You are given four identical capacitors and a 100-

If the voltage across a parallel-plate capacitor is doubled, its capacitance (a) doubles, (b) drops by half, (c) remains the same.

If the charge on an isolated spherical conductor is doubled, its self-capacitance (a) doubles, (b) drops by half, (c) remains the same.

True or false: The electrostatic energy density is uniformly distributed in the region between the conductors of a cylindrical capacitor.

If the distance between the plates of a charged and isolated parallel-plate capacitor is doubled, what is the ratio of the final stored energy to the initial stored energy?

A parallel-plate capacitor is connected to a battery. The space between the two plates is empty. If the separation between the capacitor plates is tripled while the capacitor remains connected to the battery, what is the ratio of the final stored energy to the initial stored energy?

If the capacitor of Problem 5 is disconnected from the battery before the separation between the plates is tripled, what is the ratio of the final stored energy to the initial stored energy?

True or false: (a) The equivalent capacitance of two capacitors in parallel is always greater than the larger of the two capacitance values. (b) The equivalent capacitance of two capacitors in series is always less than the least of the two capacitance values if the charges on the two plates that are connected by an otherwise isolated conductor sum to zero.

Two uncharged capacitors have capacitances and respectively, and are connected in series. This series combination is then connected across the terminals a battery. Which of the following is true? (a) The capacitor has twice the charge of the other capacitor. (b) The voltage across each capacitor is the same. (c) The energy stored by each capacitor is the same. (d) The equivalent capacitance is (e) The equivalent capacitance is 9 A dielectric is inserted between the plates of a parallelplate

pacitor, completely filling the region between the plates. Air initially filled the region between the two plates. The capacitor was connected to a battery during the entire process. True or false: 2C0 >3. 3C0 . 2C0 C 2C0 , 0 SSM (a) The capacitance value of the capacitor increases as the dielectric is inserted between the plates. (b) The charge on the capacitor plates decreases as the dielectric is inserted between the plates. (c) The electric field between the plates does not change as the dielectric is inserted between the plates. (d) The energy storage of the capacitor decreases as the dielectric is inserted between the plates.

Capacitors A and B (Figure 24-33) have identical plate areas and gap separations. The space between the plates of each capacitor is half-filled with a dielectric as shown. Which has the larger capacitance, capacitor Aor capacitor B? Explain your answer. SSM

(a) Two identical capacitors are connected in parallel. This combination is then connected across the terminals of a battery. How does the total energy stored in the parallel combination of the two capacitors compare to the total energy stored if just one of the capacitors were connected across the terminals of the same battery? (b) Two identical capacitors that have been discharged are connected in series. This combination is then connected across the terminals of a battery. How does the total energy stored in the series combination of the two capacitors compare to the total energy stored if just one of the capacitors were connected across the terminals of the same battery?

Two identical capacitors that have been discharged are connected in series across the terminals of a battery. When only one of the capacitors is connected across the terminals of the battery, the energy stored is What is the total energy stored in the two capacitors when the series combination is connected to the battery? (a) , (b) , (c) , (d) , (e) U0 U >4

Disconnect the coaxial cable from a television or other device and estimate the diameter of the inner conductor and the diameter of the shield. Assume a plausible value (see Table 24-1) for the dielectric constant of the dielectric separating the two conductors and estimate the capacitance per unit length of the cable.

ENGINEERING APPLICATION, CONTEXT-RICH You are part of an engineering research team that is designing a pulsed nitrogen laser. To create the high energy densities needed to operate such a laser, the electrical discharge from a high-voltage capacitor is used. Typically, the energy requirement per pulse (i.e., per discharge) is Estimate the capacitance required if the discharge is to create a spark across a gap of about Assume that the dielectric breakdown of nitrogen is the same as the value for normal air.

Estimate the capacitance of the Leyden jar shown in Figure 24-34. The figure of a man is one-tenth the height of an average man. SSM 1.0 cm. 100 J. SSM

An isolated conducting sphere that has a radius has an electric potential of (the potential far from the sphere is zero). (a) How much charge is on the sphere? (b) What is the self-capacitance of the sphere? (c) By how much does the selfcapacitance change if the spheres electric potential is increased to

The charge on one plate of a capacitor is and the charge on the other plate is The potential difference between the plates is What is the capacitance of the capacitor?

