1 Million+ Step-by-step solutions

(a) Find to three significant digits the charge and the mass of an ionized hydrogen atom, represented as H+. Suggestion: Begin by looking up the mass of a neutral atom on the periodic table of the elements.

(b) Find the charge and the mass of Na+, a singly ionized sodium atom.

(c) Find the charge and the average mass of a chloride ion Cl− that joins with the Na+ to make one molecule of table salt.

(d) Find the charge and the mass of Ca++ = Ca2+, a doubly ionized calcium atom.

(e) You can model the center of an ammonia molecule as an N3− ion. Find its charge and mass.

(f) The plasma in a hot star contains quadruply ionized nitrogen atoms, N4+. Find their charge and mass.

(g) Find the charge and the mass of the nucleus of a nitrogen atom.

(h) Find the charge and the mass of the molecular ion H2O−.

(b) Find the charge and the mass of Na+, a singly ionized sodium atom.

(c) Find the charge and the average mass of a chloride ion Cl− that joins with the Na+ to make one molecule of table salt.

(d) Find the charge and the mass of Ca++ = Ca2+, a doubly ionized calcium atom.

(e) You can model the center of an ammonia molecule as an N3− ion. Find its charge and mass.

(f) The plasma in a hot star contains quadruply ionized nitrogen atoms, N4+. Find their charge and mass.

(g) Find the charge and the mass of the nucleus of a nitrogen atom.

(h) Find the charge and the mass of the molecular ion H2O−.

(a) Calculate the number of electrons in a small, electrically neutral silver pin that has a mass of 10.0 g. Silver has 47 electrons per atom, and its molar mass is 107.87 g/mol.

(b) Electrons are added to the pin until the net negative charge is 1.00 mC. How many electrons are added for every 109 electrons already present?

(b) Electrons are added to the pin until the net negative charge is 1.00 mC. How many electrons are added for every 109 electrons already present?

The Nobel laureate Richard Feynman once said that if two persons stood at arm’s length from each other and each person had 1% more electrons than protons, the force of repulsion between them would be enough to lift a “weight’’ equal to that of the entire Earth. Carry out an order-of-magnitude calculation to substantiate this assertion.

Two protons in an atomic nucleus are typically separated by a distance of 2 x 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 electric force between two protons separated by 2.00 x 10−15 m?

(a) Two protons in a molecule are separated by 3.80 x 10−10 m. Find the electric force exerted by one proton on the other.

(b) How does the magnitude of this force compare to the magnitude of the gravitational force between the two protons?

(c) What If? What must be the charge-to-mass ratio of a particle if the magnitude of the gravitational force between two of these particles equals the magnitude of electric force between them?

(b) How does the magnitude of this force compare to the magnitude of the gravitational force between the two protons?

(c) What If? What must be the charge-to-mass ratio of a particle if the magnitude of the gravitational force between two of these particles equals the magnitude of electric force between them?

Two small silver spheres, each with a mass of 10.0 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 x 104 N (about 1 ton) between the spheres. (The number of electrons per atom of silver is 47, and the number of atoms per gram is Avogadro’s number divided by the molar mass of silver, 107.87 g/mol.)

Three point charges are located at the corners of an equilateral triangle as shown in Figure P23.7. Calculate the resultant electric force on the 7.00-μC charge.

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

Two identical conducting small spheres are placed with their centers 0.300m apart. One is given a charge of 12.0 nC and the other a charge of −18.0 nC.

(a) Find the electric force exerted by one sphere on the other.

(b) What If? The spheres are connected by a conducting wire. Find the electric force between the two after they have come to equilibrium.

(a) Find the electric force exerted by one sphere on the other.

(b) What If? The spheres are connected by a conducting wire. Find the electric force between the two after they have come to equilibrium.

Two small beads having positive charges 3q and q are fixed at the opposite ends of a horizontal, insulating rod, extending from the origin to the point x = d. As shown in Figure P23.10, 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?

In the Bohr Theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is 0.529 x 10-10 m.

(a) Find the electric force between the two.

(b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

(a) Find the electric force between the two.

(b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

Two identical particles, each having charge #q, are fixed in space and separated by a distance d. A third point charge -Q is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance x from the midpoint between the two fixed charges (Fig. P23.12)

(a) Show that if x is small compared with d, the motion of -Q will be simple harmonic along the perpendicular bisector. Determine the period of that motion.

