During a famous experiment in 1919, Ernest Rutherford shot

Objects are composed of atoms which are composed of charged particles (protons and electrons); however, we rarely observe the effects of the electrostatic force. Explain why we do not observe these effects.

A carbon atom can become a carbon ion if it has one or more of its electrons removed during a process called ionization. What is the net charge on a carbon atom that has had two of its electrons removed? (a) (b) (c) (d)

You do a simple demonstration for your high school physics teacher in which you claim to disprove Coulombs law. You first run a rubber comb through your dry hair, then use it to attract tiny neutral pieces of paper on the desk. You then say, Coulombs _e, _e, _2e, _2e law states that for there to be electrostatic forces of attraction between two objects, both objects need to be charged. However, the paper was not charged. So according to Coulombs law, there should be no electrostatic forces of attraction between them, yet there clearly was. You rest your case. (a) What is wrong with your assumptions? (b) Does attraction between the paper and the comb require that the net charge on the comb be negative? Explain your answer.

You have a positively charged insulating rod and two metal spheres on insulating stands. Give step-by-step directions of how the rod, without actually touching either sphere, can be used to give one of the spheres (a) a negative charge and (b) a positive charge.

(a) Two point particles that have charges of and are separated by distance Use field lines to draw a visualization of the electric field in the neighborhood of this system. (b) Draw the field lines at distances much greater than d from the charges.

Ametal sphere is positively charged. Is it possible for the sphere to electrically attract another positively charged ball? Explain your answer.

A simple demonstration of electrostatic attraction can be done by dangling a small ball of crumpled aluminum foil on a string and bringing a charged rod near the ball. The ball initially will be attracted to the rod, but once they touch, the ball will be strongly repelled from it. Explain these observations.

Two positive point charges that are equal in magnitude are fixed in place, one at and the other at on the axis. A third positive point charge is placed at an equilibrium position. (a) Where is this equilibrium position? (b) Is the equilibrium position stable if the third particle is constrained to move parallel with the axis? (c) What if it is constrained to move parallel with the axis? Explain your answer.

Two neutral conducting spheres are in contact and are supported on a large wooden table by insulated stands. A positively charged rod is brought up close to the surface of one of the spheres on the side opposite its point of contact with the other sphere. (a) Describe the induced charges on the two conducting spheres and sketch the charge distributions on them. (b) The two spheres are separated and then the charged rod is removed. The spheres are then separated far apart. Sketch the charge distributions on the separated spheres.

Three point charges, +q, +Q and -Q are placed at the corners of an equilateral triangle as shown in Figure 21-33. No other charged objects are nearby. (a) What is the direction of the net force on charge +q due to the other two charges? (b) What is the total electric force on the system of three charges? Explain.

A positively charged particle is free to move in a region with a nonzero electric field Which statement(s) must be true? (a) The particle is accelerating in the direction perpendicular to (b) The particle is accelerating in the direction of (c) The particle is moving in the direction of (d) The particle could be momentarily at rest. (e) The force on the particle is opposite the direction of (f) The particle is moving opposite the direction of

Four charges are fixed in place at the corners of a square as shown in Figure 21-34. No other charges are nearby. Which of the following statements is true? (a) \(\vec{E}\) is zero at the midpoints of all four sides of the square. (b) \(\vec{E}\) is zero at the center of the square. (c) \(\vec{E}\) is zero midway between the top two charges and midway between the bottom two charges.

Two point particles that have charges of and are separated by distance (a) Use field lines to sketch the electric field in the neighborhood of this system. (b) Draw the field lines at distances much greater than from the charges.

Three equal positive point charges (each charge ) are fixed at the vertices of an equilateral triangle that has sides of length a. The origin is at the midpoint of one side the triangle, the center of the triangle on the axis at and the vertex opposite the origin is on the axis at . (a) Express and in terms of a. (b) Write an expression for the electric field on the axis a distance from the origin on the interval 0 (c) Show that the expression you obtained in (b) gives the expected results for and for .

Amolecule has a dipole moment given by The molecule is momentarily at rest with making an angle with a uniform electric field Describe the subsequent motion of the dipole moment.

True or false: (a) The electric field of a point charge always points away from the charge. (b) The electric force on a charged particle in an electric field is always in the same direction as the field. (c) Electric field lines never intersect. (d) All molecules have dipole moments in the presence of an external electric field.

Two molecules have dipole moments of equal magnitude. The dipole moments are oriented in various configurations as shown in Figure 21-35. Determine the electric- field direction at each of the numbered locations. Explain your answers. SSM

Estimate the force required to bind the two protons in the He nucleus together. Hint: Model the protons as point charges. You will need to have an estimate of the distance between them.

