Solved: In proton-beam therapy, a high-energy beam of

What is electric potential energy?

What is the electric potential?

What potentials are especially important?

How is potential represented?

How is electric potential used?

Why is energy important in electricity?

A 2.0 cm * 2.0 cm parallel-plate capacitor with a 2.0 mm spacing is charged to {1.0 nC. First a proton and then an electron are released from rest at the midpoint of the capacitor. a. What is each particles energy? b. What is each particles speed as it reaches the plate?

A glass rod is positively charged. The figure shows an end view of the rod. A negatively charged particle moves in a circular arc around the glass rod. Is the work done on the charged particle by the rods electric field positive, negative, or zero?

A proton is fired from far away at a 1.0-mm-diameter glass sphere that has been charged to +100 nC. What initial speed must the proton have to just reach the surface of the glass?

Rank in order, from largest to smallest, the potential energies Ua to Ud of these four pairs of charges. Each + symbol represents the same amount of charge.

An interaction between two elementary particles causes an electron and a positron (a positive electron) to be shot out back to back with equal speeds. What minimum speed must each have when they are 100 fm apart in order to escape each other?

A proton is released from rest at point B, where the potential is 0 V. Afterward, the proton a. Remains at rest at B. b. Moves toward A with a steady speed. c. Moves toward A with an increasing speed. d. Moves toward C with a steady speed. e. Moves toward C with an increasing speed.

Three electrons are spaced 1.0 mm apart along a vertical line. The outer two electrons are fixed in position. a. Is the center electron at a point of stable or unstable equilibrium? b. If the center electron is displaced horizontally by a small distance, what will its speed be when it is very far away?

Rank in order, from largest to smallest, the potentials Va to Ve at the points a to e

The water molecule is a permanent electric dipole with dipole moment 6.2 * 10-30 Cm. A water molecule is aligned in an electric field with field strength 1.0 * 107 N/C. How much energy is needed to rotate the molecule 90?

Rank in order, from largest to smallest, the potential differences Vab, Vac, and Vbc between points a and b, points a and c, and points b and c.

A proton with a speed of 2.0 * 105 m/s enters a region of space in which there is an electric potential. What is the protons speed after it moves through a potential difference of 100 V? What will be the final speed if the proton is replaced by an electron?

Youve been assigned the task of measuring the speed of protons as they emerge from a small accelerator. To do so, you decide to measure how much voltage is needed across a parallel-plate capacitor to stop the protons. The capacitor you choose has a 2.0 mm plate separation and a small hole in one plate that you shoot the protons through. By filling the space between the plates with a low-density gas, you can see (with a microscope) a slight glow from the region where the protons collide with and excite the gas molecules. The width of the glow tells you how far the protons travel before being stopped and reversing direction. Varying the voltage across the capacitor gives the following data: What value will you report for the speed of the protons?

A proton is released from rest at the surface of a 1.0-cm-diameter sphere that has been charged to +1000 V. a. What is the charge of the sphere? b. What is the protons speed at 1.0 cm from the sphere?

What is the electric potential at the point indicated in FIGURE 25.29?

A thin, uniformly charged ring of radius R has total charge Q. Find the potential at distance z on the axis of the ring.

A thin plastic disk of radius R is uniformly coated with charge until it receives total charge Q. a. What is the potential at distance z along the axis of the disk? b. What is the potential energy if an electron is 1.00 cm from a 35.0-cm-diameter disk that has been charged to + 5.00 nC

a. Charge \(q_1\) is distance r from a positive point charge Q. Charge \(q_2\) = \(q_1/3\) is distance 2r from Q. What is the ratio \(U_1/U_2\) of their potential energies due to their interactions with Q? b. Charge \(q_1\) is distance s from the negative plate of a parallel-plate capacitor. Charge \(q_2\) = \(q_1/3\) is distance 2s from the negative plate. What is the ratio \(U_1/U_2\) of their potential energies?

FIGURE Q25.2 shows the potential energy of a proton 1q = +e2 and a lead nucleus 1q = +82e2. The horizontal scale is in units of femtometers, where 1 fm = 10-15 m. a. A proton is fired toward a lead nucleus from very far away. How much initial kinetic energy does the proton need to reach a turning point 10 fm from the nucleus? Explain. b. How much kinetic energy does the proton of part a have when it is 20 fm from the nucleus and moving toward it, before the collision?