(b) How fast will the charge -Q be moving when it is at the midpoint between the two fixed charges, if initially it is released at a distance a << d from the midpoint?

(a) Show that if x is small compared with d, the motion of -Q will be simple harmonic along the perpendicular bisector. Determine the period of that motion.

(b) How fast will the charge -Q be moving when it is at the midpoint between the two fixed charges, if initially it is released at a distance a << d from the midpoint?

What are the magnitude and direction of the electric field that will balance the weight of?

(a) An electron and

(b) A proton? (Use the data in Table 23.1.)

(a) An electron and

(b) A proton? (Use the data in Table 23.1.)

An object having a net charge of 24.0 *C is placed in a uniform electric field of 610 N/C directed vertically. What is the mass of this object if it “floats’’ in the field?

In Figure P23.15, determine the point (other than infinity) at which the electric field is zero.

An airplane is flying 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 a charge concentration of +40.0 C is above the plane at a height of 3 000 m within the cloud and a charge concentration of -40.0 C is at height 1 000 m, what is the electric field at the aircraft?

Two point charges are located on the x axis. The first is a charge +Q at x = - a. The second is an unknown charge located at x = + 3a. The net electric field these charges produce at the origin has a magnitude of 2keQ/a2. What are the two possible values of the unknown charge?

Three charges are at the corners of an equilateral triangle as shown in Figure P23.7.

(a) Calculate the electric field at the position of the 2.00-μC charge due to the 7.00-μC and -4.00-μC charges.

(b) Use your answer to part

(a) To determine the force on the 2.00-μC charge.

(a) Calculate the electric field at the position of the 2.00-μC charge due to the 7.00-μC and -4.00-μC charges.

(b) Use your answer to part

(a) To determine the force on the 2.00-μC charge.

Three point charges are arranged as shown in Figure P23.19.

(a) Find the vector electric field that the 6.00-nC and "3.00-nC charges together create at the origin.

(b) Find the vector force on the 5.00-nC charge.

(a) Find the vector electric field that the 6.00-nC and "3.00-nC charges together create at the origin.

(b) Find the vector force on the 5.00-nC charge.

Two 2.00-μC point charges are located on the x axis. One is at x = 1.00 m, and the other is at x = -1.00 m.

(a) Determine the electric field on the y axis at y = 0.500 m.

(b) Calculate the electric force on a -3.00-μC charge placed on the y axis at y = 0.500 m.

(a) Determine the electric field on the y axis at y = 0.500 m.

(b) Calculate the electric force on a -3.00-μC charge placed on the y axis at y = 0.500 m.

Four point charges are at the corners of a square of side a as shown in Figure P23.21.

(a) Determine the magnitude and direction of the electric field at the location of charge q. (b) What is the resultant force on q?

(a) Determine the magnitude and direction of the electric field at the location of charge q. (b) What is the resultant force on q?

Consider the electric dipole shown in Figure P23.22. Show that the electric field at a distant point on the +x axis is Ex ≈ 4keqa/x 3.

Consider n equal positive point charges each of magnitude Q/n placed symmetrically around a circle of radius R.

(a) Calculate the magnitude of the electric field at a point a distance x on the line passing through the center of the circle and perpendicular to the plane of the circle.

(b) Explain why this result is identical to that of the calculation done in Example 23.8.

(a) Calculate the magnitude of the electric field at a point a distance x on the line passing through the center of the circle and perpendicular to the plane of the circle.

(b) Explain why this result is identical to that of the calculation done in Example 23.8.

Consider an infinite number of identical charges (each of charge q) placed along the x axis at distances a, 2a, 3a, 4a, . . . , from the origin. What is the electric field at the origin due to this distribution? Suggestion: Use the fact that

A rod 14.0 cm long is uniformly charged and has a total charge of -22.0 μC. Determine the magnitude and direction of the electric field along the axis of the rod at a point 36.0 cm from its center.

A continuous line of charge lies along the x axis, extending from x = + x0 to positive infinity. The line carries charge with a uniform linear charge density 3 0. What are the magnitude and direction of the electric field at the origin?

