Solved: At the Fermi National Accelerator Laboratory

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 (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 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? Text Transcription: 12.2 cm^2 4.35 times 10^-8 C

A parallel-plate air capacitor of capacitance has a charge of magnitude 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 electric- field magnitude between the plates? (d) What is the surface charge density on each plate?

Oscilloscopes have parallel metal plates inside them to deflect the electron beam. These plates are called the deflecting plates. Typically, they are squares 3.0 cm on a side and separated by 5.0 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-V battery after the radius of each plate was doubled without changing their separation?

A parallel-plate capacitor is connected to a 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?

How far apart would parallel pennies have to be to make a capacitor? Does your answer suggest that you are justified in treating these pennies as infinite sheets? Explain

A 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 The electric field between the plates is to be no greater than 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 pC on each plate when the potential difference between the plates is 42.0 V. (a) If the area of each plate is , 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 cm, surrounded by an outer hollow conducting tube. The two conductors are separated by air, and the length of the cylinder is 12.0 cm. The capacitance is 36.7 pF. (a) Calculate the inner radius of the hollow tube. (b) When the capacitor is charged to 125 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 The inner cylinder has radius the outer one has radius and the length of each cylinder is (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 and an outer conductor of radius The two conductors are separated by vacuum, and the entire capacitor is long. (a) What is the capacitance per unit length? (b) The potential of the inner conductor is higher than that of the outer conductor. Find the charge (magnitude and sign) on both conductors.

A spherical capacitor contains a charge of when connected to a potential difference of 220 V. If its plates are separated by vacuum and the inner radius of the outer shell is calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

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

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 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 V? (b) Using the connection in part (a), how many cells must be connected together to produce the 500-V surge of the electric eel?

For the system of capacitors shown in Fig. E24.16, find the equivalent capacitance (a) between and and (b) between and

In Fig. E24.17, each capacitor has \(C=4.00\ \mu\mathrm{F}\) and \(V_{ab}=+28.0\mathrm{\ V}\). Calculate (a) the charge on each capacitor; (b) the potential difference across each capacitor; (c) the potential difference between points a and d.

In Fig. 24.8a, let and Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor

In Fig. 24.9a, let \(C_1=3.00\ \mu\mathrm{F},\quad C_2=5.00\ \mu\mathrm{F}\text{, and } V_{ab} = +52.0 \mathrm{\ V}\). Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor. Text Transcription: C_1 = 3.00 mu F, C_2 = 5.00 mu F, and V_ab = +52.0 V

In Fig. E24.20, and The capacitor network is connected to an applied potential After the charges on the capacitors have reached their final values, the charge on is (a) What are the charges on capacitors and (b) What is the applied voltage

For the system of capacitors shown in Fig. E24.21, a potential difference of 25 V is maintained across ab. (a) What is the equivalent capacitance of this system between a and b? (b) How much charge is stored by this system? (c) How much charge does the 6.5-nF capacitor store? (d) What is the potential difference across the 7.5-nF capacitor?

Figure E24.22 shows a system of four capacitors, where the potential difference across is 50.0 V. (a) Find the equivalent capacitance of this system between and (b) How much charge is stored by this combination of capacitors? (c) How much charge is stored in each of the and the capacitors?

Suppose the \(3-\muF\) capacitor in Fig. 24.10a were removed and replaced by a different one, and that this changed the equivalent capacitance between points a and b to \(8 \muF\). What would be the capacitance of the replacement capacitor? Text Transcription: 3-muF 8 muF

A parallel-plate air capacitor has a capacitance of The charge on each plate is (a) What is the potential difference between the plates? (b) If the charge is kept constant, what will be the potential difference between the plates if the separation is doubled? (c) How much work is required to double the separation?

A \(5.80-\mu \mathrm{F}\), parallel-plate, air capacitor has a plate separation of 5.00 mm and is charged to a potential difference of 400 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 mm apart. The magnitude of charge on each plate is \(0.0180\ \mu\mathrm{C}\) when the potential difference is 200 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 and separation has charges and 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 What is the change in the stored energy? (c) If is the force with which the plates attract each other, then the change in the stored energy must equal the work done in pulling the plates apart. Find an expression for (d) Explain why is not equal to where is the electric field between the plates

A parallel-plate vacuum capacitor has of energy stored in it. The separation between the plates is If the separation is decreased to 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?

For the capacitor network shown in Fig. E24.30, the potential difference across is 36 V. Find (a) the total charge stored in this network; (b) the charge on each capacitor; (c) the total energy stored in the network; (d) the energy stored in each capacitor; (e) the potential differences across each capacitor.

For the capacitor network shown in Fig. E24.31, the potential difference across is 220 V. Find (a) the total charge stored in this network; (b) the charge on each capacitor; (c) the total energy stored in the network; (d) the energy stored in each capacitor; (e) the potential difference across each capacitor.

A 0.350-m-long cylindrical capacitor consists of a solid conducting core with a radius of 1.20 mm and an outer hollow conducting tube with an inner radius of 2.00 mm. The two conductors are separated by air and charged to a potential difference of 6.00 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 stores of energy when the potential difference between the two conductors is 4.00 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 and the outer sphere has radius A potential difference of is applied to the capacitor. (a) What is the energy density at just outside the inner sphere? (b) What is the energy density at 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 capacitor is connected to a power supply that keeps a constant potential difference of 24.0 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 when there is air between the plates. The separation between the plates is (a) What is the maximum magnitude of charge that can be placed on each plate if the electric field in the region between the plates is not to exceed (b) A dielectric with 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

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 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 (a) If the sheets of paper measure 22 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 instead of the paper. What area of aluminum foil will she need for her plates to get her 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 posterboard? Explain.

The dielectric to be used in a parallel-plate capacitor has a dielectric constant of 3.60 and a dielectric strength of The capacitor is to have a capacitance of and must be able to withstand a maximum potential difference of What is the minimum area the plates of the capacitor may have?

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 the cell wall is 5.0 nm thick, and the cellwall material is air. (a) Find the magnitude of 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) A typical cell in the human body has a volume of 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

A capacitor has parallel plates of area separated by The space between the plates is filled with polystyrene (see Table 24.2). (a) Find the permittivity of polystyrene. (b) Find the maximum permissible voltage across the capacitor to avoid dielectric breakdown. (c) When the voltage equals the value found in part (b), find the surface charge density on each plate and the induced surface charge density on the surface of the dielectric.

