Two metal spheres each 1.0 cm in radius are far apart. One sphere carries 38 nC, the

Why can a bird perch on a high-voltage power line without getting electrocuted?

One proton is accelerated from rest by a uniform electric field, another proton by a nonuniform electric field. If they move through the same potential difference, how do their final speeds compare?

Would a free electron move toward higher or lower potential?

The electric field at the center of a uniformly charged ring is obviously zero, yet Example 22.6 shows that the potential at the center isnt zero. How is this possible?

Must the potential be zero at any point where the electric field is zero? Explain.

Must the electric field be zero at any point where the potential is zero? Explain.

The potential is constant throughout an entire volume. What must be true of the electric field within that volume?

In considering the potential of an infinite flat sheet, why isnt it useful to take the zero of potential at infinity?

Cherry picker trucks for working on power lines often carry electrocution hazard signs. Explain how this hazard arises and why it might be more of a danger to someone on the ground than to a worker on the truck.

Can equipotential surfaces intersect? Explain.

Is the potential at the center of a hollow, uniformly charged spherical shell higher than, lower than, or the same as at the surface?

A solid sphere contains positive charge uniformly distributed throughout its volume. Is the potential at its center higher than, lower than, or the same as at the surface?

Two equal but opposite charges form a dipole. Describe the equipotential surface on which

The electric potential in a region increases linearly with distance. What can you conclude about the electric field in this region?

How much work does it take to move a charge against a 12-V potential difference?

The potential difference between the two sides of an ordinary electric outlet is 120 V. How much energy does an electron gain when it moves from one side to the other?

It takes 45 J to move a 15-mC charge from point A to point B. Whats the potential difference

Show that 1 V/m is the same as 1 N/C.

Find the magnitude of the potential difference between two points located 1.4 m apart in a uniform 650-N/C electric field, if a line between the points is parallel to the field.

A charge of 3.1 C moves from the positive to the negative terminal of a 9.0-V battery. How much energy does the battery impart to the charge?

A proton, an alpha particle (a bare helium nucleus), and a singly ionized helium atom are accelerated through a 100-V potential difference. How much energy does each gain?

The potential difference across a typical cell membrane is about 80 mV. How much work is done on a singly ionized potassium ion moving through the membrane?

An electric field is given by where is a constant. Find the potential as a function of position, taking at

The classical picture of the hydrogen atom has the electron orbiting 0.0529 nm from the proton. Whats the electric potential associated with the protons electric field at this distance?

The potential at the surface of a 10-cm-radius sphere is 4.8 kV. Whats the spheres total charge, assuming charge is distributed in a spherically symmetric way?

Youre developing a switch for high-voltage power lines. The smallest part in your design is a 5.0-cm-diameter metal sphere. What do you specify for the maximum potential on your switch if the electric field at the spheres surface isnt to exceed the 3-MV/m breakdown field of air?

A 3.5-cm-diameter isolated metal sphere carries (a) Find the potential at the spheres surface. (b) If a proton were released from rest at the surface, what would be its speed far from the sphere?

In a uniform electric field, equipotential planes that differ by 1.0 V are 2.5 cm apart. Whats the field strength?

Figure 22.20 shows a plot of potential versus position along the x-axis. Make a plot of the x-component of the electric field for this situation.

Figure 22.21 shows some equipotentials in the plane. (a) In what region is the electric field strongest? What are (b) the direction and (c) the magnitude of the field in this region?

The electric potential in a region is with V in volts and the coordinates in meters. Find (a) the potential and (b) the components of the electric field at the point y 5 1 m, z 5 1 m.

Dielectric breakdown of air occurs at fields of 3 MV/m. Find (a) the maximum potential (measured from infinity) for the sphere of Example 22.3 before dielectric breakdown occurs at the spheres surface and (b) the charge on the sphere at this potential.

Youre an automotive engineer working on the ignition system for a new engine. Its spark plugs have center electrodes made from 2.0-mm-diameter wire. The electrode ends gradually wear to a hemispherical shape, so they behave approximately like charged spheres. Your job is to specify the minimum potential that ensures these plugs will spark in air, neglecting the presence of the second electrode.