An electron moves along the trajectory of FIGURE Q25.3 from i to f. a. Does the electric potential energy increase, decrease, or stay the same? Explain. b. Is the electrons speed at f greater than, less than, or equal to its speed at i? Explain

Two protons are launched with the same speed from point 1 inside the parallel-plate capacitor of FIGURE Q25.4. Points 2 and 3 are the same distance from the negative plate. a. Is U1S2, the change in potential energy along the path 1 S 2, larger than, smaller than, or equal to U1S3? b. Is the protons speed v2 at point 2 larger than, smaller than, or equal to v3? Explain.

Rank in order, from most positive to most negative, the potential energies Ua to Uf of the six electric dipoles in the uniform electric field of FIGURE Q25.5. Explain.

FIGURE Q25.6 shows the electric potential along the x-axis. a. Draw a graph of the potential energy of a 0.1 C charged particle. Provide a numerical scale for both axes. b. If the charged particle is shot toward the right from x = 1 m with 1.0 J of kinetic energy, where is its turning point? Use your graph to explain

A capacitor with plates separated by distance d is charged to a potential difference VC. All wires and batteries are disconnected, then the two plates are pulled apart (with insulated handles) to a new separation of distance 2d. a. Does the capacitor charge Q change as the separation increases? If so, by what factor? If not, why not? b. Does the electric field strength E change as the separation increases? If so, by what factor? If not, why not? c. Does the potential difference VC change as the separation increases? If so, by what factor? If not, why not?

Rank in order, from largest to smallest, the electric potentials Va to Ve at points a to e in FIGURE Q25.8. Explain.

FIGURE Q25.9 shows two points inside a capacitor. Let V = 0 V at the negative plate. a. What is the ratio V2/V1 of the electric potentials? Explain. b. What is the ratio E2/E1 of the electric field strengths?

FIGURE Q25.10 shows two points near a positive point charge. a. What is the ratio V2/V1 of the electric potentials? Explain. b. What is the ratio E2/E1 of the electric field strengths?

FIGURE Q25.11 shows three points near two point charges. The charges have equal magnitudes. For each part, rank in order, from most positive to most negative, the potentials Va to Vc. 1 2 1 mm 3 mm FigureQ25.10 a (a) b c a (b) b c

Reproduce FIGURE Q25.12 on your paper. Then draw a dot (or dots) on the figure to show the position (or positions) at which the electric potential is zero.

The electric field strength is 20,000 N/C inside a parallel-plate capacitor with a 1.0 mm spacing. An electron is released from rest at the negative plate. What is the electrons speed when it reaches the positive plate?

The electric field strength is 50,000 N/C inside a parallel-plate capacitor with a 2.0 mm spacing. A proton is released from rest at the positive plate. What is the protons speed when it reaches the negative plate?

A proton is released from rest at the positive plate of a parallelplate capacitor. It crosses the capacitor and reaches the negative plate with a speed of 50,000 m/s. What will be the final speed of an electron released from rest at the negative plate?

A proton is released from rest at the positive plate of a parallelplate capacitor. It crosses the capacitor and reaches the negative plate with a speed of 50,000 m/s. The experiment is repeated with a He+ ion (charge e, mass 4 u). What is the ions speed at the negative plate?

What is the potential energy of the electron-proton interactions in FIGURE EX25.5? The electrons are fixed and cannot move.

What is the electric potential energy of the group of charges in FIGURE EX25.6?

What is the electric potential energy of the group of charges in FIGURE EX25.7?

Two positive point charges are 5.0 cm apart. If the electric potential energy is 72 mJ, what is the magnitude of the force between the two charges?

A water molecule perpendicular to an electric field has 1.0 * 10-21 J more potential energy than a water molecule aligned with the field. The dipole moment of a water molecule is 6.2 * 10-30 C m. What is the strength of the electric field?

FIGURE EX25.10 shows the potential energy of an electric dipole. Consider a dipole that oscillates between {60. a. What is the dipoles mechanical energy? b. What is the dipoles kinetic energy when it is aligned with the electric field?

What is the speed of a proton that has been accelerated from rest through a potential difference of -1000 V?

What is the speed of an electron that has been accelerated from rest through a potential difference of 1000 V?

What potential difference is needed to accelerate an electron from rest to a speed of 2.0 * 106 m/s?

What potential difference is needed to accelerate a He+ ion (charge +e, mass 4 u) from rest to a speed of 2.0 * 106 m/s?

A proton with an initial speed of 800,000 m/s is brought to rest by an electric field. a. Did the proton move into a region of higher potential or lower potential? b. What was the potential difference that stopped the proton?