Apopular classroom demonstration consists of rubbing a plastic rod with fur to give the rod charge, and then placing the rod near an empty soda can that is on its side (Figure 21-36). Explain why the can will roll toward the rod.

Sparks in air occur when ions in the air are accelerated to such a high speed by an electric field that when the ions impact on neutral gas molecules, the neutral molecules become ions. If the electric field strength is large enough, the ionized collision products are themselves accelerated and produce more ions on impact, and so forth. This avalanche of ions is what we call a spark. (a) Assume that an ion moves, on average, exactly one mean free path through the air before hitting a molecule. If the ion needs to acquire approximately 1.0 eV of kinetic energy in order to ionize a molecule, estimate the minimum strength of the electric field required at standard room pressure and temperature. Assume that the cross-sectional area of an air molecule is about (b) How does the strength of the electric field in Part (a) depend on temperature? (c) How does the 2.0 strength of the electric field in Part (a) depend on pressure?

A plastic rod is rubbed against a wool shirt, thereby acquiring a charge of How many electrons are transferred from the wool shirt to the plastic rod?

A charge equal to the charge of Avogadros number of protons is called a faraday. Calculate the number of coulombs in a faraday.

What is the total charge of all of the protons in 1.00 kg of carbon?

Suppose a cube of aluminum which is 1.00 cm on a side accumulates a net charge of (a) What percentage of the electrons originally in the cube was removed? (b) By what percentage has the mass of the cube decreased because of this removal?

During a process described by the photoelectric effect, ultraviolet light can be used to charge a piece of metal. (a) If such light is incident on a slab of conducting material and electrons are ejected with enough energy that they escape the surface of the metal, how long before the metal has a net charge of if electrons are ejected per second? (b) If 1.3 eV is needed to eject an electron from the surface, what is the power rating of the light beam? (Assume this process is 100% efficient.)

A point charge is at the origin and a point charge is on the axis at (a) Find the electric force on charge (b) Find the electric force on (c) How would your answers for Parts (a) and (b) differ if were

Three point charges are on the axis: is at is at the origin, and is at Find the electric force on

A point charge and a point charge are a distance apart. Where should a third point charge be placed so that the electric force on that third charge is zero?

A point charge and a point charge are a distance apart. Where should a third point charge be placed so that the electric force on that third charge is zero?

Three point charges, each of magnitude 3.00 nC, are at separate corners of a square of edge length 5.00 cm. The two point charges at opposite corners are positive, and the third point charge is negative. Find the force exerted by these point charges on a fourth point charge at the remaining corner.

A point charge of is on the axis at and a second point charge of is on the axis at Find the electric force on a point charge of on the axis at

Apoint particle that has a charge of is located at the origin. A second point particle that has a charge of is at A third point particle, an electron, is at a point that has coordinates Find the values of and such that the electron is in equilibrium.

A point particle that has a charge of is located at the origin; a second point particle that has a charge of is located at x _ 0, y _ 0.10 m; and a third point particle that has a 2.0 mC charge of is located at Find the total electric force on each of the three point charges.

A point particle that has a charge of is located at and a point particle that has a charge is located at The electric force on a point particle at that has a charge of is Determine the charge

Five identical point charges, each having charge are equally spaced on a semicircle of radius as shown in Figure 21-37. Find the force (in terms of and ) on a charge located equidistant from the five other charges. SSM

The structure of the molecule is approximately that of an equilateral tetrahedron, with three ions forming the base and an ion at the apex of the tetrahedron. The length of each side is Calculate the electric force that acts on each ion.

A point charge of is at the origin. What are the magnitude and direction of the electric field on the axis at (a) and (b) (c) Sketch the function versus for both positive and negative values of (Remember that is negative when points in the direction.)

Two point charges, each are on the axis; one point charge is at the origin and the other is at Find the electric field on the axis at (a) (b) (c) and (d) (e) At what point on the axis is the electric field zero? (f) Sketch versus for

When a 2.0-nC point charge is placed at the origin, it experiences an electric force of in the direction. (a) What is the electric field at the origin? (b) What would be the electric force on a point charge placed at the origin? (c) If this force is due to the electric field of a point charge on the axis at what is the value of that charge?

The electric field near the surface of Earth points downward and has a magnitude of (a) Compare magnitude of the upward electric force on an electron with the magnitude of gravitational force on the electron. (b) What charge should be placed on a Ping-Pong ball of mass 2.70 g so that the electric force balances the weight of the ball near Earths surface?