A constant potential difference of is maintained between the terminals of a parallel-plate, air capacitor. (a) A sheet of Mylar is inserted between the plates of the capacitor, completely filling the space between the plates. When this is done, how much additional charge flows onto the positive plate of the capacitor (see Table 24.1)? (b) What is the total induced charge on either face of the Mylar sheet? (c) What effect does the Mylar sheet have on the electric field between the plates? Explain how you can reconcile this with the increase in charge on the plates, which acts to increase the electric field

When a 360-nF air capacitor is connected to a power supply, the energy stored in the capacitor is 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 (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 pF when the volume between the plates is filled with air. The plates are circular, with radius 3.00 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 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 capacitance when the volume between the plates is filled with air. The plates are circular, with radius The capacitor is connected to a battery, and a charge of magnitude 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 (a) What is the dielectric constant 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 separated by of Teflon. (a) Calculate the charge on the plates when they are charged to a potential difference of (b) Use Gausss law (Eq. 24.23) to calculate the electric field inside the Teflon. (c) Use Gausss 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 with an average light power output of (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 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 square, spaced apart. It is connected to a 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 what are the answers to parts (a)(d)?

Suppose the battery in Problem 24.48 remains connected while the plates are pulled apart. What are the answers then to parts (a)(d) after the plates have been pulled apart?

Cell membranes (the walled enclosure around a cell) are typically about 7.5 nm thick. They are partially permeable to allow charged material to pass in and out, as needed. Equal but opposite charge densities build up on the inside and outside faces of such a membrane, and these charges prevent additional charges from passing through the cell wall. We can model a cell membrane as a parallel-plate capacitor, with the membrane itself containing proteins embedded in an organic material to give the membrane a dielectric constant of about 10. (See Fig. P24.50.) (a) What is the capacitance per square centimeter of such a cell wall? (b) In its normal resting state, a cell has a potential difference of 85 mV across its membrane. What is the electric field inside this membrane?

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 mm, the outer cylinder has radius 3.10 mm, and the length of each cylinder is 36.0 cm. If the potential difference between the surfaces of the two cylinders is 80.0 V, what is the magnitude of the electric field at a point between the two cylinders that is a distance of 2.80 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 and the separation between the plates is before the key is depressed. (a) Calculate the capacitance before the key is depressed. (b) If the circuitry can detect a change in capacitance of how far must the key be depressed before the circuitry detects its depression?

A capacitor is charged to a potential difference of 800 V. The terminals of the charged capacitor are then connected to those of an uncharged 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.

In Fig. 24.9a, let \(C_1=9.0\ \mu\mathrm{F}\), \(C_2=4.0\ \mu\mathrm{F}\), and \(V_{ab}=36\mathrm{\ V}\). Suppose the charged capacitors are disconnected from the source and from each other, and then reconnected to each other with plates of opposite sign together. By how much does the energy of the system decrease?

For the capacitor network shown in Fig. P24.55, the potential difference across is 12.0 V. Find (a) the total energy stored in this network and (b) the energy stored in the capacitor.

Several \(0.25-\mu \mathrm{F}\) capacitors are available. The voltage across each is not to exceed 600 V. You need to make a capacitor with capacitance \(0.25 \ \mu \mathrm{F}\) to be connected across a potential difference of 960 V. (a) Show in a diagram how an equivalent capacitor with the desired properties can be obtained. (b) No dielectric is a perfect insulator that would not permit the flow of any charge through its volume. Suppose that the dielectric in one of the capacitors in your diagram is a moderately good conductor. What will happen in this case when your combination of capacitors is connected across the 960-V potential difference?

. In Fig. P24.57, and The applied potential is (a) What is the equivalent capacitance of the network between points and (b) Calculate the charge on each capacitor and the potential difference across each capacitor

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 nF; (b) 450 nF; (c) 25 nF; (d) 75 nF.

In Fig. E24.20, \(C_{1}=3.00 \mu \mathrm{F} \text { and } V_{a b}=150 \mathrm{~V}\). The charge on capacitor \(C_1\) is \(150 \mu \mathrm{C}\) and the charge on \(C_3\) is \(450 \mu \mathrm{C}\). What are the values of the capacitances of \(C_2\) and \(C_3\)?

The capacitors in Fig. P24.60 are initially uncharged and are connected, as in the diagram, with switch S open. The applied potential difference is (a) What is the potential difference (b) What is the potential difference across each capacitor after switch is closed? (c) How much charge flowed through the switch when it was closed?

Three capacitors having capacitances of 8.4, 8.4, and \(4.2\mu F\) are connected in series across a 36-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?

The charge center of a thundercloud, drifting above the earths surface, contains of negative charge. Assuming the charge center has a radius of and modeling the charge center and the earths 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.

In Fig. P24.63, each capacitance is and each capacitance is (a) Compute the equivalent capacitance of the network between points and (b) Compute the charge on each of the three capacitors nearest and when (c) With across and compute

Each combination of capacitors between points and in Fig. P24.64 is first connected across a battery, charging the combination to These combinations are then connected to make the circuits shown. When the switch is thrown, a surge of charge for the discharging capacitors flows to trigger the signal device. How much charge flows through the signal device in each case?

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 V when placed across the capacitor. When a dielectric is inserted between the plates, completely filling the space, the voltmeter reads 11.5 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 onethird of the space between the plates?

An air capacitor is made by using two flat plates, each with area separated by a distance Then a metal slab having thickness (less than ) 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 when the metal slab is not present. (c) Discuss what happens to the capacitance in the limits and

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 km. Compare your results to the capacitance of typical capacitors used in electronic circuits, which ranges from 10 pF to 100 pF.

A potential difference is applied across the capacitor network of Fig. E24.17. If and C1 = C2 = 4.00 mF what must the capacitance be if the network is to store of electrical energy?

The earth can be considered as a single-conductor capacitor (see Problem 24.67). It can also be considered in combination with a charged layer of the atmosphere, the ionosphere, as a spherical capacitor with two plates, the surface of the earth being the negative plate. The ionosphere is at a level of about and the potential difference between earth and ionosphere is about Calculate: (a) the capacitance of this system; (b) the total charge on the capacitor; (c) the energy stored in the system.