An electron with an initial speed of 500,000 m/s is brought to rest by an electric field. a. Did the electron move into a region of higher potential or lower potential? b. What was the potential difference that stopped the electron?

Through what potential difference must a proton be accelerated to reach the speed it would have by falling 100 m in vacuum?

In proton-beam therapy, a high-energy beam of protons is fired at a tumor. As the protons stop in the tumor, their kinetic energy breaks apart the tumors DNA, thus killing the tumor cells. For one patient, it is desired to deposit 0.10 J of proton energy in the tumor. To create the proton beam, protons are accelerated from rest through a 10,000 kV potential difference. What is the total charge of the protons that must be fired at the tumor?

A student wants to make a very small particle accelerator using a 9.0 V battery. What speed will (a) a proton and (b) an electron have after being accelerated from rest through the 9.0 V potential difference?

Physicists often use a different unit of energy, the electron volt, when dealing with energies at the atomic level. One electron volt, abbreviated eV, is defined as the amount of kinetic energy gained by an electron upon accelerating through a 1.0 V potential difference. a. What is 1.0 electron volt in joules? b. What is the speed of a proton with 5000 eV of kinetic energy?

Show that 1 V/m = 1 N/C

a. What is the potential of an ordinary AA or AAA battery? (If youre not sure, find one and look at the label.) b. An AA battery is connected to a parallel-plate capacitor having 4.0 cm * 4.0 cm plates spaced 1.0 mm apart. How much charge does the battery supply to each plate?

A 3.0-cm-diameter parallel-plate capacitor has a 2.0 mm spacing. The electric field strength inside the capacitor is 1.0 * 105 V/m. a. What is the potential difference across the capacitor? b. How much charge is on each plate?

Two 2.00 cm * 2.00 cm plates that form a parallel-plate capacitor are charged to {0.708 nC. What are the electric field strength inside and the potential difference across the capacitor if the spacing between the plates is (a) 1.00 mm and (b) 2.00 mm?

Two 2.0-cm-diameter disks spaced 2.0 mm apart form a parallel-plate capacitor. The electric field between the disks is 5.0 * 105 V/m. a. What is the voltage across the capacitor? b. An electron is launched from the negative plate. It strikes the positive plate at a speed of 2.0 * 107 m/s. What was the electrons speed as it left the negative plate?

In FIGURE EX25.26, a proton is fired with a speed of 200,000 m/s from the midpoint of the capacitor toward the positive plate. a. Show that this is insufficient speed to reach the positive plate. b. What is the protons speed as it collides with the negative plate?

a. What is the electric potential at points A, B, and C in FIGURE EX25.27? b. What are the potential differences VAB = VB - VA and VCB = VB - VC

A 1.0-mm-diameter ball bearing has 2.0 * 109 excess electrons. What is the ball bearings potential?

In a semiclassical model of the hydrogen atom, the electron orbits the proton at a distance of 0.053 nm. a. What is the electric potential of the proton at the position of the electron? b. What is the electrons potential energy?

What is the electric potential at the point indicated with the dot in FIGURE EX25.30?

What is the electric potential at the point indicated with the dot in FIGURE EX25.31?

The electric potential at the dot in FIGURE EX25.32 is 3140 V. What is charge q?

A -2.0 nC charge and a +2.0 nC charge are located on the x-axis at x = -1.0 cm and x = +1.0 cm, respectively. a. Other than at infinity, is there a position or positions on the x-axis where the electric field is zero? If so, where? b. Other than at infinity, at what position or positions on the x-axis is the electric potential zero? c. Sketch graphs of the electric field strength and the electric potential along the x-axis

Two point charges qa and qb are located on the x-axis at x = a and x = b. FIGURE EX25.34 is a graph of V, the electric potential. a. What are the signs of qa and qb? b. What is the ratio qa/qb ? c. Draw a graph of Ex, the x-component of the electric field, as a function of x.

The two halves of the rod in FIGURE EX25.35 are uniformly charged to {Q. What is the electric potential at the point indicated by the dot?

A 5.0-cm-diameter metal ball has a surface charge density of 10 mC/m2 . How much work is required to remove one electron from this ball?

Two point charges 2.0 cm apart have an electric potential energy -180 mJ. The total charge is 30 nC. What are the two charges?

A -10.0 nC point charge and a +20.0 nC point charge are 15.0 cm apart on the x-axis. a. What is the electric potential at the point on the x-axis where the electric field is zero? b. What is the magnitude of the electric field at the point on the x-axis, between the charges, where the electric potential is zero?