Two isolated conducting spheres of equal radius have charges and respectively. Their centers are separated by a distance that is large compared to their radius. Estimate the capacitance of this unusual capacitor.

(a) The potential difference between the plates of a capacitor is How much energy is stored in the capacitor? (b) How much additional energy is required to increase the potential difference between the plates from to SSM 100 V 200 V?

The charges on the plates of a 10- capacitor are (a) How much energy is stored in the capacitor? (b) If charge is transferred until the charges on the plates are equal to how much stored energy remains?

(a) Find the energy stored in a 20.0- capacitor when the charges on the plates are (b) How much additional energy is stored if charges are increased from to

What is the maximum electric energy density in a region containing dry air at standard conditions?

An air-gap parallel-plate capacitor that has a plate area of and a separation of is charged to (a) What is the electric field between the plates? (b) What is the electric energy density between the plates? (c) Find the total energy by multiplying your answer from Part (b) by the volume between the plates. (d) Determine the capacitance of this arrangement. (e) Calculate the total energy from and compare your answer with your result from Part (c).

A solid metal sphere has radius of and a concentric metal spherical shell has an inside radius of The solid sphere has a charge (a) Estimate the energy stored in the electric field in the region between the spheres. Hint: You can treat the spheres essentially as parallel flat slabs separated by (b) Estimate the capacitance of this two-sphere system. (c) Estimate the total energy stored in the electric field from and compare it to your answer in Part (a).

A parallel-plate capacitor has plates of area and is connected across the terminals of a battery. After some time has passed, the capacitor is disconnected from the battery. When the plates are then moved farther apart, the charge on each plate remains constant but the potential difference between the plates increases by (a) What is the magnitude of the charge on each plate? (b) Do you expect the energy stored in the capacitor to increase, decrease, or remain constant as the plates are moved this way? Explain your answer. (c) Support your answer to Part (b) by determining the change in stored energy in the capacitor due to the movement of the plates.

(a) How many 1.00- capacitors connected in parallel would it take to store a total charge of if the potential difference across each capacitor is Diagram the parallel combination. (b) What would be the potential difference across this parallel combination? (c) If the capacitors in Part (a) are discharged, connected in series, and then energized until the potential difference across each is equal to find the charge on each capacitor and the potential difference across the connection.

A3.00- capacitor and a 6.00- capacitor are discharged and then connected in series, and the series combination is then connected in parallel with an 8.00- capacitor. Diagram this combination. What is the equivalent capacitance of this combination?

Three capacitors are connected in a triangle as shown in Figure 24-35. Find an expression for the equivalent capacitance between points and in terms of the three capacitance values.

A \(10.0-\mu \mathrm{F}\) capacitor and a \(20.0-\mu \mathrm{F}\) capacitor are connected in parallel across the terminals of a 6.00-V battery. (a) What is the equivalent capacitance of this combination? (b) What is the potential difference across each capacitor? (c) Find the charge on each capacitor. (d) Find the energy stored in each capacitor.

A 10.0- capacitor and a 20.0- capacitor are discharged and then connected in series. The series combination is then connected across the terminals of a 6.00- battery. (a) What is the equivalence capacitance of this combination? (b) Find the charge on each capacitor. (c) Find the potential difference across each capacitor. (d) Find the energy stored in each capacitor.

Three identical capacitors are connected so that their maximum equivalent capacitance, which is is obtained. (a) Determine how the capacitors are connected and diagram the combination. (b) There are three additional ways to connect all three capacitors. Diagram these three ways and determine the equivalent capacitances for each arrangement.

For the circuit shown in Figure 24-36, the capacitors were each discharged before being connected to the voltage source. Find (a) the equivalent capacitance of the combination, (b) the charge stored on the positively charged plate of each capacitor, (c) the voltage across each capacitor, and (d) the energy stored in each capacitor.

(a) Show that the equivalent capacitance of two capacitors in series can be written (b) Using only this formula and some algebra, show that must always be less than and and hence must be less than the smaller of the two values. (c) Show that the equivalent capacitance of three capacitors in series can be written (d) Using only this formula and some algebra, show that must always be less than each of and and hence must be less than the least of the three values.