Two point charges and both have a charge equal to and are on the axis at and respectively. (a) What are the magnitude and direction of the electric field on the axis at (b) What is the force exerted on a third charge when it is placed on the axis at

A point charge of is located on the axis at and a second point charge of is located on the axis at Where should a third charge of be placed so that the electric field at the origin is zero?

A point charge is located at and a point charge is located at (a) Find the magnitude and direction of the electric field at (b) Calculate the magnitude and direction of the electric force on an electron that is placed at

Two equal positive charges are on the axis; one point charge is at and the other is at (a) Show that on the axis the component of the electric field is given by (b) Show that near the origin, where is much smaller than (c) Show that for values of much larger than Explain why a person might expect this result even without deriving it by taking the appropriate limit.

A point charge is located at and a point charge is located at (a) Find the magnitude and direction of the electric field at (b) Find the magnitude and direction of the force on a proton placed at

Two positive point charges, each having charge , are on the axisone at and the other at . (a) Show that the electric field strength on the axis is greatest at and by computing and setting the derivative equal to zero. (b) Sketch the function versus using your results for Part (a) of this problem and the facts that is approximately when is much smaller than and is approximately when is much larger than

Two point particles, each having a charge sit on the base of an equilateral triangle that has sides of length as shown in Figure 21-38. Athird point particle that has a charge equal to sits at the apex of the triangle. Where must a fourth point particle that has a charge equal to be placed in order that the electric field at the center of the triangle be zero? (The center is in the plane of the triangle and equidistant from the three vertices.) SSM

Two point particles, each having a charge equal to sit on the base of an equilateral triangle that has sides of length as shown in Figure 21-38. A third point particle that has a charge equal to sits at the apex of the triangle. A fourth point particle that has charge is placed at the midpoint of the baseline making the electric field at the center of the triangle equal to zero. What is the value of (The center is in the plane of the triangle and equidistant from all three vertices.)

Two equal positive point charges are on the axis; one is at and the other is at The electric field at the origin is zero. A test charge placed at the origin will therefore be in equilibrium. (a) Discuss the stability of the equilibrium for a positive test charge by considering small displacements from equilibrium along the axis and small displacements along the axis. (b) Repeat Part (a) for a negative test charge. (c) Find the magnitude and sign of a charge that when placed at the origin results in a net force of zero on each of the three charges.

Two positive point charges are on the axis at and A bead of mass and charge slides without friction along a taut thread that runs along the axis. Let be the position of the bead. (a) Show that for the bead experiences a linear restoring force (a force that is proportional to and directed toward the equilibrium position at and therefore undergoes simple harmonic motion. (b) Find the period of the motion.

The acceleration of a particle in an electric field depends on (the charge-to-mass ratio of the particle). (a) Compute for an electron. (b) What are the magnitude and direction of the acceleration of an electron in a uniform electric field that has a magnitude of (c) Compute the time it takes for an electron placed at rest in a uniform electric field that has a magnitude of to reach a speed (When the speed of an electron approaches the speed of light relativistic kinematics must be used to calculate its motion, but at speeds of or less, nonrelativistic kinematics is sufficiently accurate for most purposes.) (d) How far does the electron travel in that time?

The acceleration of a particle in an electric field depends on (the charge-to-mass ratio of the particle). (a) Compute q/m for an electron. (b) What are the magnitude and direction of the acceleration of an electron in a uniform electric field that has a magnitude of 100 N/C? (c) Compute the time it takes for an electron placed at rest in a uniform electric field that has a magnitude of 100 N/C to reach a speed 0.01c. (When the speed of an electron approaches the speed of light relativistic kinematics must be used to calculate its motion, but at speeds of 0.01c or less, non relativistic kinematics is sufficiently accurate for most purposes.) (d) How far does the electron travel in that time?

An electron has an initial velocity of in the direction. It enters a region that has a uniform electric field (a) Find the acceleration of the electron. (b) How long does it take for the electron to travel 10.0 cm in the direction in the region that has the field? (c) Through what angle, and in what direction, is the electron deflected while traveling the 10.0 cm in the direction?

An electron is released from rest in a weak electric field given by After the electron has traveled a vertical distance of what is its speed? (Do not neglect the gravitational force on the electron.)

A 2.00-g charged particle is released from rest in a region that has a uniform electric field After traveling a distance of 0.500 m in this region, the particle has a kinetic energy of 0.120 J. Determine the charge of the particle.