A +3.0 nC charge is at x = 0 cm and a -1.0 nC charge is at x = 4 cm. At what point or points on the x-axis is the electric potential zero?

A -3.0 nC charge is on the x-axis at x = -9 cm and a +4.0 nC charge is on the x-axis at x = 16 cm. At what point or points on the y-axis is the electric potential zero?

Two small metal cubes with masses 2.0 g and 4.0 g are tied together by a 5.0-cm-long massless string and are at rest on a frictionless surface. Each is charged to +2.0 \(\mu C\). a. What is the energy of this system? b. What is the tension in the string? c. The string is cut. What is the speed of each cube when they are far apart? Hint: There are two conserved quantities. Make use of both.

The four 1.0 g spheres shown in FIGURE P25.42 are released simultaneously and allowed to move away from each other. What is the speed of each sphere when they are very far apart?

A protons speed as it passes point A is 50,000 m/s. It follows the trajectory shown in FIGURE P25.43. What is the protons speed at point B? ||Living cells pump singly ionized sodium ions, Na+, from the inside of the cell to the outside to maintain a membrane potential Vmembrane = Vin - Vout = - 70 mV. It is called pumping because work must be done to move a positive ion from the negative inside of the cell to the positive outside, and it must go on continuously because sodium ions leak back through the cell wall by diffusion. a. How much work must be done to move one sodium ion from the inside of the cell to the outside? b. At rest, the human body uses energy at the rate of approximately 100 W to maintain basic metabolic functions. It has been estimated that 20% of this energy is used to operate the sodium pumps of the body. Estimateto one significant figurethe number of sodium ions pumped per second.

An arrangement of source charges produces the electric potential V = 5000x2 along the x-axis, where V is in volts and x is in meters. What is the maximum speed of a 1.0 g, 10 nC charged particle that moves in this potential with turning points at {8.0 cm?

A proton moves along the x-axis where some arrangement of charges has produced the potential V1x2 = V0 sin12px/l2, where V0 = 5000 V and l = 1.0 mm. a. What minimum speed must the proton have at x = 0 to move down the axis without being reflected? b. What is the maximum speed reached by a proton that at x = 0 has the speed you calculated in part a?

The electron gun in an old TV picture tube accelerates electrons between two parallel plates 1.2 cm apart with a 25 kV potential difference between them. The electrons enter through a small hole in the negative plate, accelerate, then exit through a small hole in the positive plate. Assume that the holes are small enough not to affect the electric field or potential. a. What is the electric field strength between the plates? b. With what speed does an electron exit the electron gun if its entry speed is close to zero? Note The exit speed is so fast that we really need to use the theory of relativity to compute an accurate value. Your answer to part b is in the right range but a little too big

A room with 3.0-m-high ceilings has a metal plate on the floor with V = 0 V and a separate metal plate on the ceiling. A 1.0 g glass ball charged to +4.9 nC is shot straight up at 5.0 m/s. How high does the ball go if the ceiling voltage is (a) +3.0 * 106 V and (b) -3.0 * 106 V?

A group of science and engineering students embarks on a quest to make an electrostatic projectile launcher. For their first trial, a horizontal, frictionless surface is positioned next to the 12-cm-diameter sphere of a Van de Graaff generator, and a small, 5.0 g plastic cube is placed on the surface with its center 2.0 cm from the edge of the sphere. The cube is given a positive charge, and then the Van de Graaff generator is turned on, charging the sphere to a potential of 200,000 V in a negligible amount of time. How much charge does the plastic cube need to achieve a final speed of a mere 3.0 m/s? Does this seem like a practical projectile launcher?

Two 2.0 g plastic buttons each with +50 nC of charge are placed on a frictionless surface 2.0 cm (measured between centers) on either side of a 5.0 g button charged to +250 nC. All three are released simultaneously. a. How many interactions are there that have a potential energy? b. What is the final speed of each button?

What is the escape speed of an electron launched from the surface of a 1.0-cm-diameter glass sphere that has been charged to 10 nC?

An electric dipole has dipole moment p. If r W s, where s is the separation between the charges, show that the electric potential of the dipole can be written V = 1 4pP0 p cos u r2 where r is the distance from the center of the dipole and u is the angle from the dipole axis.

Three electrons form an equilateral triangle 1.0 nm on each side. A proton is at the center of the triangle. What is the potential energy of this group of charges?

