In Figure P15.27, determine the point (other than innity)

A charge of 4.5  10
9 C is located 3.2 m from a charge of
2.8  10
9 C. Find the electrostatic force exerted by one charge on the other.

The Moon and Earth are bound together by gravity. If, instead, the force of attraction were the result of each having a charge of the same magnitude but opposite in sign, nd the quantity of charge that would have to be placed on each to produce the required force.

An alpha particle (charge 2.0e) is sent at high speed toward a gold nucleus (charge 79e). What is the electrical force acting on the alpha particle when it is 2.0  10
14 m from the gold nucleus?

Four point charges are situated at the corners of a square with sides of length a, as in Figure P15.4. Find the expression for the resultant force on the positive charge q.

The nucleus of 8Be, which consists of 4 protons and 4 neutrons, is very unstable and spontaneously breaks into two alpha particles (helium nuclei, each consisting of 2 protons and 2 neutrons). (a) What is the force between the two alpha particles when they are 5.00  10
15 m apart, and (b) what will be the magnitude of the acceleration of the alpha particles due to this force? Note that the mass of an alpha particle is 4.0026 u.

A molecule of DNA (deoxyribonucleic acid) is 2.17  m long. The ends of the molecule become singly ionized negative on one end, positive on the other. The helical molecule acts like a spring and compresses 1.00% upon becoming charged. Determine the effective spring constant of the molecule.

Suppose that 1.00g of hydrogen is separated into electrons and protons. Suppose also that the protons are placed at the Earths North Pole and the electrons are placed at the South Pole. What is the resulting compressional force on the Earth?

An electron is released a short distance above the surface of the Earth. A second electron directly below it exerts an electrostatic force on the rst electron just great enough to cancel the gravitational force on it. How far below the rst electron is the second?

Two identical conducting spheres are placed with their centers 0.30m apart. One is given a charge of 12  10
9 C, the other a charge of
18  10
9 C. (a) Find the electrostatic force exerted on one sphere by the other. (b) The spheres are connected by a conducting wire. Find the electrostatic force between the two after equilibrium is reached.

Calculate the magnitude and direction of the Coulomb force on each of the three charges shown in Figure P15.10.

Three charges are arranged as shown in Figure P15.11. Find the magnitude and direction of the electrostatic force on the charge at the origin.

Three charges are arranged as shown in Figure P15.12. Find the magnitude and direction of the electrostatic force on the 6.00-nC charge.

Three point charges are located at the corners of an equilateral triangle as in Figure P15.13. Calculate the net electric force on the 7.00 C charge.

Two small beads having positive charges 3q and q are xed at the opposite ends of a horizontal insulating rod, extending from the origin to the point x  d. As shown in Figure P15.14, a third small charged bead is free to slide on the rod. At what position is the third bead in equilibrium? Can it be in stable equilibrium?

Two small metallic spheres, each of mass 0.20 g, are suspended as pendulums by light strings from a common point as shown in Figure P15.15. The spheres are given the same electric charge, and it is found that they come to equilibrium when each string is at an angle of 5.0 with the vertical. If each string is 30.0 cm long, what is the magnitude of the charge on each sphere?

A charge of 6.00  10
9 C and a charge of
3.00  10
9 C are separated by a distance of 60.0cm. Find the position at which a third charge, of 12.0  10
9 C, can be placed so that the net electrostatic force on it is zero.

An object with a net charge of 24  C is placed in a uniform electric eld of 610 N/C, directed vertically. What is the mass of the object if it oats in the electric eld?

(a) Determine the electric eld strength at a point 1.00 cm to the left of the middle charge shown in Figure P15.10. (b) If a charge of
2.00  C is placed at this point, what are the magnitude and direction of the force on it?

An airplane is ying through a thundercloud at a height of 2 000 m. (This is a very dangerous thing to do because of updrafts, turbulence, and the possibility of electric discharge.) If there are charge concentrations of 40.0 C at a height of 3 000 m within the cloud and
40.0 C at a height of 1 000 m, what is the electric eld at the aircraft?

An electron is accelerated by a constant electric eld of magnitude 300 N/C. (a) Find the acceleration of the electron. (b) Use the equations of motion with constant acceleration to nd the electrons speed after 1.00  10
8 s, assuming it starts from rest.

A Styrofoam ball covered with a conducting paint has a mass of 5.0  10
3 kg and has a charge of 4.0  C. What electric eld directed upward will produce an electric force on the ball that will balance its weight?

Each of the protons in a particle beam has a kinetic energy of 3.25  10
15 J. What are the magnitude and direction of the electric eld that will stop these protons in a distance of 1.25 m?

