The earth is a good conductor and has a radius of 6380 \(\mathrm{km} .\) Compare your results to the capacitance of typical capacitors used in electronic circuits, which ranges from 10 \(\mathrm{pF}\) to 100 \(\mathrm{pF} .\)

A parallel-plate capacitor is made from two plates 12.0 \(\mathrm{cm}\) on each side and 4.50 \(\mathrm{mm}\) apart. Half of the space between these plates contains only air, but the other half is filled with Plexiglas' of dielectric constant 3.40 (Fig. P24.72). An 18.0-V battery is connected across the plates. (a) What is the capacitance of this combination? (Hint: Can you think of this capacitor as equivalent to two capacitors in parallel? (b) How much energy is stored in the capacitor? (c) If we remove the Plexiglas' but change nothing else, how much energy will be stored in the capacitor?