A large metal sphere has three times the diameter of a smaller sphere and carries three times the charge. Both spheres are isolated, so their surface charge densities are uniform. Compare (a) the potentials and (b) the electric field strengths at their surfaces.

Two points A and B lie 15 cm apart in a uniform electric field, with the path AB parallel to the field. If the potential difference is 840 V, whats the field strength?

The electric field within a cell membrane is approximately 8.0 MV/m and is essentially uniform. If the membrane is 10 nm thick, whats the potential difference across the membrane?

Whats the potential difference between the terminals of a battery that can impart to each electron that moves between the terminals?

Whats the charge on an ion that gains when it moves through a potential difference of 2500 V?

Two flat metal plates are a distance d apart, where d is small compared with the plate size. If the plates carry surface charge densities show that the magnitude of the potential difference between them is

An electron passes point A moving at 6.5 Mm/s. At point B its come to a stop. Find the potential difference

A 5.0-g object carries It acquires speed v when accelerated from rest through a potential difference V. If a 2.0-g object acquires twice the speed under the same circumstances, whats its charge?

Points A and B lie 20 cm apart on a line extending radially from a point charge Q, and the potentials at these points are and Find Q and the distance r between A and the charge.

A sphere of radius R carries negative charge of magnitude Q, distributed in a spherically symmetric way. Find the escape speed for a proton at the spheres surfacethat is, the speed that would enable the proton to escape to arbitrarily large distances starting at the spheres surface.

Proton-beam therapy is preferable to X rays for cancer treatment because protons deliver most of their energy to the tumor, with less damage to healthy tissue. A cyclotron used to accelerate protons for cancer treatment repeatedly passes the protons through a 15-kV potential difference. (a) How many passes are needed to bring the protons kinetic energy to ? (b) Whats the resulting proton energy in electronvolts?

A thin spherical shell has radius R and total charge Q distributed uniformly over its surface. Find the potential at its center.

A solid sphere of radius R carries charge Q distributed uniformly throughout its volume. Find the potential difference from the spheres surface to its center. (Hint: Consult Example 21.1.)

Find the potential as a function of position in the electric field E where a is a constant and V 5 0 at x 5 0.

Your radio station needs a new coaxial cable to connect the transmitter and antenna. One possible cable consists of a 2.0-mmdiameter inner conductor and an outer conductor with diameter 1.6 cm and negligible thickness (Fig. 22.22); the maximum safe potential difference between the conductors is 2 kV. In your application, the conductors carry charge densities Will this cable work for you?

The potential difference between the surface of a 3.0-cm-diameter power line and a point 1.0 m distant is 3.9 kV. Find the line charge density on the power line.

Three equal charges q form an equilateral triangle of side a. Find the potential at the center of the triangle.

A charge lies at the origin and at Find two points on the x-axis where

Two identical charges q lie on the x-axis at (a) Find an expression for the potential at all points in the x-y plane. (b) Show that your result reduces to the potential of a point charge for distances large compared with a.

A dipole of moment consists of two charges separated by far less than 10 cm. Find the potential 10 cm from the dipole (a) on its axis, (b) at to its axis, and (c) on its perpendicular bisector.

A thin plastic rod 20 cm long carries 3.2 nC distributed uniformly over its length. (a) If the rod is bent into a ring, find the potential at its center. (b) If its bent into a semicircle, find the potential at the center (i.e., at the center of the circle of which the semicircle is part).

A thin ring of radius R carries charge distributed uniformly over three-fourths of its circumference, and over the rest. Find the potential at the rings center.

The potential at the center of a uniformly charged ring is 45 kV, and 15 cm along the ring axis the potential is 33 kV. Find the rings radius and total charge.

The annulus shown in Fig. 22.23 carries a uniform surface charge density Find an expression for the potential at an arbitrary point P on its axis.

The potential in a region is where a is a constant. (a) Determine the electric field in the region. (b) Sketch some equipotentials and field lines.