The inner cylinder of a long, cylindrical capacitor has radius and linear charge density It is surrounded by a coaxial cylindrical conducting shell with inner radius and linear charge density (see Fig. 24.6). (a) What is the energy density in the region between the conductors at a distance from the axis? (b) Integrate the energy density calculated in part (a) over the volume between the conductors in a length of the capacitor to obtain the total electric-field energy per unit length. (c) Use Eq. (24.9) and the capacitance per unit length calculated in Example 24.4 (Section 24.1) to calculate Does your result agree with that obtained in part (b)?

A capacitor has a potential difference of \(2.25 \times 10^3 \ \mathrm V\) between its plates. A short aluminum wire with initial temperature \(23.0^{\circ} \mathrm{C}\) is connected between the plates of the capacitor and all the energy stored in the capacitor goes into heating the wire. The wire has mass 12.0 g. If no heat is lost to the surroundings and the final temperature of the wire is \(34.2^{\circ} \mathrm{C}\), what is the capacitance of the capacitor?

A parallel-plate capacitor is made from two plates 12.0 cm on each side and 4.50 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?

A parallel-plate capacitor has square plates that are 8.00 cm on each side and 3.80 mm apart. The space between the plates is completely filled with two square slabs of dielectric, each 8.00 cm on a side and 1.90 mm thick. One slab is pyrex glass and the other is polystyrene. If the potential difference between the plates is 86.0 V, how much electrical energy is stored in the capacitor?

A fuel gauge uses a capacitor to determine the height of the fuel in a tank. The effective dielectric constant changes from a value of 1 when the tank is empty to a value of the dielectric constant of the fuel, when the tank is full. The appropriate electronic circuitry can determine the effective dielectric constant of the combined air and fuel between the capacitor plates. Each of the two rectangular plates has a width and a length (Fig. P24.74). The height of the fuel between the C3 plates is You can ignore any fringing effects. (a) Derive an expression for as a function of (b) What is the effective dielectric constant for a tank full, full, and full if the fuel is gasoline (c) Repeat part (b) for methanol (d) For which fuel is this fuel gauge more practical?

Three square metal plates A, B, and C, each 12.0 cm on a side and 1.50 mm thick, are arranged as in Fig. P24.75. The plates are separated by sheets of paper 0.45 mm thick and with dielectric constant 4.2. The outer plates are connected together and connected to point b. The inner plate is connected to point a. (a) Copy the diagram and show by plus and minus signs the charge distribution on the plates when point a is maintained at a positive potential relative to point b. (b) What is the capacitance between points a and b?

The parallel-plate air capacitor in Fig. P24.76 consists of two horizontal conducting plates of equal area \(A\). The bottom plate rests on a fixed support, and the top plate is suspended by four springs with spring constant \(k\), positioned at each of the four corners of the top plate as shown in the figure. When uncharged, the plates are separated by a distance \(z_0\). A battery is connected to the plates and produces a potential difference \(V\) between them. This causes the plate separation to decrease to \(z\). Neglect any fringing effects. (a) Show that the electrostatic force between the charged plates has a magnitude \(\epsilon_0 A V^2 / 2 z^2\). (Hint: See Exercise 24.27.) (b) Obtain an expression that relates the plate separation \(z\) to the potential difference \(V\). The resulting equation will be cubic in \(z\). (c) Given the values \(A=0.300 \mathrm{~m}^2, z_0=1.20 \mathrm{~mm}\), \(k=25.0 \mathrm{~N} / \mathrm{m}\), and \(V=120 \mathrm{~V}\), find the two values of \(z\) for which the top plate will be in equilibrium. (Hint: You can solve the cubic equation by plugging a trial value of \(z\) into the equation and then adjusting your guess until the equation is satisfied to three significant figures. Locating the roots of the cubic equation graphically can help you pick starting values of \(z\) for this trial-and-error procedure. One root of the cubic equation has a nonphysical negative value.) (d) For each of the two values of \(z\) found in part (c), is the equilibrium stable or unstable? For stable equilibrium a small displacement of the object will give rise to a net force tending to return the object to the equilibrium position. For unstable equilibrium a small displacement gives rise to a net force that takes the object farther away from equilibrium.

Two square conducting plates with sides of length are separated by a distance A dielectric slab with constant with dimensions is inserted a distance into the space between the plates, as shown in Fig. P24.77. (a) Find the capacitance of this system. (b) Suppose that the capacitor is connected to a battery that maintains a constant potential difference between the plates. If the dielectric slab is inserted an additional distance into the space between the plates, show that the change in stored energy is (c) Suppose that before the slab is pulls the slab into the capacitor. (e) Figure 24.16 shows that the force in fact pulls the slab into the capacitor. Explain why the result of part (b) gives an incorrect answer for the direction of this force, and calculate the magnitude of the force. (This method does not require knowledge of the nature of the fringing field.) moved by the plates are disconnected from the battery, so that the charges on the plates remain constant. Determine the magnitude of the charge on each plate, and then show that when the slab is moved farther into the space between the plates, the stored energy changes by an amount that is the negative of the expression for given in part (b). (d) If is the force exerted on the slab by the charges on the plates, then should equal the work done against this force to move the slab a distance Thus Show that applying this expression to the result of part (b) suggests that the electric force on the slab pushes it out of the capacitor, while the result of part (c) suggests that the force

An isolated spherical capacitor has charge +Q on its inner conductor (radius \(r_a\)) and charge -Q on its outer conductor (radius \(r_b\)). Half of the volume between the two conductors is then filled with a liquid dielectric of constant K, as shown in cross section in Fig. P24.78. (a) Find the capacitance of the half-filled capacitor. (b) Find the magnitude of \(\vec{E}\) in the volume between the two conductors as a function of the distance r from the center of the capacitor. Give answers for both the upper and lower halves of this volume. (c) Find the surface density of free charge on the upper and lower halves of the inner and outer conductors. (d) Find the surface density of bound charge on the inner (\(r=r_a\)) and outer (\(r=r_b\)) surfaces of the dielectric. (e) What is the surface density of bound charge on the flat surface of the dielectric? Explain.