A uniformly charged ring of radius 10.0 cm has a total charge of 75.0 μC. Find the electric field on the axis of the ring at

(a) 1.00 cm,

(b) 5.00 cm,

(c) 30.0 cm, and

(d) 100 cm from the center of the ring

(a) 1.00 cm,

(b) 5.00 cm,

(c) 30.0 cm, and

(d) 100 cm from the center of the ring

A line of charge starts at x = + x0 and extends to positive infinity. The linear charge density is A = A0x0/x. Determine the electric field at the origin.

Show that the maximum magnitude Emax of the electric field along the axis of a uniformly charged ring occurs at x = a/√2 (see Fig. 23.18) and has the value Q/ (6√3πЄ0a2)

A uniformly charged disk of radius 35.0 cm carries charge with a density of 7.90 & 10"3 C/m2. Calculate the electric field on the axis of the disk at

(a) 5.00 cm,

(b) 10.0 cm,

(c) 50.0 cm, and

(d) 200 cm from the center of the disk

(a) 5.00 cm,

(b) 10.0 cm,

(c) 50.0 cm, and

(d) 200 cm from the center of the disk

Example 23.9 derives the exact expression for the electric field at a point on the axis of a uniformly charged disk. Consider a disk, of radius R = 3.00 cm, having a uniformly distributed charge of + 5.20 μC.

(a) Using the result of Example 23.9, compute the electric field at a point on the axis and 3.00 mm from the center. What If? Compare this answer with the field computed from the near-field approximation E = 2/2) 0.

(b) Using the result of Example 23.9, compute the electric field at a point on the axis and

30.0 cm from the center of the disk. What If? Compare this with the electric field obtained by treating the disk as a +5.20-μC point charge at a distance of 30.0 cm.

(a) Using the result of Example 23.9, compute the electric field at a point on the axis and 3.00 mm from the center. What If? Compare this answer with the field computed from the near-field approximation E = 2/2) 0.

(b) Using the result of Example 23.9, compute the electric field at a point on the axis and

30.0 cm from the center of the disk. What If? Compare this with the electric field obtained by treating the disk as a +5.20-μC point charge at a distance of 30.0 cm.

The electric field along the axis of a uniformly charged disk of radius R and total charge Q was calculated in Example 23.9. Show that the electric field at distances x that are large compared with R approaches that of a point charge Q = aπR2. (Suggestion: First show that x/(x2 + R2)1/2 = (1 + R2/x2)-1/2 and use the binomial expansion (1 + ∂) n ≈ 1 + n∂ when ∂ << 1.)

A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle as shown in Figure P23.33. The rod has a total charge of - 7.50 μC. Find the magnitude and direction of the electric field at O, the center of the semicircle.

(a) Consider a uniformly charged thin-walled right circular cylindrical shell having total charge Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder as shown in Figure P23.34. (Suggestion: Use the result of Example 23.8 and treat the cylinder as a collection of ring charges.)

(b) What If? Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Use the result of Example 23.9 to find the field it creates at the same point.

(b) What If? Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Use the result of Example 23.9 to find the field it creates at the same point.

A thin rod of length ℓ and uniform charge per unit length 3 lies along the x axis, as shown in Figure P23.35.

(a) Show that the electric field at P, a distance y from the rod along its perpendicular bisector, has no x component and is given by E = 2ke3 sin +0/y.

(b) What If? Using your result to part (a), show that the field of a rod of infinite length is E = 2keA/y. (Suggestion: First calculate the field at P due to an element of length dx, which has a charge 3 dx. Then change variables from x to +, using the relationships x = y tan θ and dx = y sec2 θ d θ, and integrate over θ.)

(a) Show that the electric field at P, a distance y from the rod along its perpendicular bisector, has no x component and is given by E = 2ke3 sin +0/y.

(b) What If? Using your result to part (a), show that the field of a rod of infinite length is E = 2keA/y. (Suggestion: First calculate the field at P due to an element of length dx, which has a charge 3 dx. Then change variables from x to +, using the relationships x = y tan θ and dx = y sec2 θ d θ, and integrate over θ.)

Three solid plastic cylinders all have radius 2.50 cm and length 6.00 cm. One

(a) Carries charge with uniform density 15.0 nC/m2 everywhere on its surface. Another

(b) Carries charge with the same uniform density on its curved lateral surface only. The third

(c) Carries charge with uniform density 500 nC/m3 throughout the plastic Find the charge of each cylinder.