A proton accelerates from rest in a uniform electric eld of 640 N/C. At some later time, its speed is 1.20  106 m/s. (a) Find the magnitude of the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in that interval? (d) What is its kinetic energy at the later time?

Three charges are at the corners of an equilateral triangle, as shown in Figure P15.24. Calculate the electric eld at a point midway between the two charges on the x-axis.

Three identical charges (q 
5.0  C) lie along a circle of radius 2.0 m at angles of 30, 150, and 270, as shown in Figure P15.25. What is the resultant electric eld at the center of the circle?

Two point charges lie along the y -axis. A charge of q1 
9.0  C is at y  6.0 m, and a charge of q2 
8.0  C is at y 
4.0 m. Locate the point (other than innity) at which the total electric eld is zero.

In Figure P15.27, determine the point (other than innity) at which the total electric eld is zero.

Figure P15.28 shows the electric eld lines for two point charges separated by a small distance. (a) Determine the ratio q1/q2. (b) What are the signs of q1 and q2?

(a) Sketch the electric eld lines around an isolated point charge q  0. (b) Sketch the electric eld pattern around an isolated negative point charge of magnitude
2q.

(a) Sketch the electric eld pattern around two positive point charges of magnitude 1  C placed close together. (b) Sketch the electric eld pattern around two negative point charges of
2  C, placed close together. (c) Sketch the pattern around two point charges of 1  C and
2  C, placed close together.

Two point charges are a small distance apart. (a) Sketch the electric eld lines for the two if one has a charge four times that of the other and both charges are positive. (b) Repeat for the case in which both charges are negative.

(a) Sketch the electric eld pattern set up by a positively charged hollow sphere. Include regions inside and regions outside the sphere. (b) A conducting cube is given a positive charge. Sketch the electric eld pattern both inside and outside the cube.

Refer to Figure 15.20. The charge lowered into the center of the hollow conductor has a magnitude of 5  C. Find the magnitude and sign of the charge on the inside and outside of the hollow conductor when the charge is as shown in (a) Figure 15.20a; and (b) Figure 15.20b; (c) Figure 15.20c; and (d) Figure 15.20d.

The dome of a Van de Graaff generator receives a charge of 2.0  10
4 C. Find the strength of the electric eld (a) inside the dome; (b) at the surface of the dome, assuming it has a radius of 1.0 m; and (c) 4.0 m from the center of the dome. [Hint: See Section 15.6 to review properties of conductors in electrostatic equilibrium. Also, use the fact that the points on the surface are outside a spherically symmetric charge distribution; the total charge may be considered to be located at the center of the sphere.]

If the electric eld strength in air exceeds 3.0  106 N/C, the air becomes a conductor. Using this fact, determine the maximum amount of charge that can be carried by a metal sphere 2.0 m in radius. (See the hint in Problem 34.)

In the Millikan oil drop experiment, an atomizer (a sprayer with a ne nozzle) is used to introduce many tiny droplets of oil between two oppositely charged parallel metal plates. Some of the droplets pick up one or more excess electrons. The charge on the plates is adjusted so that the electric force on the excess electrons exactly balances the weight of the droplet. The idea is to look for a droplet that has the smallest electric force and assume that it has only one excess electron. This strategy lets the observer measure the charge on the electron. Suppose we are using an electric eld of 3  104 N/C. The charge on one electron is about 1.6  10
19 C. Estimate the radius of an oil drop of density 858 kg/m3 for which its weight could be balanced by the electric force of this eld on one electron. (Problem 36 is courtesy of E.F. Redish. For more problems of this type, visit http://www.physics.umd.edu/perg/.)

A Van de Graaff generator is charged so that the electric eld at its surface is 3.0  104 N/C. Find (a) the electric force exerted on a proton released at its surface and (b) the acceleration of the proton at that instant of time.

A at surface having an area of 3.2 m2 is rotated in a uniform electric eld of magnitude E  6.2  105 N/C. Determine the electric ux through this area (a) when the electric eld is perpendicular to the surface and (b) when the electric eld is parallel to the surface.

An electric eld of intensity 3.50 kN/C is applied along the x-axis. Calculate the electric ux through a rectangular plane 0.350 m wide and 0.700 m long if (a) the plane is parallel to the yz-plane; (b) the plane is parallel to the xy-plane; and (c) the plane contains the y-axis, and its normal makes an angle of 40.0 with the x-axis.