A charged particle leaves the origin with a speed of at an angle of above the axis. A uniform electric field, given by exists throughout the region. Find such that the particle will cross the axis at if the particle is (a) an electron and (b) a proton.

An electron starts at the position shown in Figure 21-39 with an initial speed at to the axis. The electric field is in the direction and has a magnitude of The black lines in the figure are charged metal plates. On which plate and at what location will the electron strike? SSM

ENGINEERING APPLICATION An electron that has a kinetic energy equal to is moving to the right along the axis of a cathode-ray tube as shown in Figure 21-40. An electric field exists in the region between the deflection plates, and no electric field ( exists outside this region. (a) How far is the electron from the axis of the tube when it exits the region between the plates? (b) At what angle is the electron moving, with respect to the axis, after exiting the region between the plates? (c) At what distance from the axis will the electron strike the fluorescent screen?

Two point charges, and are separated by (a) What is the magnitude of the dipole moment of this pair of charges? (b) Sketch the pair and show the direction of the dipole moment.

Adipole of moment is placed in a uniform electric field that has a magnitude of What is the magnitude of the torque on the dipole when (a) the dipole is aligned with the electric field, (b) the dipole is transverse to (perpendicular to) the electric field, and (c) the direction of dipole makes an angle of with the direction of electric field? (d) Defining the potential energy to be zero when the dipole is transverse to the electric field, find the potential energy of the dipole for the orientations specified in Parts (a) and (c).

Show that it is only possible to place one isolated proton in an ordinary empty coffee cup by considering the following situation. Assume the first proton is fixed at the bottom of the cup. Determine the distance directly above this proton where a second proton would be in equilibrium. Compare this distance to the depth of an ordinary coffee cup to complete the argument.

Point charges of and are located on the axis at and respectively. Calculate the electric field on the axis at and at Are there any points on the axis where the magnitude of the electric field is zero? If so, where are those points?

Point charges of and are located on the axis at and respectively. (a) Calculate the electric field strength at (b) Estimate the electric field strength at by modeling the two charges as an electric dipole located at the origin and using (Equation 21-10). Compare your result with the result obtained in Part (a), and explain the reason for the difference between the two results.

A fixed point charge of is connected by strings to point charges of and as shown in Figure 21-41. Find the tensions and

A positive charge is to be divided into two positive point charges and Show that, for a given separation the force exerted by one charge on the other is greatest if

A point charge is located on the axisat and a point charge is located at The electric force on a point charge of is zero if that charge is placed at and is 126 N in the direction if placed at x _ 8.00 cm. Determine the charge Q.

Two point particles separated by 0.60 m have a total charge of (a) If the two particles repel each other with a force of 80 N, what is the charge on each of the two particles? (b) If the two particles attract each other with a force of 80 N, what are the charges on the two particles?

A point particle that has charge and unknown mass is released from rest in a region that has a uniform electric field that is directed vertically downward. The particle hits the ground at a speed where is the initial height of the particle. Find in terms of and

A rigid 1.00-m-long rod is pivoted about its center (Figure 21-42). A charge is placed on one end of the rod, and a charge is placed a distance directly below it. (a) What is the force exerted by on (b) What is the torque (measured about the rotation axis) due to that force? (c) To counterbalance the attraction between the two charges, we hang a block 25.0 cm from the pivot as shown. What value should we choose for the mass of the block? (d) We now move the block and hang it a distance of 25.0 cm from the balance point, on the same side of the balance as the charge. Keeping the same, and the same, what value should we choose for to keep this apparatus in balance? SSM

Two point charges are located at and at Two other point charges, each with charge are located at and at (Figure 21-43). The electric field at due to the presence of the four charges is Determine Q.

Two point charges have a total charge equal to and are separated by (a) Find the charge of each particle if the particles repel each other with a force of 120 N. (b) Find the force on each particle if the charge on each particle is

Two point charges have a total charge equal to and are separated by (a) Find the charge of each particle if the particles attract each other with a force of 120 N. (b) Find the force on each particle if the charge on each particle is

A point charge of is located at the origin; a point charge of is located on the axis at a third point charge is located on the axis at The electric force on the charge is 240 N in the direction. (a) Determine the charge (b)With this configuration of three charges, at what location(s) is the electric field zero?