A 2.0-mm-diameter glass bead is positively charged. The potential difference between a point 2.0 mm from the bead and a point 4.0 mm from the bead is 500 V. What is the charge on the bead?

Your lab assignment for the week is to measure the amount of charge on the 6.0-cm-diameter metal sphere of a Van de Graaff generator. To do so, youre going to use a spring with spring constant 0.65 N/m to launch a small, 1.5 g bead horizontally toward the sphere. You can reliably charge the bead to 2.5 nC, and your plan is to use a video camera to measure the beads closest approach to the edge of the sphere as you change the compression of the spring. Your data are as follows: Compression (cm) Closest approach (cm) 1.6 5.5 1.9 2.6 2.2 1.6 2.5 0.4 Use an appropriate graph of the data to determine the spheres charge in nC. You can assume that the beads motion is entirely horizontal, that the spring is so far away that the bead has no interaction with the sphere as its launched, and that the approaching bead does not alter the charge distribution on the sphere.

A proton is fired from far away toward the nucleus of an iron atom. Iron is element number 26, and the diameter of the nucleus is 9.0 fm. What initial speed does the proton need to just reach the surface of the nucleus? Assume the nucleus remains at rest

A proton is fired from far away toward the nucleus of a mercury atom. Mercury is element number 80, and the diameter of the nucleus is 14.0 fm. If the proton is fired at a speed of 4.0 * 107 m/s, what is its closest approach to the surface of the nucleus? Assume the nucleus remains at rest.

In the form of radioactive decay known as alpha decay, an unstable nucleus emits a helium-atom nucleus, which is called an alpha particle. An alpha particle contains two protons and two neutrons, thus having mass m = 4 u and charge q = 2e. Suppose a uranium nucleus with 92 protons decays into thorium, with 90 protons, and an alpha particle. The alpha particle is initially at rest at the surface of the thorium nucleus, which is 15 fm in diameter. What is the speed of the alpha particle when it is detected in the laboratory? Assume the thorium nucleus remains at rest

One form of nuclear radiation, beta decay, occurs when a neutron changes into a proton, an electron, and a neutral particle called a neutrino: n S p+ + e- + n where n is the symbol for a neutrino. When this change happens to a neutron within the nucleus of an atom, the proton remains behind in the nucleus while the electron and neutrino are ejected from the nucleus. The ejected electron is called a beta particle. One nucleus that exhibits beta decay is the isotope of hydrogen 3 H, called tritium, whose nucleus consists of one proton (making it hydrogen) and two neutrons (giving tritium an atomic mass m = 3 u). Tritium is radioactive, and it decays to helium: 3 H S 3 He + e- + n. a. Is charge conserved in the beta decay process? Explain. b. Why is the final product a helium atom? Explain. c. The nuclei of both 3 H and 3 He have radii of 1.5 * 10-15 m. With what minimum speed must the electron be ejected if it is to escape from the nucleus and not fall back?

Two 10-cm-diameter electrodes 0.50 cm apart form a parallelplate capacitor. The electrodes are attached by metal wires to the terminals of a 15 V battery. After a long time, the capacitor is disconnected from the battery but is not discharged. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes a. Right after the battery is disconnected? b. After insulating handles are used to pull the electrodes away from each other until they are 1.0 cm apart? c. After the original electrodes (not the modified electrodes of part b) are expanded until they are 20 cm in diameter?

Two 10-cm-diameter electrodes 0.50 cm apart form a parallelplate capacitor. The electrodes are attached by metal wires to the terminals of a 15 V battery. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes a. While the capacitor is attached to the battery? b. After insulating handles are used to pull the electrodes away from each other until they are 1.0 cm apart? The electrodes remain connected to the battery during this process. c. After the original electrodes (not the modified electrodes of part b) are expanded until they are 20 cm in diameter while remaining connected to the battery?

Electrodes of area A are spaced distance d apart to form a parallel-plate capacitor. The electrodes are charged to {q. a. What is the infinitesimal increase in electric potential energy dU if an infinitesimal amount of charge dq is moved from the negative electrode to the positive electrode? b. An uncharged capacitor can be charged to {Q by transferring charge dq over and over and over. Use your answer to part a to show that the potential energy of a capacitor charged to {Q is Ucap = 1 2 Q VC.

a. Find an algebraic expression for the electric field strength E0 at the surface of a charged sphere in terms of the spheres potential V0 and radius R. b. What is the electric field strength at the surface of a 1.0-cm-diameter marble charged to 500 V?