Problem 1DQ Equation (24.2) shows that the capacitance of a parallel-plate capacitor becomes larger as the plate separation decreases. However, there is a practical limit to how small can be made, which places limits on how large can be. Explain what sets the limit on (Hint: What happens to the magnitude of the electric field as \(d \rightarrow 0\)?) Equation Transcription: Text Transcription: d-0

Problem 1E 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 4.00 X 106 V/m. What is (a) the potential difference between the plates; (b) the area of each plate; (c) the capacitance?

Problem 2DQ Suppose several different parallel-plate capacitors are charged up by a constant-voltage source. Thinking of the actual movement and position of the charges on an atomic level, why does it make sense that the capacitances are proportional to the surface areas of the plates? Why does it make sense that the capacitances are inversely proportional to the distance between the plates?

The plates of a parallel-plate capacitor are 3.28 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?

Problem 3DQ Suppose the two plates of a capacitor have different areas. When the capacitor is charged by connecting it to a battery, do the charges on the two plates have equal magnitude, or may they be different? Explain your reasoning.

Problem 3E A parallel-plate air capacitor of capacitance 245 pF has a charge of magnitude 0.148 µC 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 electric-field magnitude between the plates? (d) What is the surface charge density on each plate?

Problem 4DQ At the Fermi National Accelerator Laboratory (Fermilab) in Illinois, protons are accelerated around a ring 2 km in radius to speeds that approach that of light The energy for this is stored in capacitors the size of a house. When these capacitors are being charged, they make a very loud creaking sound. What is the origin of this sound?

Problem 7E How far apart would parallel pennies have to be to make a 1.00- pF capacitor? Does your answer suggest that you are justified in treating these pennies as infinite sheets? Explain.

Problem 8DQ Two parallel-plate capacitors, identical except that one has twice the plate separation of the other, are charged by the same voltage source. Which capacitor has a stronger electric field between the plates? Which capacitor has a greater charge? Which has greater energy density? Explain your reasoning.

Problem 8E 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 X 102 V. The electric field between the plates is to be no greater than 1.00 X 104 N/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.

Problem 10E A cylindrical capacitor consists of a solid inner conducting core with radius 0.250 cm, surrounded by an outer hollow conducting tube. The two conductors are separated by air, and the length of the cylinder is 12.0 cm. The capacitance is 36.7 pF. (a) Calculate the inner radius of the hollow tube. (b) When the capacitor is charged to 125 V, what is the charge per unit length ? on the capacitor?

Problem 11DQ As shown in Table 24.1, water has a very large dielectric constant \(K=80.4\). Why do you think water is not commonly used as a dielectric in capacitors? Equation Transcription: Text Transcription: K=80.4

Problem 11E 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 pC. The inner cylinder has radius 0.50 mm, the outer one has radius 5.00 mm, and the length of each cylinder is 18.0 cm. (a) What is the capacitance? (b) What applied potential difference is necessary to produce these charges on the cylinders?

Problem 12DQ Is dielectric strength the same thing as dielectric constant? Explain any differences between the two quantities. Is there a simple relationship between dielectric strength and dielectric constant (see Table 24.2)?

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

Problem 13DQ A capacitor made of aluminum foil strips separated by Mylar film was subjected to excessive voltage, and the resulting dielectric breakdown melted holes in the Mylar. After this, the capacitance was found to be about the same as before, but the breakdown voltage was much less. Why?

Problem 13E A spherical capacitor contains a charge of 3.30 nC when connected to a potential difference of 220 V. If its plates are separated by vacuum and the inner radius of the outer shell is 4.00 cm, calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

Problem 14DQ Suppose you bring a slab of dielectric close to the gap between the plates of a charged capacitor, preparing to slide it between the plates. What force will you feel? What does this force tell you about the energy stored between the plates once the dielectric is in place, compared to before the dielectric is in place?

Problem 14E A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has radius 15.0 cm and the capacitance is 116 pF. (a) What is the radius of the outer sphere? (b) If the potential difference between the two spheres is 220 V, what is the magnitude of charge on each sphere?

Problem 15DQ The freshness of fish can be measured by placing a fish between the plates of a capacitor and measuring the capacitance. How does this work? (Hint: As time passes, the fish dries out. See Table 24.1.)

Problem 15E 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 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 V? (b) Using the connection in part (a), how many cells must be connected together to produce the 500-V surge of the electric eel?

Problem 16DQ Electrolytic capacitors use as their dielectric an extremely thin layer of nonconducting oxide between a metal plate and a conducting solution. Discuss the advantage of such a capacitor over one constructed using a solid dielectric between the metal plates.

For the system of capacitors shown in Fig. E24.16, find the equivalent capacitance (a) between b and c, and (b) between a and c.

Problem 17DQ In terms of the dielectric constant K, what happens to the electric flux through the Gaussian surface shown in Fig. 24.22 when the dielectric is inserted into the previously empty space between the plates? Explain.

In Fig. E24.17, each capacitor has \(C=4.00 \mu \mathrm{F}\) and \(V_{a b}=+28.0 \mathrm{~V}\) Calculate (a) the charge on each capacitor; (b) the potential difference across each capacitor; (c) the potential difference between points \(a\) and \(d\).

Problem 18DQ A parallel-plate capacitor is connected to a power supply that maintains a fixed potential difference between the plates. (a) If a sheet of dielectric is then slid between the plates, what happens to (i) the electric field between the plates, (ii) the magnitude of charge on each plate, and (iii) the energy stored in the capacitor? (b) Now suppose that before the dielectric is inserted, the charged capacitor is disconnected from the power supply. In this case, what happens to (i) the electric field between the plates, (ii) the magnitude of charge on each plate, and (iii) the energy stored in the capacitor? Explain any differences between the two situations.

Problem 18E In Fig. 24.8a, let \(C_{1}=3.00 \mu\)F, \(C_{2}=5.00\)F, and \(V_{a b}=+52.0\) V. Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor. Equation Transcription: Text Transcription: C_1=3.00 uF C_2=5.00 uF V_ab=+52.0

Problem 19DQ Liquid dielectrics that have polar molecules (such as water) always have dielectric constants that decrease with increasing temperature. Why?

Problem 24E A parallel-plate air capacitor has a capacitance of 920 pF. The charge 011 each plate is 2.55 µC. (a) What is the potential difference between the plates? (b) If the charge is kept constant, what will be the potential difference between the plates if the separation is doubled? (c) How much work is required to double the separation?