Two point particles, each of mass and charge are suspended from a common point by threads of length Each thread makes an angle with the vertical as shown in Figure 21-44. (a) Show that where is the Coulomb constant. (b) Find if and

Suppose that in Problem 74 and (a) What is the angle that each string makes with the vertical if ? (b) What is the angle that each string makes with the vertical if one particle has a charge of and the other has a charge of

Four point charges of equal magnitude are arranged at the corners of a square of side as shown in Figure 21-45. (a) Find the magnitude and direction of the force exerted on the charge in the lower left corner by the other three charges. (b) Show that the electric field at the midpoint of one of the sides of the square is directed along that side toward the negative charge and has a magnitude given by

Figure 21-46 shows a dumbbell consisting of two identical small particles, each of mass attached to the ends of a thin (massless) rod of length that is pivoted at its center. The particles have charges of and and the dumbbell is located in a uniform electric field Show that for small values of the angle between the direction of the dipole and the direction of the electric field, the system displays a rotational form of simple harmonic motion, and obtain an expression for the period of that motion. SSM

For the dumbbell in Problem 77, let and The dumbbell is initially at rest and makes an angle of with the axis. The dumbbell is then released, and when it is momentarily aligned with the electric field, its kinetic energy is Determine the magnitude of .

An electron (charge mass ) and a positron (charge mass ) revolve around their common center of mass under the influence of their attractive coulomb force. Find the speed of each particle in terms of and their separation distance

A simple pendulum of length 1.0 m and mass is placed in a uniform, electric field that is directed vertically upward. The bob has a charge of The period of the pendulum is 1.2 s. What are the magnitude and direction of

A point particle of mass and charge is constrained to move vertically inside a narrow, frictionless cylinder (Figure 21-47). At the bottom of the cylinder is a point charge having the same sign as q. (a) Show that the particle whose mass is m will be in equilibrium at a height (b) Show that if the particle is displaced from its equilibrium position by a small amount and released, it will exhibit simple harmonic motion with angular frequency

Two neutral molecules on the axis attract each other. Each molecule has a dipole moment and these dipole moments are on the direction and are separated by a distance Derive an expression for the force of attraction in terms of and

Two equal positive point charges are on the axis at and (a) Obtain an expression for the electric field on the axis as a function of . (b) A bead of mass which has a charge moves along the axis on a thin frictionless taut thread. Find the electric force that acts on the bead as a function of and determine the sign of such that this force always points away from the origin. (c) The bead is initially at rest at the origin. If it is given a slight nudge in the direction, how fast will the bead be traveling the instant the net force on it is a maximum? (Assume any effects due to gravity are negligible.)

A gold nucleus is \(100 \ \mathrm {fm} (1 \ \mathrm {fm}=10^{-15} \ \mathrm m)\) from a proton, which initially is at rest. When the proton is released, it speeds away because of the repulsion that it experiences due to the charge on the gold nucleus. What is the proton’s speed a large distance (assume to be infinity) from the gold nucleus? (Assume the gold nucleus remains stationary.)

During a famous experiment in 1919, Ernest Rutherford shot doubly ionized helium nuclei (also known as alpha particles) at a gold foil. He discovered that virtually all of the mass of an atom resides in an extremely compact nucleus. Suppose that during such an experiment, an alpha particle far from the foil has a kinetic energy of 5.0 MeV. If the alpha particle is aimed directly at the gold nucleus, and the only force acting on it is the electric force of repulsion exerted on it by the gold nucleus, how close will it approach the gold nucleus before turning back? That is, what is the minimum center-tocenter separation of the alpha particle and the gold nucleus?

During the Millikan experiment used to determine the charge on the electron, a charged polystyrene microsphere is released in still air in a known vertical electric field. The charged microsphere will accelerate in the direction of the net force until it reaches terminal speed. The charge on the microsphere is determined by measuring the terminal speed. During one such experiment, the microsphere has radius and the field has a magnitude The magnitude of the drag force on the sphere is given by where is the speed of the sphere and is the viscosity of air Polystyrene has density (a) If the electric field is pointing down and the polystyrene microsphere is rising with a terminal speed of what is the charge on the sphere? (b) How many excess electrons are on the sphere? (c) If the direction of the electric field is reversed but its magnitude remains the same, what is the new terminal speed?

In Problem 86, there is a description of the Millikan experiment used to determine the charge on the electron. During the experiment, a switch is used to reverse the direction of the electric field without changing its magnitude, so that one can measure the terminal speed of the microsphere both as it is moving upward and as it is moving downward. Let represent the terminal speed when the particle is moving up and the terminal speed when moving down. (a) If we let show that where is the microspheres net charge. For the purpose of determining what advantage does measuring both and have over measuring only one terminal speed? (b) Because charge is quantized, can only change by steps of magnitude , where is an integer. Using the data from Problem 86, calculate . SSM