The electric eld everywhere on the surface of a thin spherical shell of radius 0.750 m is measured to be equal to 890 N/C and points radially toward the center of the sphere. (a) What is the net charge within the spheres surface? (b) What can you conclude about the nature and distribution of the charge inside the spherical shell?

A 40-cm-diameter loop is rotated in a uniform electric eld until the position of maximum electric ux is found. The ux in that position is measured to be 5.2  105 Nm2/C. Calculate the electric eld strength in this region.

A point charge of 5.00  C is located at the center of a sphere with a radius of 12.0 cm. Determine the electric ux through the surface of the sphere.

A point charge q is located at the center of a spherical shell of radius a that has a charge
q uniformly distributed on its surface. Find the electric eld (a) for all points outside the spherical shell and (b) for a point inside the shell a distance r from the center.

Use Gausss law and the fact that the electric eld inside any closed conductor in electrostatic equilibrium is zero to show that any excess charge placed on the conductor must reside on its surface.

An innite plane conductor has charge spread out on its surface as shown in Figure P15.45. Use Gausss law to show that the electric eld at any point outside the conductor is given by E   /  0, where  is the charge per unit area on the conductor. [Hint: Choose a Gaussian surface in the shape of a cylinder with one end inside the conductor and one end outside the conductor.]

Show that the electric eld just outside the surface of a good conductor of any shape is given by E   /  0, where  is the charge per unit area on the conductor. [Hint: The electric eld just outside the surface of a charged conductor is perpendicular to its surface.]

Two protons in an atomic nucleus are typically separated by a distance of 2  10
15 m. The electric repulsion force between the protons is huge, but the attractive nuclear force is even stronger and keeps the nucleus from bursting apart. What is the magnitude of the electrical force between two protons separated by 2.00  10
15 m?

In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton. The radius of the orbit is 0.53  10
10 m. (a) Find the electrostatic force acting on each particle. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

Three point charges are aligned along the x-axis as shown in Figure P15.49. Find the electric eld at the position x 2.0 m, y  0.

A small 2.00-g plastic ball is suspended by a 20.0-cm-long string in a uniform electric eld, as shown in Figure P15.50. If the ball is in equilibrium when the string makes a 15.0 angle with the vertical as indicated, what is the net charge on the ball?

(a) Two identical point charges q are located on the y-axis at y a and y 
a. What is the electric eld along the x -axis at x  b? (b) A circular ring of charge of radius a has a total positive charge Q distributed uniformly around it. The ring is in the x  0 plane with its center at the origin. What is the electric eld along the x axis at x  b due to the ring of charge? [Hint: Consider the charge Q to consist of many pairs of identical point charges positioned at the ends of diameters of the ring.]

A positively charged bead having a mass of 1.00 g falls from rest in a vacuum from a height of 5.00 m in a uniform vertical electric eld with a magnitude of 1.00  104 N/C. The bead hits the ground at a speed of 21.0 m/s. Determine (a) the direction of the electric eld (upward or downward), and (b) the charge on the bead.

A solid conducting sphere of radius 2.00 cm has a charge of 8.00  C. A conducting spherical shell of inner radius 4.00 cm and outer radius 5.00 cm is concentric with the solid sphere and has a charge of
4.00  C. Find the electric eld at (a) r  1.00 cm, (b) r  3.00 cm, (c) r  4.50 cm, and (d) r  7.00 cm from the center of this charge conguration.

Two small silver spheres, each with a mass of 100 g, are separated by 1.00 m. Calculate the fraction of the electrons in one sphere that must be transferred to the other in order to produce an attractive force of 1.00  104 N (about a ton) between the spheres. (The number of electrons per atom of silver is 47, and the number of atoms per gram is Avogadros number divided by the molar mass of silver, 107.87 g/mol.)

A vertical electric eld of magnitude 2.00  104 N/C exists above the Earths surface on a day when a thunderstorm is brewing. A car with a rectangular size of 6.00 m by 3.00 m is traveling along a roadway sloping downward at 10.0. Determine the electric ux through the bottom of the car.

A 2.00 C charged 1.00-g cork ball is suspended vertically on a 0.500-m-long light string in the presence of a uniform downward-directed electric eld of magnitude E  1.00  105 N/C. If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum. (a) Determine the period of the balls oscillation. (b) Should gravity be included in the calculation for part (a)? Explain.

Two 2.0-g spheres are suspended by 10.0-cm-long light strings (Fig. P15.57). A uniform electric eld is applied in the x-direction. If the spheres have charges of
5.0  10
8 C and 5.0  10
8 C, determine the electric eld intensity that enables the spheres to be in equilibrium at 

Two point charges like those in Figure P15.58 are called an electric dipole. Show that the electric eld at a distant point along the x-axis is given by Ex  4keqa/x3.