For the circuit shown in Figure 24-37 find (a) the equivalent capacitance between the terminals, (b) the charge stored on the positively charged plate of each capacitor, (c) the voltage across each capacitor, and (d) the total stored energy.

Five identical capacitors of capacitance are connected in a so-called bridge network, as shown in Figure 24-38. (a) What is the equivalent capacitance between points a and b? (b) Find the equivalent capacitance between points and if the capacitor at the center is replaced by a capacitor that has a capacitance of 10C0 .

You and your laboratory team have been given a project by your electrical engineering professor. Your team must design a network of capacitors that has an equivalent capacitance of and breakdown voltage of The restriction is that your team must use only 2.00- capacitors that have individual breakdown voltages of Diagram the combination.

Find the different equivalent capacitances that can be obtained by using two or three of the following capacitors: a 1.00- capacitor, a 2.00- capacitor, and a 4.00- capacitor.

What is the equivalent capacitance (in terms of which is the capacitance of one of the capacitors) of the infinite ladder of capacitors shown in Figure 24-39?

A parallel-plate capacitor has a capacitance of and a plate separation of (a) What is the maximum potential difference between the plates, so that dielectric breakdown of the air between the plates does not occur? (b) How much charge is stored at this potential difference?

An electric field of exists between the circular plates of a parallel-plate capacitor that has a plate separation of (a) What is the potential difference across the capacitor plates? (b) What plate radius is required if the positively charged plate is to have a charge of

A parallel-plate, air-gap capacitor has a capacitance of The plates are apart. (a) What is the area of each plate? (b) What is the potential difference between the plates if the positively charged plate has a charge of (c) What is the stored energy? (d) What is the maximum energy this capacitor can store before dielectric breakdown of the air between the plates occurs?

Design a parallel-plate capacitor that has air between its plates and that can be charged to a maximum potential difference of before dielectric breakdown occurs. (a) What is the minimum possible separation between the plates? (b) What minimum area must each plate of the capacitor have?

In preparation for an experiment that you will do in your introductory nuclear physics lab, you are shown the inside of a Geiger tube. You measure the radius and the length of the central wire of the Geiger tube to be and respectively. The outer surface of the tube is a conducing cylindrical shell that has an inner radius of The shell is coaxial with the wire and has the same length Calculate (a) the capacitance of your tube, assuming that the gas in the tube has a dielectric constant of 1.00, and (b) the value of the linear charge density on the wire when the potential difference between the wire and shell is of

A cylindrical capacitor consists of a long wire that has a radius a length and a charge The wire is enclosed by a coaxial outer cylindrical shell that has an inner radius length and charge (a) Find expressions for the electric field and energy density as a function of the distance from the axis. (b) How much energy resides in a region between the conductors that has a radius a thickness and a volume (c) Integrate your expression from Part (b) to find the total energy stored in the capacitor. Compare your result with that obtained by using the formula in conjunction with the known expression for the capacitance of a cylindrical capacitor.

Three concentric, thin, long conducting cylindrical shells have radii of and The space between the shells is filled with air. The innermost and outermost shells are connected at one end by a conducting wire. Find the capacitance per unit length of this configuration.

ENGINEERING APPLICATION A goniometer is a precise instrument for measuring angles. A capacitive goniometer is shown in Figure 24-40a. Each plate of the variable capacitor (Figure 24-40b) consists of a flat metal semicircle that has an inner radius and an outer radius The plates share a common rotation axis, and the width of the air gap separating the plates is Calculate the capacitance as a function of the angle u and the parameters given.

ENGINEERING APPLICATION A capacitive pressure gauge is shown in Figure 24-41. Each plate has an area The plates are separated by a material that has a dielectric constant a thickness and a Youngs modulus If a pressure increase of is applied to the plates, derive an expression for the change in capacitance.

Model Earth as a conducting sphere. (a) What is its selfcapacitance? (b) Assume the magnitude of the electric field at Earths surface is What charge density does this correspond to? Express this value in fundamental charge units per square centimeter.