Use Equation 22.6 to calculate the electric field on the perpendicular bisector of a dipole, and show that your result is equivalent to Equation 20.6a.

Use the result of Example 22.6 to determine the on-axis field of a charged ring, and verify that your answer agrees with the result of Example 20.6.

The electric potential in a region is where and R are constants and r is the radial distance from the origin. Find expressions for the magnitude and direction of the electric field in this region.

Two metal spheres each 1.0 cm in radius are far apart. One sphere carries 38 nC, the other nC. (a) Whats the potential on each? (b) If the spheres are connected by a thin wire, what will be the potential on each once equilibrium is reached? (c) How much charge moves between the spheres in order to achieve equilibrium?

Two 5.0-cm-diameter conducting spheres are 8.0 m apart, and each carries Determine (a) the potential on each sphere, (b) the field strength at the surface of each sphere, (c) the potential midway between the spheres, and (d) the potential difference between the spheres.

A 2.0-cm-radius metal sphere carries 75 nC and is surrounded by a concentric spherical conducting shell of radius 10 cm carrying (a) Find the potential difference between shell and sphere. (b) How would your answer change if the shells charge were nC?

A sphere of radius R carries a nonuniform but spherically symmetric volume charge density that results in an electric field in the sphere given by where is a constant. Find the potential difference from the spheres surface to its center.

The potential as a function of position in a region is with x in meters and V in volts. Find (a) all points on the x-axis where (b) an expression for the electric field, and (c) all points on the x-axis where

A conducting sphere 5.0 cm in radius carries 60 nC. Its surrounded by a concentric spherical conducting shell of radius 15 cm carrying nC. (a) Find the potential at the spheres surface, taking at infinity. (b) Repeat for the case when the shell carries nC.

Show that the result of Example 22.8 approaches the field of a point charge for (Hint: Youll need to apply the binomial approximation from Appendix A to the expression )

The potential on the axis of a uniformly charged disk at 5.0 cm from the disk center is 150 V; the potential 10 cm from the disk center is 110 V. Find the disk radius and its total charge.

A uranium nucleus (mass 238 u, charge ) decays, emitting an alpha particle (mass 4 u, charge ) and leaving a thorium nucleus (mass 234 u, charge ). At the instant the alpha particle leaves the nucleus, the centers of the two are 7.4 fm apart and essentially at rest. Treating each particle as a spherical charge distribution, find their speeds when theyre a great distance apart.

A disk of radius a carries nonuniform surface charge density where is a constant. (a) Find the potential at an arbitrary point x on the disk axis, where is the disk center. (b) Use the result of (a) to find the electric field on the disk axis, and (c) show that the field reduces to an expected form for

An open-ended cylinder of radius a and length carries charge q spread uniformly over its surface. Find the potential at the center of the cylinder. (Hint: Treat the cylinder as a stack of charged rings, and integrate.)

A line charge extends along the x-axis from to Its line charge density is where is a constant. Find an expression for the potential on the x-axis for Check that your expression reduces to an expected result for

Repeat Problem 73 for the charge distribution (Hint: What does this charge distribution resemble at large distances?)

Youre sizing a new electric transmission line, and you can save money with thinner wire. The potential difference between the line and the ground, 60 m below, is 115 kV. The field at the wire surface cannot exceed 25% of the 3-MV/m breakdown field in air. Neglecting charges in the ground itself, what minimum wire diameter do you specify? (Hint: Youll have to do a numerical calculation.)

From the equipotentials, you can infer that the hearts electrical structure resembles that of a a. uniform charged sheet. b. dipole. c. point charge. d. uniformly charged sphere.

The electric field in the vicinity of the heart points approximately a. from upper left to lower right. b. from lower left to upper right. c. from upper right to lower left. d. from lower right to upper left.

The electric field is strongest in the region marked a. A. b. B. c. C. d. D.

The electric field in region A is approximately a. 20 b. 2 mN/C. c. 20 mN/C. d. 2 kN/C.