Problem 25E A 5.80-µF, parallel-plate, air capacitor has a plate separation of 5.00 mm and is charged to a potential difference of 400 V. Calculate the energy density in the region between the plates, in units of J/m3.

Problem 26E An air capacitor is made from two flat parallel plates 1.50 mm apart. The magnitude of charge on each plate is 0.0180 µC when the potential difference is 200 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 X 106 V/m.) (d) When the charge is 0.0180 µC, what total energy is stored?

For the capacitor network shown in Fig. E24.30, the potential difference across is 36 V. Find (a) the total charge stored in this network; (b) the charge on each capacitor; (c) the total energy stored in the network; (d) the energy stored in each capacitor; (e) the potential differences across each capacitor.

Problem 31E For the capacitor network shown in Fig. E24.31, the potential difference \(a b\) across is 220 V. Find (a) the total charge stored in this network; (b) the charge on each capacitor; (c) the total energy stored in the network; (d) the energy stored in each capacitor; (e) the potential difference across each capacitor. Equation Transcription: Text Transcription: ab 35 nF 75 nF

Problem 32E A 0.350-m-long cylindrical capacitor consists of a solid conducting core with a radius of 1.20 mm and an outer hollow conducting tube with an inner radius of 2.00 mm. The two conductors are separated by air and charged to a potential difference of 6.00 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.

Problem 33E A cylindrical air capacitor of length 15.0 m stores 3.20 X 10-9 J of energy when the potential difference between the two conductors is 4.00 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 cm, and the outer sphere has radius 14.8 cm. A potential difference of 120V is applied to the capacitor. (a) What is the energy density at r = 12.6 cm, just outside the inner sphere? (b) What is the energy density at r = 14.7 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-µF capacitor is connected to a power supply that keeps a constant potential difference of 24.0 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?

Problem 36E A parallel-plate capacitor has capacitance C0 = 5.00 pF when there is air between the plates. The separation between the plates is 1.50 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 × 104 V/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 × 104 V/m?

Problem 37E Two parallel plates have equal and opposite charges. When the space between the plates is evacuated, the electric field is E = 3.20 X 105 V/m. When the space is filled with dielectric, the electric field is E = 2.50 X 105 V/m. (a) What is the charge density on each surface of the dielectric? (b) What is the dielectric constant?

Problem 38E 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 mm. (a) If the sheets of paper measure 22 X 28 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 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 poster-board to use as a dielectric. Will she need a larger or smaller area of Teflon than of posterboard? Explain.

Problem 42E A constant potential difference of 12 V is maintained between the terminals of a 0.25-\(\mu \mathrm{F}\), parallel-plate, air capacitor. (a) A sheet of Mylar is inserted between the plates of the capacitor, completely filling the space between the plates. When this is done, how much additional charge flows onto the positive plate of the capacitor (see Table 24.1)? (b) What is the total induced charge on either face of the Mylar sheet? (c) What effect does the Mylar sheet have on the electric field between the plates? Explain how you can reconcile this with the increase in charge on the plates, which acts to increase the electric field. Equation Transcription: Text Transcription: uF

Problem 43E When a 360-nF air capacitor (1 nF = 10-9 F) is connected to a power supply, the energy stored in the capacitor is 1.85 X 10-5 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 X 10-5 J. (a) What is the potential difference between the capacitor plates? (b) What is the dielectric constant of the slab?

Problem 44E A parallel-plate capacitor has capacitance C = 12.5 pF when the volume between the plates is filled with air. The plates are circular, with radius 3.00 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 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 be-tween the plates before and after the dielectric has been inserted?

Problem 45E A parallel-plate capacitor has the volume between its plates filled with plastic with dielectric constant K. The magnitude of the charge on each plate is Q. Each plate has area A, and the distance between the plates is d. (a) Use Gauss’s law as stated in Eq. (24.23) to calculate the magnitude of the electric field in the dielectric. (b) Use the electric field determined in part (a) to calculate the potential difference between the two plates. (c) Use the result of part (b) to determine the capacitance of the capacitor. Compare your result to Eq. (24.12).

Problem 46E A parallel-plate capacitor has plates with area \(0.0225 \mathrm{~m}^{2}\) separated by 1.00 mm of Teflon. (a) Calculate the charge on the plates when they are charged to a potential difference of 12.0 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. Equation Transcription: Text Transcription: 0.225 m^2

Problem 47P 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 \(\frac{1}{675}\)S with an average light power output of \(2.70 \times 10^{5}\) 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 V when the stored energy equals the value calculated in part (a). What is the capacitance? Equation Transcription: Text Transcription: 1/675 2.70x10^5

Problem 55P For the capacitor network shown in Fig. P24.55, the potential difference across \(a b\) is 12.0 V. Find (a) the total energy stored in this network and (b) the energy stored in the 4.80-\(\mu \mathrm{F}\) capacitor . Equation Transcription: Text Transcription: ab uF 8.60 uF 4.80 uF 6.20 uF 11.8 uF 3.50 uF

Problem 60P The capacitors in Fig. P24.60 are initially uncharged and are connected, as in the diagram, with switch S open. The applied potential difference is \(V_{a b}=+210\)V. (a) What is the potential difference \(V_{c d}\)? (b) What is the potential difference across each capacitor after switch S is closed? (c) How much charge flowed through the switch when it was closed? Equation Transcription: Text Transcription: V_ab=+210 V_cd 3.00 uF 6.00 uF 6.00 uF 3.00 uF

Problem 61P Three capacitors having capacitances of 8.4, 8.4, and 4.2 µF are connected in series across a 36-V potential difference. (a) What is the charge on the 4.2-µF 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?

Problem 62P Capacitance of a Thundercloud. The charge center of a thundercloud, drifting 3.0 km above the earth’s surface, contains 20 C of negative charge. Assuming the charge center has a radius of 1.0 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.

In Fig. P24.63, each capacitance \(C_{1} \text { is } 6.9 \mu \mathrm{F}\) and each capacitance \(C_{2} \text { is } 4.6 \mu \mathrm{F}\). (a) Compute the equivalent capacitance of the network between points and (b) Compute the charge on each of the three capacitors nearest a and b when \(V_{a b}=420 \mathrm{~V}\). (c) With 420 V across a and b compute \(V_{c d}\).