Two spherical drops of mercury each have a charge of 0.10 nC and a potential of 300 V at the surface. The two drops merge to form a single drop. What is the potential at the surface of the new drop?

A Van de Graaff generator is a device for generating a large electric potential by building up charge on a hollow metal sphere. A typical classroom-demonstration model has a diameter of 30 cm. a. How much charge is needed on the sphere for its potential to be 500,000 V? b. What is the electric field strength just outside the surface of the sphere when it is charged to 500,000 V?

FIGURE P25.66 shows two uniformly charged spheres. What is the potential difference between points a and b? Which point is at the higher potential? Hint: The potential at any point is the superposition of the potentials due to all charges.

Two positive point charges q are located on the y-axis at y = {1 2 s.a. Find an expression for the potential along the x-axis. b. Draw a graph of V versus x for - 6 x 6 . For comparison, use a dotted line to show the potential of a point charge 2q located at the origin.

The arrangement of charges shown in FIGURE P25.68 is called a linear electric quadrupole. The positive charges are located at y = {s. Notice that the net charge is zero. Find an expression for the electric potential on the y-axis at distances y W s. Give your answer in terms of the quadrupole moment, Q = 2qs2 .

FIGURE P25.69 shows a thin rod of length L and charge Q. Find an expression for the electric potential a distance x away from the center of the rod on the axis of the rod.

FIGURE P25.69 shows a thin rod of length L and charge Q. Find an expression for the electric potential a distance z away from the center of rod on the line that bisects the rod

FIGURE P25.71 shows a thin rod with charge Q that has been bent into a semicircle of radius R. Find an expression for the electric potential at the center.

A disk with a hole has inner radius Rin and outer radius Rout. The disk is uniformly charged with total charge Q. Find an expression for the on-axis electric potential at distance z from the center of the disk. Verify that your expression has the correct behavior when Rin S 0.

The wire in FIGURE P25.73 has linear charge density l. What is the electric potential at the center of the semicircle?

In Problems 74 through 76 you are given the equation(s) used to solve a problem. For each of these, a. Write a realistic problem for which this is the correct equation(s). b. Finish the solution of the problem.19.0 * 109 Nm2 /C2 2q1q2 0.030 m = 90 * 10-6 J q1 + q2 = 40 nC

In Problems 74 through 76 you are given the equation(s) used to solve a problem. For each of these, a. Write a realistic problem for which this is the correct equation(s). b. Finish the solution of the problem.2 11.67 * 10-27 kg212.5 * 106 m/s22 + 0 = 1 2 11.67 * 10-27 kg2vi 2 + 19.0 * 109 Nm2 /C2 212.0 * 10-9 C211.60 * 10-19 C2 0.0010 m

In Problems 74 through 76 you are given the equation(s) used to solve a problem. For each of these, a. Write a realistic problem for which this is the correct equation(s). b. Finish the solution of the problem.19.0 * 109 Nm2 /C2 213.0 * 10-9 C2 0.030 m + 19.0 * 109 Nm2 /C2 213.0 * 10-9 C2 10.030 m2 + d

An electric dipole consists of 1.0 g spheres charged to {2.0 nC at the ends of a 10-cm-long massless rod. The dipole rotates on a frictionless pivot at its center. The dipole is held perpendicular to a uniform electric field with field strength 1000 V/m, then released. What is the dipoles angular velocity at the instant it is aligned with the electric field?

A proton and an alpha particle 1q = +2e, m = 4 u2 are fired directly toward each other from far away, each with an initial speed of 0.010c. What is their distance of closest approach, as measured between their centers?

Bead A has a mass of 15 g and a charge of -5.0 nC. Bead B has a mass of 25 g and a charge of -10.0 nC. The beads are held 12 cm apart (measured between their centers) and released. What maximum speed is achieved by each bead?

Two 2.0-mm-diameter beads, C and D, are 10 mm apart, measured between their centers. Bead C has mass 1.0 g and charge 2.0 nC. Bead D has mass 2.0 g and charge -1.0 nC. If the beads are released from rest, what are the speeds vC and vD at the instant the beads collide?

A thin rod of length L and total charge Q has the nonuniform linear charge distribution l1x2 = l0x/L, where x is measured from the rods left end. a. What is l0 in terms of Q and L? b. What is the electric potential on the axis at distance d left of the rods left end?

A hollow cylindrical shell of length L and radius R has charge Q uniformly distributed along its length. What is the electric potential at the center of the cylinder?