(a) Carries charge with uniform density 15.0 nC/m2 everywhere on its surface. Another

(b) Carries charge with the same uniform density on its curved lateral surface only. The third

(c) Carries charge with uniform density 500 nC/m3 throughout the plastic Find the charge of each cylinder.

Eight solid plastic cubes, each 3.00 cm on each edge, are glued together to form each one of the objects (i, ii, iii, and iv) shown in Figure P23.37.

(a) Assuming each object carries charge with uniform density 400 nC/m3 throughout its volume; find the charge of each object.

(b) Assuming each object carries charge with uniform density 15.0 nC/m2 everywhere on its exposed surface; find the charge on each object.

(c) Assuming charge is placed only on the edges where perpendicular surfaces meet, with uniform density 80.0 pC/m, find the charge of each object.

(a) Assuming each object carries charge with uniform density 400 nC/m3 throughout its volume; find the charge of each object.

(b) Assuming each object carries charge with uniform density 15.0 nC/m2 everywhere on its exposed surface; find the charge on each object.

(c) Assuming charge is placed only on the edges where perpendicular surfaces meet, with uniform density 80.0 pC/m, find the charge of each object.

A positively charged disk has a uniform charge per unit area as described in Example 23.9. Sketch the electric field lines in a plane perpendicular to the plane of the disk passing through its center.

A negatively charged rod of finite length carries charge with a uniform charge per unit length. Sketch the electric field lines in a plane containing the rod.

Figure P23.40 shows the electric field 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) Determine the ratio q1/q2.

(b) What are the signs of q1 and q2?

Three equal positive charges q are at the corners of an equilateral triangle of side a as shown in Figure P23.41.

(a) Assume that the three charges together create an electric field. Sketch the field lines in the plane of the charges. Find the location of a point (other than 4) where the electric field is zero.

(b) What are the magnitude and direction of the electric field at P due to the two charges at the base?

(a) Assume that the three charges together create an electric field. Sketch the field lines in the plane of the charges. Find the location of a point (other than 4) where the electric field is zero.

(b) What are the magnitude and direction of the electric field at P due to the two charges at the base?

An electron and a proton are each placed at rest in an electric field of 520 N/C. Calculate the speed of each particle 48.0 ns after being released.

A proton accelerates from rest in a uniform electric field of 640 N/C. At some later time, its speed is 1.20 x 106 m/s (non-relativistic, because v is much less than the speed of light).

(a) Find the acceleration of the proton.

(b) How long does it take the proton to reach this speed?

(c) How far has it moved in this time?

(d) What is its kinetic energy at this time?

(a) Find the acceleration of the proton.

(b) How long does it take the proton to reach this speed?

(c) How far has it moved in this time?

(d) What is its kinetic energy at this time?

A proton is projected in the positive x direction into a region of a uniform electric field E = - 6.00 x 105i N/C at t = 0. The proton travels 7.00 cm before coming to rest.

Determine

(a) The acceleration of the proton,

(b) Its initial speed, and

(c) The time at which the proton comes to rest.

Determine

(a) The acceleration of the proton,

(b) Its initial speed, and

(c) The time at which the proton comes to rest.

The electrons in a particle beam each have a kinetic energy K. What are the magnitude and direction of the electric field that will stop these electrons in a distance d?

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 field with a magnitude of 1.00 x 104 N/C. The bead hits the ground at a speed of 21.0 m/s.
Determine

(a) The direction of the electric field (up or down), and

(b) The charge on the bead.

(a) The direction of the electric field (up or down), and

(b) The charge on the bead.

A proton moves at 4.50 x 105 m/s in the horizontal direction. It enters a uniform vertical electric field with a magnitude of 9.60 x 103 N/C. Ignoring any gravitational effects, find (a) The time interval required for the proton to travel 5.00 cm horizontally,

(b) Its vertical displacement during the time interval in which it travels 5.00 cm horizontally, and

(c) The horizontal and vertical components of its velocity after it has traveled 5.00 cm horizontally.

(b) Its vertical displacement during the time interval in which it travels 5.00 cm horizontally, and

(c) The horizontal and vertical components of its velocity after it has traveled 5.00 cm horizontally.

Two horizontal metal plates, each 100 mm square, are aligned 10.0 mm apart, with one above the other. They are given equal-magnitude charges of opposite sign so that a uniform downward electric field of 2 000 N/C exists in the region between them. A particle of mass 2.00 x 10-16 kg and with a positive charge of 1.00 x 10-6 C leaves the center of the bottom negative plate with an initial speed of 1.00 & 105 m/s at an angle of 37.0° above the horizontal. Describe the trajectory of the particle. Which plate does it strike? Where does it strike, relative to its starting point?