Problem 64P Each combination of capacitors between points a and b in Fig. P24.64 is first connected across a 120-V battery, charging the combination to 120 V. These combinations are then connected to make the circuits shown. When the switch S is thrown, a surge of charge for the discharging capacitors flows to trigger the signal device. How much charge flows through the signal device in each case? Equation Transcription: Text Transcription: 10.0 uF 20.0 uF 30.0 uF 10.0 uF 20.0 uF 30.0 uF

Problem 65P 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 V when placed across the capacitor. When a dielectric is inserted between the plates, completely filling the space, the voltmeter reads 11.5 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?

Problem 66P An air capacitor is made by using two flat plates, each with area A, separated by a distance d. Then a metal slab having thickness (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 \(a \rightarrow d\). Equation Transcription: Text Transcription: C_0 a right arrow 0 a right arrow d

Problem 67P 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 km. Compare your results to the capacitance of typical capacitors used in electronic circuits, which ranges from 10 pF to 100 pF

Problem 68P A potential difference \(V_{a b}=48.0\) V is applied across the capacitor network of Fig. E24.17. If \(C_{1}=C_{2}=4.00 \mu \mathrm{F}\)F and \(C_{4}=8.00 \mu\)F, what must the capacitance \(C_{3}\) be if the network is to store \(2.90 \times 10^{-3}\) J of electrical energy? Equation Transcription: Text Transcription: V_ab=48.0 C_1=C_2=4.00 uF C_3 2.90x10^-3

Problem 72P A parallel-plate capacitor is made from two plates 12.0 cm on each side and 4.50 mm apart. Half of the space between these plates contains only air, but the other half is filled with \(\text { Plexiglas® }\) 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? Equation Transcription: Plexiglas® Plexiglas® Text Transcription: Plexiglas(R) Plexiglas(R)

Problem 73P A parallel-plate capacitor has square plates that are 8.00 cm on each side and 3.80 mm apart. The space between the plates is completely filled with two square slabs of dielectric, each 8.00 cm on a side and 1.90 mm thick. One slab is Pyrex glass and the other is polystyrene. If the potential difference between the plates is 86.0 V, how much electrical energy is stored in the capacitor?

Problem 74P A fuel gauge uses a capacitor to determine the height of the fuel in a tank. The effective dielectric constant \(K_{e f f}\) changes from a value of 1 when the tank is empty to a value of ????, the dielectric constant of the fuel, when the tank is full. The appropriate electronic circuitry can determine the effective dielectric constant of the combined air and fuel between the capacitor plates. Each of the two rectangular plates has a width and a length ???? (Fig. P24.74). The height of the fuel between the plates is h.You can ignore any fringing effects. (a) Derive an expression for \(K_{e f f}\) as a function of h. (b) What is the effective dielectric constant for a tank \(\frac{1}{4}\) full, \(\frac{1}{2}\) full, and \(\frac{3}{4}\) full if the fuel is gasoline \((K=1.95)\) ? (c) Repeat part (b) for methanol \((K=33.0)\). (d) For which fuel is this fuel gauge more practical? Equation Transcription: Text Transcription: K_eff K_eff 1/4 1/2 3/4 (K=1.95) (K=33.0)

Problem 75P Three square metal plates ????, ????, and ????, each 12.0 cm on a side and 1.50 mm thick, are arranged as in Fig. P24.75. The plates are separated by sheets of paper 0.45 mm thick and with dielectric constant 4.2. The outer plates are connected together and connected to point b. The inner plate is connected to point a (a) Copy the diagram and show by plus and minus signs the charge distribution on the plates when point a is maintained at a positive potential relative to point b (b) What is the capacitance between point a and b?

Problem 76CP The parallel-plate air capacitor in Fig. P24.76 consists of two horizontal conducting plates of equal area ????. The bottom plate rests on a fixed support, and the top plate is suspended by four springs with spring constant ????, positioned at each of the four corners of the top plate as shown in the figure. When uncharged, the plates are separated by a distance \(z_{0}\). A battery is connected to the plates and produces a potential difference ???? between them. This causes the plate separation to decrease to ????. Neglect any fringing effects. (a) Show that the electrostatic force between the charged plates has a magnitude \(\epsilon_{0} A V^{2} / 2 z^{2}\). (Hint: See Exercise 24.27.) (b) Obtain an expression that relates the plate separation ???? to the potential difference V. The resulting equation will be cubic in ????. (c) Given the values \(A=0.300 \mathrm{~m}^{2}\), \(z_{0}=1.20\) mm, \(k=25.0 \mathrm{~N} / \mathrm{m}\), and \(V=120\) V, find the two values of ???? for which the top plate will be in equilibrium. (Hint: You can solve the cubic equation by plugging a trial value of ???? into the equation and then adjusting your guess until the equation is satisfied to three significant figures. Locating the roots of the cubic equation graphically can help you pick starting values of ???? for this trial-and-error procedure. One root of the cubic equation has a nonphysical negative value.) (d) For each of the two values of ???? found in part (c), is the equilibrium stable or unstable? For stable equilibrium a small displacement of the object will give rise to a net force tending to return the object to the equilibrium position. For unstable equilibrium a small displacement gives rise to a net force that takes the object farther away from equilibrium. Equation Transcription: Text Transcription: z_0 epsilon_0AV^2/2z^2 A=0.300 m^2 z_0=1.20 k=25.0 N/m V=120