A spherical capacitor consists of a thin spherical shell that has a radius and a thin, concentric spherical shell that has a radius where (a) Show that the capacitance is given by (b) Show that when the radii of the shells are nearly equal, the capacitance is approximately given by the expression for the capacitance of a parallel-plate capacitor, where is the area of the sphere and

A spherical capacitor is composed of an inner sphere which has a radius and a charge and an outer concentric spherical thin shell which has a radius and a charge (a) Find the electric field and the energy density as a function of where is the distance from the center of the sphere, for (b) Calculate the energy associated with the electrostatic field in a spherical shell between the conductors that has a radius a thickness and a volume (c) Integrate your expression from Part (b) to find the total energy and compare your result with the result obtained using

An isolated conducting sphere of radius has a charge distributed uniformly over its surface. Find the distance from the center of the sphere such that half the total electrostatic energy of the system is associated with the electric field beyond that distance.

A 2.00- capacitor is energized to a potential difference of The wires connecting the capacitor to the battery are then disconnected from the battery and connected across a second, initially uncharged capacitor. The potential difference across the 2.00- capacitor then drops to What is the capacitance of the second capacitor?

A 100- capacitor and a 400- capacitor are both charged to They are then disconnected from the voltage source and are connected together, positive plate to negative plate and negative plate to positive plate. (a) Find the resulting potential difference across each capacitor. (b) Find the energy dissipated when the connections are made.

Two capacitors, one that has a capacitance of and one that has a capacitance of are first discharged and then are connected in series. The series combination is then connected across the terminals of a 12.0- battery. Next, they are carefully disconnected so that they are not discharged and they are then reconnected to each otherpositive plate to positive plate and negative plate to negative plate. (a) Find the potential difference across each capacitor after they are reconnected. (b) Find the energy stored in the capacitors before they are disconnected from the battery, and find the energy stored after they are reconnected.

A 1.2- capacitor is charged to After charging, the capacitor is disconnected from the voltage source and is connected across the terminals of a second capacitor that had previously been discharged. The final voltage across the 1.2- capacitor is (a) What is the capacitance of the second capacitor? (b) How much energy was dissipated when the connection was made?

A 12- capacitor and a capacitor of unknown capacitance are both charged to After charging, the two capacitors are disconnected from the voltage source. The capacitors are then connected to each otherpositive plate to negative plate and negative plate to positive plate. The final voltage across the terminals of the 12- capacitor is (a) What is the capacitance of the second capacitor? (b) How much energy was dissipated when the connection was made?

Two capacitors, one that has a capacitance of and one that has a capacitance of are connected in parallel. The parallel combination is then connected across the terminals of a 12.0- battery. Next, they are carefully disconnected so that they are not discharged. They are then reconnected to each other the positive plate of each capacitor connected to the negative plate of the other. (a) Find the potential difference across each capacitor after they are reconnected. (b) Find the energy stored in the capacitors before they are disconnected from the battery, and find the energy stored after they are reconnected.

A capacitor is charged to and then removed from the battery and connected to an uncharged 50- capacitor. (a) What is the new charge on each capacitor? (b) Find the energy stored in the 20- capacitor before it is disconnected from the battery, and the energy stored in the two capacitors after they are connected to each other. Does the stored energy increase or decrease when the two capacitors are connected to each other?

Capacitors 1, 2, and 3 have capacitances equal to and respectively. The capacitors are connected in parallel, and the parallel combination is connected across the terminals of a source. The capacitors are then disconnected from both the voltage source and each other, and are connected to three switches as shown in Figure 24-42. (a) What is the potential difference across each capacitor when switches and are closed but switch remains open? (b) After switch is closed, what is the final charge on the leftmost plate of each capacitor? (c) Give the final potential difference across each capacitor after switch S is closed. SSM 3

A capacitor has a capacitance and a charge on its positively charged plate. A student connects one terminal of the capacitor to a terminal of an identical capacitor whose plates are electrically neutral. When the remaining two terminals are connected, charge flows until electrostatic equilibrium is reestablished and both capacitors have charge on them. Compare the total energy initially stored in the one capacitor to the total energy stored in the two capacitors in which electrostatic equilibrium is reestablished. If there is less energy afterward, where do you think the missing energy went? Hint: Wires that transport charge can heat up, which is called Joule heating and is discussed in detail in Chapter 25.