Protons are projected with an initial speed vi = 9.55 x 103 m/s into a region where a uniform electric field E = -720j N/C is present, as shown in Figure P23.49. The protons are to hit a target that lies at a horizontal distance of 1.27 mm from the point where the protons cross the plane and enter the electric field in Figure P23.49. Find

(a) The two projection angles + that will result in a hit and

(b) The total time of flight (the time interval during which the proton is above the plane in Figure P23.49) for each trajectory.

(a) The two projection angles + that will result in a hit and

(b) The total time of flight (the time interval during which the proton is above the plane in Figure P23.49) for each trajectory.

Two known charges, -12.0 μC and 45.0 μC, and an unknown charge are located on the x axis. The charge -12.0 μC is at the origin, and the charge 45.0 μC is at x = 15.0 cm. The unknown charge is to be placed so that each charge is in equilibrium under the action of the electric forces exerted by the other two charges. Is this situation possible? Is it possible in more than one way? Find the required location, magnitude, and sign of the unknown charge.

A uniform electric field of magnitude 640 N/C exists between two parallel plates that are 4.00 cm apart. A proton is released from the positive plate at the same instant that an electron is released from the negative plate.

(a) Determine the distance from the positive plate at which the two pass each other. (Ignore the electrical attraction between the proton and electron.)

(b) What If? Repeat part (a) for a sodium ion (Na+) and a chloride ion (Cl-).

(a) Determine the distance from the positive plate at which the two pass each other. (Ignore the electrical attraction between the proton and electron.)

(b) What If? Repeat part (a) for a sodium ion (Na+) and a chloride ion (Cl-).

Three point charges are aligned along the x axis as shown in Figure P23.52. Find the electric field at

(a) The position (2.00, 0) and

(b) The position (0, 2.00).

(a) The position (2.00, 0) and

(b) The position (0, 2.00).

A researcher studying the properties of ions in the upper atmosphere wishes to construct an apparatus with the following characteristics: Using an electric field, a beam of ions, each having charge q, mass m, and initial velocity vi, is turned through an angle of 90° as each ion undergoes displacement Ri + Rj . The ions enter a chamber as shown in Figure P23.53, and leave through the exit port with the same speed they had when they entered the chamber. The electric field acting on the ions is to have constant magnitude.

(a) Suppose the electric field is produced by two concentric cylindrical electrodes not shown in the diagram, and hence is radial. What magnitude should the field have? What If?

(b) If the field is produced by two flat plates and is uniform in direction, what value should the field have in this case?

(a) Suppose the electric field is produced by two concentric cylindrical electrodes not shown in the diagram, and hence is radial. What magnitude should the field have? What If?

(b) If the field is produced by two flat plates and is uniform in direction, what value should the field have in this case?

A small, 2.00-g plastic ball is suspended by a 20.0-cm-long string in a uniform electric field as shown in Figure P23.54. If the ball is in equilibrium when the string makes a 15.0° angle with the vertical, what is the net charge on the ball?

A charged cork ball of mass 1.00 g is suspended on a light string in the presence of a uniform electric field as shown in Figure P23.55. When E = (3.00i + 5.00j) x 105 N/C, the ball is in equilibrium at 0 = 37.0°. Find

(a) The charge on the ball and

(b) The tension in the string.

(a) The charge on the ball and

(b) The tension in the string.

A charged cork ball of mass m is suspended on a light string in the presence of a uniform electric field as shown in Figure P23.55. When E = (Ai + Bj) N/C, where A and B are positive numbers, the ball is in equilibrium at the angle θ. Find

(a) The charge on the ball and

(b) The tension in the string.

(a) The charge on the ball and

(b) The tension in the string.

Four identical point charges (q = +10.0 μC) are located on the corners of a rectangle as shown in Figure P23.57. The dimensions of the rectangle are L = 60.0 cm and W = 15.0 cm. Calculate the magnitude and direction of the resultant electric force exerted on the charge at the lower left corner by the other three charges.