Problem 77CP Two square conducting plates with sides of length L are separated by a distance D. A dielectric slab with constant K with dimensions \(L \times L \times D\) is inserted a distance x into the space between the plates, as shown in Fig. P24.77. (a) Find the capacitance C of this system. (b) Suppose that the capacitor is connected to a battery that maintains a constant potential difference V between the plates. If the dielectric slab is inserted an additional distance d? into the space between the plates, show that the change in stored energy is \(d U=+\frac{(K-1) \epsilon_{0} V^{2} L}{2 D} d x\) (c) Suppose that before the slab is moved by \(d x\), the plates are disconnected from the battery, so that the charges on the plates remain constant. Determine the magnitude of the charge on each plate, and then show that when the slab is moved \(d x\) farther into the space between the plates, the stored energy changes by an amount that is the negative of the expression for \(d U\) given in part (b). (d) If F is the force exerted on the slab by the charges on the plates, then \(d U\) should equal the work done against this force to move the slab a distance \(d x\). Thus \(d U=-F d x\). Show that applying this expression to the result of part (b) suggests that the electric force on the slab pushes it out of the capacitor, while the result of part (c) suggests that the force pulls the slab into the capacitor. (e) Figure 24.16 shows that the force in fact pulls the slab into the capacitor. Explain why the result of part (b) gives an incorrect answer for the direction of this force, and calculate the magnitude of the force. (This method does not require knowledge of the nature of the fringing field.) Equation Transcription: Text Transcription: LXLXD dx dU=+(K-1)epsilon_0V^2L over 2D dx dx dx dU dx dU=-Fdx

Capacitance of an Oscilloscope. Oscilloscopes have parallel metal plates inside them to deflect the electron beam. These plates are called the deflecting plates. Typically, they are squares 3.0 cm on a side and separated by 5.0 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.)

In the parallel-plate capacitor of Fig. 24.2, suppose the plates are pulled apart so that the separation is much larger than the size of the plates. (a) Is it still accurate to say that the electric field between the plates is uniform? Why or why not? (b) In the situation shown in Fig. 24.2, the potential difference between the plates is \(V_{a b}=Q d / E_{0} A\) If the plates are pulled apart as described above, is more or less than this formula would indicate? Explain your reasoning. (c) With the plates pulled apart as described above, is the capacitance more than, less than, or the same as that given by Eq. (24.2)? Explain your reasoning. Equation transcription: Text transcription: V{a b}=Q d / E{0} A

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-V battery after the radius of each plate was doubled without changing their separation?

Problem 6DQ A parallel-plate capacitor is charged by being connected to a battery and is kept connected to the battery. The separation between the plates is then doubled. How does the electric field change? The charge on the plates? The total energy? Explain.

Problem 6E A 10.0- µ.F 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?

Problem 7DQ A parallel-plate capacitor is charged by being connected to a battery and is then disconnected from the battery. The separation between the plates is then doubled. How does the electric field change? The potential difference? The total energy? Explain.

Problem 9DQ The charged plates of a capacitor attract each other, so to pull the plates farther apart requires work by some external force. What becomes of the energy added by this work? Explain.

Problem 9E A parallel-plate air capacitor is to store charge of magnitude 240.0 pC on each plate when the potential difference between the plates is 42.0 V. (a) If the area of each plate is 6.80 cm2, 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?

The two plates of a capacitor are given charges \(\pm Q\). The capacitor is then disconnected from the charging device so that the charges on the plates can’t change, and the capacitor is immersed in a tank of oil. Does the electric field between the plates increase, decrease, or stay the same? Explain your reasoning. How can this field be measured? Equation transcription: Text transcription: pm Q

In Fig. 24.9a, let \(C_1=3.00\ \mu F,\ C_2-5.00\ \mu F\), and \(V_{ab}=+52.0\) V. Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

A conductor is an extreme case of a dielectric, since if an electric field is applied to a conductor, charges are free to move within the conductor to set up "induced charges." What is the dielectric constant of a perfect conductor? Is it \(K=0, K \rightarrow \infty\), or something in between? Explain your reasoning. Equation transcription: Text transcription: K=0, K rightarrow infty

In Fig. E24.20, \(C_{1}=6.00 \mu F, C_{2}=3.00 \mu F\), and \(C_{3}=5.00 \mu F\). The capacitor network is connected to an applied potential \(V_{a b}\). After the charges on the capacitors have reached their final values, the charge on \(C_{2}\) is \(40.0 \mu C\). (a) What are the charges on capacitors \(C_{1}\) and \(C_{3}\)? (b) What is the applied voltage \(V_{a b}\)? Equation transcription: Text transcription: C{1}=6.00 \mu F, C{2}=3.00 \mu F C{3}=5.00 \mu F C{2} 40.0 \mu C C{1} C{3} V{a b}

For the system of capacitors shown in Fig. E24.21, a potential difference of is maintained across . (a) What is the equivalent capacitance of this system between and (b) How much charge is stored by this system? (c) How much charge does the capacitor store? (d) What is the potential difference across the capacitor?

Figure E24.22 shows a system of four capacitors, where the potential difference across is . (a) Find the equivalent capacitance of this system between and . (b) How much charge is stored by this combination of capacitors? (c) How much charge is stored in each of the \(10.0 \mu F\) and the \(9.0 \mu F\) capacitors? Equation transcription: Text transcription: 10.0 \mu F 9.0 \mu F

Suppose the 3-F capacitor in Fig. 24.10a were removed and replaced by a different one, and that this changed the equivalent capacitance between points and to 8 F.. What would be the capacitance of the replacement capacitor?

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 dx. 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 dW = Fdx done in pulling the plates apart. Find an expression for F. (d) Explain why F is not equal to QE where E is the electric field between the plates.

Problem 28E A parallel-plate vacuum capacitor has 8.38 J of energy stored in it. The separation between the plates is 2.30 mm. If the separation is decreased to 1.15 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?

Problem 29E 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?

Problem 39E The dielectric to be used in a parallel-plate capacitor has a dielectric constant of 3.60 and a dielectric strength of 1.60 X 107 V/m. The capacitor is to have a capacitance of 1.25 X 10-9 F and must be able to withstand a maximum potential difference of 5500 V. What is the minimum area the plates of the capacitor may have?

BIO 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 \(\pm 0.50 \times 10^{-3} C / m^{3}\), the cell wall is thick, and the cell wall material is air. (a) Find the magnitude of \(\overline{\bar{E}}\) 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) A typical cell in the human body has a volume of \(10^{-16} m^{3}\). 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 . Repeat parts (a) and (b) in this case. Equation transcription: Text transcription: pm 0.50 times 10^{-3} C / m^{3} overline{bar{E}} 10^{-16} m^{3}

A capacitor has parallel plates of area \(12 \mathrm{~cm}^{2}\) separated by 2.0 mm. The space between the plates is filled with polystyrene (see Table 24.2). (a) Find the permittivity of polystyrene. (b) Find the maximum permissible voltage across the capacitor to avoid dielectric breakdown. (c) When the voltage equals the value found in part (b), find the surface charge density on each plate and the induced surface charge density on the surface of the dielectric.