ENGINEERING APPLICATION, CONTEXT-RICH You are a laboratory assistant in a physics department that has budget problems. Your supervisor wants to construct inexpensive parallelplate capacitors for use in introductory laboratory experiments. The design uses polyethylene, which has a dielectric constant of 2.30, Q>2 C Q between two sheets of aluminum foil. The area of each sheet of foil is and the thickness of the polyethylene is Find the capacitance of this arrangement.

The radius and the length of the central wire in a Geiger tube are and respectively. The outer surface of the tube is a conducing cylindrical shell that has an inner radius of The shell is coaxial with the wire and has the same length The tube is filled with a gas that has a dielectric constant of 1.08 and a dielectric strength of (a) What is the maximum potential difference that can be maintained between the wire and shell? (b) What is the maximum charge per unit length on the wire?

ENGINEERING APPLICATION, CONTEXT-RICH You are a materials science engineer and your group has fabricated a new dielectric, that has a dielectric constant of 24 and a dielectric strength of Suppose you want to use this material to construct a 0.10- parallel plate capacitor that can withstand a potential difference of (a) What is the minimum plate separation required to do this? (b) What is the area of each plate at this separation?

A parallel-plate capacitor has plates separated by a distance The capacitance of this capacitor is when no dielectric is in the space between the plates. However, the space between the plates is completely filled by two different dielectrics. One dielectric has a thickness and a dielectric constant and the other dielectric has a thickness and a dielectric constant Find the capacitance of this capacitor.

Two capacitors each have two conducting plates of surface area and an air gap of width They are connected in parallel, as shown in Figure 24-43, and each has a charge on the positively charged plate. A slab that has a width an area and a dielectric constant is inserted between the plates of one of the capacitors. Calculate the new charge on the positively charged plate of that capacitor after electrostatic equilibrium is reestablished.

A parallel-plate capacitor has a plate separation and has a capacitance equal to when there is only empty space in the space between the plates. Aslab of thickness where that has a dielectric constant is placed in the space between the plates- completely covering one of the plates. What is the capacitance with the slab inserted?

BIOLOGICAL APPLICATION The membrane of the axon of a nerve cell can be modeled as a thin cylindrical shell of radius having a length of and a thickness of The membrane has a positive charge on one side and a negative charge on the other, and the membrane acts as a parallelplate capacitor of area and separation Assume the membrane is filled with a material whose dielectric constant is 3.00. (a) Find the capacitance of the membrane. If the potential difference across the membrane is find (b) the charge on the positively charged side of the membrane, and (c) the electric field strength in the membrane.

The space between the plates of a capacitor that is connected across the terminals of a battery is filled with a dielectric material. Determine the dielectric constant of the material if the induced bound charge per unit area on it is (a) 80 percent of the free charge per unit area on the plates, (b) 20 percent of the free charge per unit area on the plates, and (c) 98 percent of the free charge per unit area on the plates.

The positively charged plate of a parallel-plate capacitor has a charge equal to When the space between the plates is evacuated of air, the electric field strength between the plates is When the space is filled with a certain dielectric material, the field strength between the plates is reduced to (a) What is the dielectric constant of the material? (b) If what is the area of the plates? (c) What is the total induced bound charge on either face of the dielectric material?

Find the capacitance of the parallel-plate capacitor shown in Figure 24-44.

You are given four identical capacitors and a 100- battery. When only one of the capacitors is connected to the battery the energy stored is Combine the four capacitors in such a way that the total energy stored in all four capacitors is Describe the combination and explain your answer.

Three capacitors have capacitances of and Find the equivalent capacitance if (a) the capacitors are connected in parallel and (b) the capacitors are connected in series.

A1.00- capacitor is connected in parallel with a 2.00- capacitor, and this combination is connected in series with a 6.00- capacitor. What is the equivalent capacitance of this combination?

The voltage across a parallel-plate capacitor that has a plate separation equal to is The capacitor is disconnected from the voltage source and the separation between the plates is increased until the energy stored in the capacitor has been doubled. Determine the final separation between the plates.

