Chapter 16

A pair of parallel plates is charged by a 12-V battery. If the electric field between the plates is \(1200 \mathrm{~N} / \mathrm{C}\), how far apart are the plates?

A pair of parallel plates is charged by a 12-V battery. How much work is required to move a particle with a charge of \(-4.0 \mu C\) from the positive to the negative plate?

If it takes \(+1.6 \times 10^{-5} \mathrm{~J}\) to move a positively charged particle between two charged parallel plates, (a) what is the charge on the particle if the plates are connected to a 6.0-V battery? (b) Was it moved from the negative to the positive plate or from the positive to the negative plate?

An electron is accelerated by a uniform electric field \((1000 \mathrm{~V} / \mathrm{m})\) pointing vertically upward. Use Newton's laws to determine the electron's velocity after it moves \(0.10 \mathrm{~cm}\) from rest.

Consider two points at different distances from a positive point charge. (a) The point closer to the charge is at a (1) higher, (2) equal, (3) lower potential than the point farther away. Why? (b) How much different is the electric potential \(20 \mathrm{~cm}\) from a charge of \(5.5 \mu \mathrm{C}\) compared to \(40 \mathrm{~cm}\) from the same charge?

(a) At one-third the original distance from a positive point charge, by what factor is the electric potential changed: \((1) 1 / 3,(2) 3,(3) 1 / 9,\) or (4) \(9 ?\) Why? (b) How far from \(a+1.0-\mu C\) charge is a point with an electric potential value of \(10 \mathrm{kV} ?\) (c) How much of a change in potential would occur if the point were moved to three times that distance?

According to the Bohr model of the hydrogen atom (see Chapter 27 ), the electron can exist only in circular orbits of certain radii about a proton. (a) Will a larger orbit have (1) a higher, (2) an equal, or (3) a lower electric potential than a smaller orbit? Why? (b) Determine the potential difference between two orbits of radii \(0.21 \mathrm{nm}\) and \(0.48 \mathrm{nm}\).

It takes +6.0 J of work to move two charges from a large distance apart to \(1.0 \mathrm{~cm}\) from one another. If the charges have the same magnitude, (a) how large is each charge, and (b) what can you tell about their signs?

\(\mathrm{A}+2.0-\mu \mathrm{C}\) charge is initially \(0.20 \mathrm{~m}\) from a fixed \(-5.0-\mu C\) charge and is then moved to a position \(0.50 \mathrm{~m}\) from the fixed charge. (a) How much work is required to move the charge? (b) Does the work depend on the path through which the charge is moved?

A uniform electric field of \(10 \mathrm{kV} / \mathrm{m}\) points vertically upward. How far apart are the equipotential planes that differ by \(100 \mathrm{~V} ?\)

If the radius of the equipotential surface of a point charge is \(10.5 \mathrm{~m}\) and is at a potential of \(+2.20 \mathrm{kV}\) (compared to zero at infinity), what are the magnitude and sign of the point charge?

(a) The equipotential surfaces in the neighborhood of a positive point charge are spheres. Which sphere is associated with the higher electric potential: (1) the smaller one, (2) the larger one, or (3) they are associated with the same potential? (b) Calculate the amount of work (in electron-volts) it would take to move an electron from \(12.6 \mathrm{~m}\) to \(14.3 \mathrm{~m}\) away from \(\mathrm{a}+3.50-\mu \mathrm{C}\) point charge.

The potential difference between the cloud and ground in a typical lightning discharge may be up to 100 MV (million volts). What is the gain in kinetic energy of an electron accelerated through this potential difference? Give your answer in both electron-volts and joules. (Assume that there are no collisions.)

In a typical Van de Graaff linear accelerator, protons are accelerated through a potential difference of \(20 \mathrm{MV}\). What is their kinetic energy if they started from rest? Give your answer in (a) \(\mathrm{eV},\) (b) \(\mathrm{keV},\) (c) \(\mathrm{MeV}\), (d) \(\mathrm{GeV},\) and (e) joules.

Calculate the voltage required to accelerate a beam of protons initially at rest, and calculate their speed if they have a kinetic energy of (a) \(3.5 \mathrm{eV},\) (b) \(4.1 \mathrm{keV}\), and (c) \(8.0 \times 10^{-16} \mathrm{~J}\)

How much charge flows through a 12-V battery when a \(2.0-\mu \mathrm{F}\) capacitor is connected across its terminals?

What plate separation is required for a parallel plate capacitor to have a capacitance of \(9.00 \mathrm{nF}\) if the plate area is \(0.425 \mathrm{~m}^{2}\) ?

(a) For a parallel plate capacitor with a fixed plate separation distance, a larger plate area results in (1) a larger capacitance value, (2) an unchanged capacitance value, (3) a smaller capacitance value. (b) A 2.50 -nF parallel plate capacitor has a plate area of \(0.514 \mathrm{~m}^{2}\). If the plate area is doubled, what is the new capacitance value?

A 12.0 -V battery remains connected to a parallel plate capacitor with a plate area of \(0.224 \mathrm{~m}^{2}\) and a plate separation of \(5.24 \mathrm{~mm}\). (a) What is the charge on the capacitor? (b) How much energy is stored in the capacitor? (c) What is the electric field between its plates?

Current state-of-the-art capacitors are capable of storing many times the energy of older ones. Such a capacitor, with a capacitance of \(1.0 \mathrm{~F}\), is able to light a small 0.50 - \(W\) bulb at steady full power for 5.0 s before it quits. What is the terminal voltage of the battery that charged the capacitor?