A parallel-plate air capacitor is made by using two plates 16 cm square, spaced 3.7 mm apart. It is connected to a 12-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric Held 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 mm are the answers to parts (a)?(d)?

Problem 49P Suppose the battery in Problem remains connected while the plates are pulled apart. What are the answers, then to parts (a)-(d) after the plates have been pulled apart? Problem: A parallel-plate air capacitor is made by using two plates 16 cm square, spaced 3.7 mm apart. It is connected to a 12-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric Held between the plates? (d) What 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 mm are the answers to parts (a)-(d)?

Cell Membranes. Cell membranes (the walled enclosure around a cell) are typically about 7.5 nm thick. They are partially permeable to allow charged material to pass in and out, as needed. Equal but opposite charge densities build up on the inside and outside faces of such a membrane, and these charges prevent additional charges from passing through the cell wall. We can model a cell membrane as a parallel-plate capacitor, with the membrane itself containing proteins embedded in an organic material to give the membrane a dielectric constant of about 10. (See Fig. P24.50.) (a) What is the capacitance per square centimeter of such a cell wall? (b) In its normal resting state, a cell has a potential difference of 85 mV across its membrane. What is the electric field inside this membrane?

Problem 51P 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 mm, the outer cylinder has radius 3.10 mm, and the length of each cylinder is 36.0 cm. If the potential difference between the surfaces of the two cylinders is 80.0 V, what is the magnitude or the electric field al a point between the two cylinders that is. a distance of 2.80 mm from their common axis and midway between the ends of the cylinders?

Problem 52P 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 mm2, and the separation between the plates is 0.700 mm before the key is depressed. (a) Calculate the capacitance before the key is depressed. (b) If the circuitry can detect a change in capacitance of 0.250 pF, how far must the key be depressed before the circuitry detects its depression?

Problem 53P A 20.0-µF 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-µF 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.

In Fig. 24.9a, let \(\text { let } C_{1}=9.0 \mu \mathrm{F}, \quad C_{2}=4.0 \mu \mathrm{F}\), and \(V_{a b}=36\) Suppose the charged capacitors are disconnected from the source and from each other, and then reconnected to each other with plates of opposite sign together. By how much does the energy of the system decrease?

Several \(0.25-\mu \mathrm{F}\) capacitors are available. The voltage each is not to exceed 600 V. You need to make a capacitor with capacitance \(0.25-\mu \mathrm{F}\) to be connected across a potential difference of 960 V. (a) Show in a diagram how an equivalent capacitor with the desired properties can be obtained. (b) No dielectric is a perfect insulator that would not permit the flow of any charge through its volume. Suppose that the dielectric in one of the capacitors in your diagram is a moderately good conductor. What will happen in this case when your combination of capacitors is connected across the 960-V potential difference?

In Fig. P24.57, \(C_{1}=C_{3}=8.4 \mu F\) and \(C_{2}=C_{3}=C_{4}=4.2 \mu F\). The applied potential is \(V_{a b}=220 V\). (a) What is the equivalent capacitance of the network between points and (b) Calculate the charge on each capacitor and the potential difference across each capacitor. Equation transcription: Text transcription: C{1}=C{3}=8.4 \mu F C{2}=C{3}=C{4}=4.2 \mu F V{a b}=220 V

Problem 58P 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 nF; (b) 450 nF; (c) 25 nF; (d) 75 nF.

In Fig. E24.20, \(C_{1}=3.00 \mu \mathrm{F} \text { and } V_{a b}=150 \mathrm{~V}\). The charge on capacitor \(C_1\) is \(150 \mu \mathrm{C}\) and the charge on \(C_3\) is \(450 \mu \mathrm{C}\). What are the values of the capacitances of \(C_2\) and \(C_3\)?

Problem 69P Earth-Ionosphere Capacitance. The earth can be considered as a single-conductor capacitor. It can also be considered in combination with a charged layer of the atmosphere, the ionosphere, as a spherical capacitor with two plates, the surface of the earth being the negative plate. The ionosphere is at a level of about 70 km, and the potential difference between earth and ionosphere is about 350,000 V. Calculate: (a) the capacitance of this system; (b) the total charge 011 the capacitor; (c) the energy stored in the system.

The inner cylinder of a long, cylindrical capacitor has radius \(r_a\) and linear charge density \(+\lambda\). It is surrounded by a coaxial cylindrical conducting shell with inner radius \(r_{b}\) and linear charge density \(-\lambda\) (see Fig. 24.6). (a) What is the energy density in the region between the conductors at a distance from the axis? (b) Integrate the energy density calculated in part (a) over the volume between the conductors in a length of the capacitor to obtain the total electric-field energy per unit length. (c) Use Eq. (24.9) and the capacitance per unit length calculated in Example 24.4 (Section 24.1) to calculate U/L Does your result agree with that obtained in part (b)?

Problem 71P A capacitor has a potential difference of 2.25 × 103 V between its plates. A short aluminum wire with initial temperature 23.0°C is connected between the plates of the capacitor and all the energy stored in the capacitor goes into heating the wire. The wire has mass 12.0 g. If 110 heat is lost to the surroundings and the final temperature of the wire is 34.2°C, what is the capacitance of the capacitor?

An isolated spherical capacitor has charge +Q on its inner conductor \(\left(\text{radius }r_a\right)\) and charge -Q on its outer conductor \(\left(\text{radius }r_b\right)\). Half of the volume between the two conductors is then filled with a liquid dielectric of constant K, as shown in the cross section in Fig. P24.78. (a) Find the capacitance of the half-filled capacitor (b) Find the magnitude of \(\overrightarrow{\boldsymbol{E}}\) in the volume between the two conductors as a function of the distance r from the center of the capacitor. Give answers for both the upper and lower halves of this volume. (c) Find the surface density of free charge on the upper and lower halves of the inner and outer conductors. (d) Find the surface density of bound charge on the inner \(\left(r=r_{a}\right)\) and outer \(\left(r=r_{b}\right)\) surfaces of the dielectric. (e) What is the surface density of bound charge on the flat surface of the dielectric? Explain.