Determine the equivalent capacitance, in terms of of each of the combinations of capacitors shown in Figure 24-45. C0

Figure 24-46 shows four capacitors connected in the arrangement known as a capacitance bridge. The capacitors are initially uncharged. What must the relation between the four capacitances be so that the potential difference between points and d remains zero when a voltage V is applied between points a and b?

The plates of a parallel-plate capacitor are separated by distance and each plate has area The capacitor is charged to a potential difference and then disconnected from the voltage source. The plates are then pulled apart until the separation is Find (a) the new capacitance, (b) the new potential difference, and (c) the new stored energy. (d) How much work was required to change the plate separation from to

A parallel-plate capacitor has capacitance when there is no dielectric in the space between the plates. The space between the plates is then filled with a material that has a dielectric constant of When a second capacitor of capacitance is connected in series with the first, the capacitance of the series combination is Find in terms of

A parallel combination of two identical 2.00- parallelplate capacitors (no dielectric is in the space between the plates) is connected to a 100- battery. The battery is then removed and the separation between the plates of one of the capacitors is doubled. Find the charge on the positively charged plate of each of the capacitors.

A parallel-plate capacitor that has no dielectric in the space between the plates has a capacitance and a plate separation Two dielectric slabs that have dielectric constants of and respectively, are then inserted between the plates as shown in Figure 24-47. Each slab has a thickness and has area the same area as each capacitor plate. When the charge on the positively charged capacitor plate is find (a) the electric field in each dielectric and (b) the potential difference between the plates. (c) Show that the capacitance of the system after the slabs are inserted is given by (d) Show that is the equivalent capacitance of a series combination of two capacitors, each having plates of area and a gap width equal to The space between the plates of one is filled with a material that has a dielectric constant equal to and the space between the plates of the other is filled with a material that has a dielectric constant equal to

The plates of a parallel-plate capacitor are separated by distance and each plate has area Ametal slab of thickness and area is inserted between the plates in such a way that the slab is parallel with the capacitor plates. (a) Show that the new capacitance is given by regardless of the distance between the metal slab and the positively charged plate. (b) Show that this arrangement can be modeled as a capacitor that has plate separation in series with a capacitor of plate separation where a _ b _ d _ d0 b, .

A parallel-plate capacitor that has plate area is filled with two dielectrics of equal size, as shown in Figure 24-48. (a) Show that this system can be modeled as two capacitors that are connected in parallel and each have an area (b) Show that the capacitance is given by where is the capacitance if there were no dielectric materials in the space between the plates.

A parallel-plate capacitor that has no dielectric in the space between the plates has a plate area and a gap width A charge is on the positively charged plate. (a) Find the stored electrostatic energy as a function of (b) Find the increase in energy due to an increase in plate separation from (c) If is the force exerted by one plate on the other, the work needed to move one plate a distance is Show that (d) Show that the force in Part (c) equals where is the charge on one plate and is the electric field between the plates. Give a conceptual explanation for the factor in this result.

A rectangular parallelplate capacitor that has a length and a width has a dielectric that has a width partially inserted a distance between the plates, as shown in Figure 24-49. (a) Find the capacitance as a function of Neglect edge effects. (b) Show that your answer gives the expected results for and

An electrically isolated capacitor that has a charge on its positively charged plate is partly filled with a dielectric substance as shown in Figure 24-49. The capacitor consists of two rectangular plates that have edge lengths and and are separated by distance The dielectric is inserted into the gap a distance (a) What is the energy stored in the capacitor? Hint: The capacitor can be modeled as two capacitors connected in parallel. (b) Because the energy of the capacitor decreases as increases, the electric field must be doing work on the dielectric, meaning that there must be an electric force pulling it in. Calculate this force by examining how the stored energy varies with (c) Express the force in terms of the capacitance and potential difference between the plates. (d) From where does this force originate?