Two parallel plates have a capacitance value of \(0.17 \mu \mathrm{F}\) when they are \(1.5 \mathrm{~mm}\) apart. They are connected permanently to a 100 -V power supply. If you pull the plates out to a distance of \(4.5 \mathrm{~mm},\) (a) what is the electric field between them? (b) By how much has the capacitor's charge changed? (c) By how much has its energy storage changed? (d) Repeat these calculations assuming the power supply is disconnected before you pull the plates further apart.

A capacitor has a capacitance of \(50 \mathrm{pF}\), which increases to \(150 \mathrm{pF}\) when a dielectric material is between its plates. What is the dielectric constant of the material?

A 50 -pF capacitor is immersed in silicone oil, which has a dielectric constant of \(2.6 .\) When the capacitor is connected to a 24 - \(V\) battery, \((\) a) what will be the charge on the capacitor? (b) How much energy is stored in the capacitor?

co A parallel plate capacitor has a capacitance of \(1.5 \mu \mathrm{F}\) with air between the plates. The capacitor is connected to a 12-V battery and charged. The battery is then removed. When a dielectric is placed between the plates, a potential difference of \(5.0 \mathrm{~V}\) is measured across the plates. (a) What is the dielectric constant of the material? (b) What happened to the energy storage in the capacitor: (1) it increased, (2) it decreased, or (3) it stayed the same? (c) By how much did the energy storage of this capacitor change when the dielectric was inserted?

An air-filled parallel plate capacitor has rectangular plates with dimensions of \(6.0 \mathrm{~cm} \times 8.0 \mathrm{~cm} .\) It is connected to a 12-V battery. While the battery remains connected, a sheet of 1.5 -mm-thick Teflon (dielectric constant of 2.1 ) is inserted and completely fills the space between the plates. (a) While the dielectric was being inserted, (a) charge flowed onto the capacitor, (2) charge flowed off the capacitor, (3) no charge flowed. (b) Determine the change in the charge storage of this capacitor because of the dielectric insertion. (c) Determine the change in energy storage in this capacitor because of the dielectric insertion. (d) By how much was the battery's stored energy changed?

What is the equivalent capacitance of two capacitors with capacitances of \(0.40 \mu \mathrm{F}\) and \(0.60 \mu \mathrm{F}\) when they are connected (a) in series and (b) in parallel?

Two identical capacitors are connected in series and their equivalent capacitance is \(1.0 \mu \mathrm{F}\). What is each one's capacitance value? Repeat the calculation if, instead, they were connected in parallel.

(a) Two capacitors can be connected to a battery in either a series or parallel combination. The parallel combination will require (1) more, (2) equal, (3) less energy from a battery than the series combination. Why? (b) Two uncharged capacitors, one with a capacitance of \(0.75 \mu \mathrm{F}\) and the other with that of \(0.30 \mu \mathrm{F}\) are connected in series to a 12-V battery. Then the capacitors are disconnected, discharged, and reconnected to the same battery in parallel. Calculate the energy loss of the battery in both cases.

(a) Three capacitors of equal capacitance are connected in parallel to a battery, and together they acquire a certain total charge \(Q\) from that battery. Will the charge on each capacitor be \((1) Q,(2) 3 Q,\) or \((3) Q / 3 ?\) (b) Three capacitors of \(0.25 \mu \mathrm{F}\) each are connected in parallel to a 12-V battery. What is the charge on each capacitor? (c) How much total charge was acquired from the battery?

(a) If you are given three identical capacitors, you can obtain (1) three, (2) five, (3) seven different capacitance values. (b) If the three capacitors each have a capacitance of \(1.0 \mu \mathrm{F}\), what are the different values of equivalent capacitance?

What are the maximum and minimum equivalent capacitances that can be obtained by combinations of three capacitors of \(1.5 \mu \mathrm{F}, 2.0 \mu \mathrm{F},\) and \(3.0 \mu \mathrm{F} ?\)

A tiny dust particle in the form of a long thin needle has charges of \(\pm 7.14 \mathrm{pC}\) on its ends. The length of the particle is \(3.75 \mu \mathrm{m}\). (a) Which location is at a higher potential: (1) \(7.65 \mu \mathrm{m}\) above the positive end, (2) \(5.15 \mu \mathrm{m}\) above the positive end, or (3) both locations are at the same potential? (b) Compute the potential at the two points in part (a). (c) Use your answer from part (b) to determine the work needed to move an electron from the near point to the far point.

A vacuum tube has a vertical height of \(50.0 \mathrm{~cm}\). An electron leaves from the top at a speed of \(3.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\) downward and is subjected to a "typical" Earth field of \(150 \mathrm{~V} / \mathrm{m}\) downward. (a) Use energy methods to determine whether it reaches the bottom surface of the tube. (b) If it does, with what speed does it hit? If not, how close does it come to the bottom surface? (c) How does the gravitational force on the electron compare to the electric force on it, both in magnitude and in direction?

A capacitor \((5.70 \mu \mathrm{F})\) is connected in a series arrangement with a second capacitor \((2.30 \mu \mathrm{F})\) and a \(12-\mathrm{V}\) battery. (a) How much charge is stored on each capacitor? (b) What is the voltage drop across each capacitor? The battery is then removed, leaving the two capacitors isolated. (c) If the smaller capacitor's capacitance is now doubled, by how much does the charge on each and the voltage across each change?