A charged cork ball of mass 1.00 g is suspended on a light string in the presence of a uniform electric eld as in Figure P15.59. When the electric eld has an x-component of 3.00  105 N/C and a y-component of 5.00  105 N/C, the ball is in equilibrium at   37.0. Find (a) the charge on the ball and (b) the tension in the string.

A point charge of magnitude 5.00  C is at the origin of a coordinate system, and a charge of
4.00  C is at the point x  1.00 m. There is a point on the x-axis, at x less than innity, where the electric eld goes to zero. (a) Show by conceptual arguments that this point cannot be located between the charges. (b) Show by conceptual arguments that the point cannot be at any location between x  0 and negative innity. (c) Show by conceptual arguments that the point must be between x  1.00 m and x  positive innity. (d) Use the values given to nd the point and show that it is consistent with your conceptual argument.

Two hard rubber spheres of mass 15 g are rubbed vigorously with fur on a dry day and are then suspended from a rod with two insulating strings of length 5.0 cm. They are observed to hang at equilibrium as shown in Figure P15.61, each at an angle of 10 with the vertical. Estimate the amount of charge that is found on each sphere. (Problem 61 is courtesy of E.F. Redish. For more problems of this type, visit http://www.physics.umd.edu/perg/.)

Three identical point charges, each of mass m  0.100 kg, hang from three strings, as shown in Figure P15.62. If the lengths of the left and right strings are each L  30.0 cm, and if the angle

Each of the electrons in a particle beam has a kinetic energy of 1.60  10
17 J. (a) What is the magnitude of the uniform electric eld (pointing in the direction of the electrons movement) that will stop these electrons in a distance of 10.0 cm? (b) How long will it take to stop the electrons? (c) After the electrons stop, what will they do? Explain.

Protons are projected with an initial speed v0  9 550 m/s into a region where a uniform electric eld E  720 N/C is present (Fig. P15.64). The protons are to hit a target that lies a horizontal distance of 1.27 mm from the point where the protons are launched. Find (a) the two projection angles  that will result in a hit and (b) the total duration of ight for each of the two trajectories.

The following are a number of experiments that you can perform to investigate static electricity. (a) Attach two inated balloons to the ends of a light string having a length of about 2 m. Tape the center of the string to the top of an open doorway as in Figure A15.1. Note that the balloons touch each other as they hang freely. Now rub each balloon several times with a wool cloth, and let them hang freely once again. Why are the balloons no longer touching each other, but are now separated? (b) Rub an inated balloon with a piece of wool and press it against a wall. Note that the balloon adheres to the wall. Why? (c) Rub nylon hose with a plastic dry-cleaner bag and observe how the hose expands like a balloon. Why does it do so? (d) Tear some paper into very small pieces. Comb your hair and then bring the comb close to the pieces of paper. Notice that they accelerate toward the comb. How does the magnitude of the electric force compare with the magnitude of the gravitational force exerted on the paper? Keep watching, and you might see a few pieces jump away from the comb. They do not fall away; they are clearly repelled. What causes this repulsion?

For this experiment, you will need two 20-cm strips of transparent tape. (The mass of each is about 65 mg.) Fold about 1 cm of tape over at one end of each strip to create a handle. Press both pieces of tape side by side onto a tabletop, rubbing your nger back and forth across the strips. Quickly pull the strips off the surface so that they become charged. Hold the tape handles together, and the strips will repel each other, forming an inverted V shape. Measure the angle between the pieces and estimate the excess charge on each strip. Assume that the charges act as if they were located at the center of mass of each strip.

Rub an inated balloon with a piece of wool cloth and place the balloon near a ne stream of water falling from a faucet. The stream of water will deect toward the balloon. Why? Vary the distance between the balloon and the stream, and observe the displacement of the water stream for different distances. What is the relationship between the displacement of the stream and the distance of separation?

If you have access to a Van de Graaff generator, here are a few interesting things to try: (a) Stack several aluminum pie plates on top of the generator and then turn the generator on. What do you think is going to happen? Why? Try it. (b) Tape one pie plate on top of the generator and pour in some puffed rice or pieces of paper. What do you think will happen when you turn the generator on? Why? Try it and see. (c) Tape a glass beaker to the top of the generator and pour in some puffed rice. What do you think will happen when you turn the generator on? Try it. (d) Bring a uorescent bulb close to a charged generator and observe what happens. Why does it happen?