Aspherical capacitor consists of a solid conducting sphere that has a radius and a charge and a concentric conducting spherical shell that has an inner radius and a charge The space between the two is filled with two different dielectric materials of dielectric constants and The boundary between the two dielectrics occurs a distance from the center. (a) Calculate the electric field in the regions and (b) Integrate the expression to obtain the potential difference, between the two conductors. (c)Use to obtain an expression for the capacitance of this system. (d) Show that your answer from Part (c) simplifies to the expected one if equals

A capacitance balance is shown in Figure 24-50. The balance has a weight attached on one side and a capacitor that has a variable gap width on the other side. Assume the upper plate of the capacitor has negligible mass. When the capacitor potential difference between the plates is the attractive force between the plates balances the weight of the hanging mass. (a) Is the balance stable? That is, if we balance it out, and then move the plates a little closer together, will they snap shut or move back to the equilibrium point? (b) Calculate the value of required to balance an object of mass assuming the plates are separated by distance and have area Hint: A useful relation is that the force between the plates is equal to the derivative of the stored electrostatic energy with respect to the plate separation.

ENGINEERING APPLICATION, CONTEXT-RICH You are an intern at an engineering company that makes capacitors used for energy storage in pulsed lasers. Your manager asks your team to construct a parallel-plate, air-gap capacitor that will store of energy. (a) What minimum volume is required between the plates of the capacitor? (b) Suppose you have developed a dielectric that has a dielectric strength of and a dielectric constant of 5.00. What volume of this dielectric, between the plates of the capacitor, is required for it to be able to store of energy?

Consider two parallel-plate capacitors, and that are connected in parallel. The capacitors are identical except that has a dielectric inserted between its plates. A battery is connected across the combination until electrostatic equilibrium is established, and then the battery is disconnected. (a) What is the charge on each capacitor? (b) What is the total stored energy of the capacitors? (c) The dielectric is removed from What is the final stored energy of the capacitors? (d) What is the final voltage across the two capacitors?

A capacitor is constructed of two coaxial conducting thin cylindrical shells of radii and which have a length Acharge of is on the inner cylinder, and a charge of is on the outer cylinder. The region between the two cylinders is filled with a material that has a dielectric constant (a) Find the potential difference between the cylinders. (b) Find the density of the free charge on the inner cylinder and the outer cylinder. (c) Find the bound charge density on the inner cylindrical surface of the dielectric and on the outer cylindrical surface of the dielectric. (d) Find the total stored energy. (e) If the dielectric will move without friction, how much mechanical work is required to remove the dielectric cylindrical shell?

Before switch S is closed, as shown in Figure 24-51, the voltage across the terminals of the switch is and the voltage across the capacitor labeled is The capacitance of is The total energy stored in the two capacitors is After closing the switch, the voltage across each capacitor is and the energy stored by the two capacitors has dropped to Determine the capacitance of and the charge on that capacitor before the switch was closed.

An air-filled parallel-plate capacitor that has gap width has plates that each have an area The capacitor is charged to a potential difference and is then removed from the voltage source. A dielectric slab that has a dielectric constant of 2.00, a thickness and an area is then inserted, as shown in Figure 24-52. Let s1 be the free charge density at the conductordielectric surface, and (a) Explain why the electric field must have the same value inside the dielectric as in the free space between the plates. (b) Show that (c) Show that the final capacitance (after the slab is inserted) is 1.50 times the capacitance when the capacitor is filled with air. (d) Show that the final potential difference is (e) Show that energy stored after the slab is inserted is only two-thirds of the energy stored before insertion.

A capacitor has rectangular plates of length and width The top plate is inclined at a small angle, as shown in Figure 24-53. The plate separation varies from at the left to at the right, where is much less than or Calculate the capacitance of this arrangement. Hint: Break the problem up into a parallel combination. Choose strips of width and length b to approximate small (differential) capacitors (each having a value of ). Each will have a plate area of b and separation distance Then argue that these differential capacitors are connected in parallel.

Not all dielectrics that separate the plates of a capacitor are rigid. For example, the membrane of a nerve axon is a lipid bilayer that has a finite compressibility. Consider a parallel-plate capacitor whose plate separation is maintained by a material that has a dielectric constant of 3.00, a dielectric strength of and a Youngs modulus for compressive stress of When the potential difference between the capacitor plates is zero, the thickness of the dielectric is equal to and the capacitance of the capacitor is given by (a) Derive an expression for the capacitance as a function of the potential difference between the capacitor plates. (b) What is the maximum value of this potential difference? (Assume that the dielectric constant and the dielectric strength do not change under compression.)