Inez is putting up decorations for her sister’s quinceañera (fifteenth birthday party). She ties three light silk ribbons together to the top of a gateway and hangs a rubber balloon from each ribbon (Fig. P23.58). To include the effects of the gravitational and buoyant forces on it, each balloon can be modeled as a particle of mass 2.00 g, with its center 50.0 cm from the point of support. To show off the colors of the balloons, Inez rubs the whole surface of each balloon with her woolen scarf, to make them hang separately with gaps between them. The centers of the hanging balloons form a horizontal equilateral triangle with sides 30.0 cm long. What is the common charge each balloon carries?

Two identical metallic blocks resting on a frictionless horizontal surface are connected by a light metallic spring having a spring constant k as shown in Figure P23.59a and an unstretched length Li. A total charge Q is slowly placed on the system, causing the spring to stretch to an equilibrium length L, as shown in Figure P23.59b. Determine the value of Q, assuming that all the charge resides on the blocks and modeling the blocks as point charges.

Consider a regular polygon with 29 sides. The distance from the center to each vertex is a. identical charges q are placed at 28 vertices of the polygon. A single charge Q is placed at the center of the polygon. What is the magnitude and direction of the force experienced by the charge Q? (Suggestion: You may use the result of Problem 63 in Chapter 3.)

Identical thin rods of length 2a carry equal charges +Q uniformly distributed along their lengths. The rods lie along the x axis with their centers separated by a distance b - 2a (Fig. P23.61). Show that the magnitude of the force exerted by the left rod on the right one is given by

Two small spheres, each of mass 2.00 g, are suspended by light strings 10.0 cm in length (Fig. P23.62). A uniform electric field is applied in the x direction. The spheres have charges equal to -5.00 x 10-8 C and +5.00 x 10-8 C. Determine the electric field that enables the spheres to be in equilibrium at an angle 0 = 10.0°.

A line of positive charge is formed into a semicircle of radius R = 60.0 cm as shown in Figure P23.63. The charge per unit length along the semicircle is described by the expression 3 = 30 cos +. The total charge on the semicircle is 12.0 μC. Calculate the total force on a charge of 3.00 μC placed at the center of curvature.

Three charges of equal magnitude q are fixed in position at the vertices of an equilateral triangle (Fig. P23.64). A fourth charge Q is free to move along the positive x axis under the influence of the forces exerted by the three fixed charges. Find a value for s for which Q is in equilibrium. You will need to solve a transcendental equation.

Two small spheres of mass m are suspended from strings of length ℓ that are connected at a common point. One sphere has charge Q; the other has charge 2Q. The strings make angles θ1 and θ2 with the vertical.

(a) How are θ1 and θ2 related?

(b) Assume θ1 and θ2 are small. Show that the distance r between the spheres is given by

(a) How are θ1 and θ2 related?

(b) Assume θ1 and θ2 are small. Show that the distance r between the spheres is given by

Four identical particles, each having charge +q, are fixed at the corners of a square of side L. A fifth point charge -Q lies a distance z along the line perpendicular to the plane of the square and passing through the center of the square (Fig. P23.66)

(a) Show that the force exerted by the other four charges on -Q is Note that this force is directed toward the center of the square whether z is positive (-Q above the square) or negative (-Q below the square).

(b) If z is small compared with L, the above expression reduces to F = - (constant) zk. Why does this imply that the motion of the charge -Q is simple harmonic, and what is the period of this motion if the mass of -Q is m?

(a) Show that the force exerted by the other four charges on -Q is Note that this force is directed toward the center of the square whether z is positive (-Q above the square) or negative (-Q below the square).

(b) If z is small compared with L, the above expression reduces to F = - (constant) zk. Why does this imply that the motion of the charge -Q is simple harmonic, and what is the period of this motion if the mass of -Q is m?

A 1.00-g cork ball with charge 2.00 *C is suspended vertically on a 0.500-m-long light string in the presence of a uniform, downward-directed electric field of magnitude E = 1.00 x 105 N/C. If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum.

(a) Determine the period of this oscillation.

(b) Should gravity be included in the calculation for part

(a)? Explain.

(a) Determine the period of this oscillation.

(b) Should gravity be included in the calculation for part

(a)? Explain.

Two identical beads each have a mass m and charge q. When placed in a hemispherical bowl of radius R with frictionless, non-conducting walls, the beads move, and at equilibrium they are a distance R apart (Fig. P23.68). Determine the charge on each bead.

Eight point charges, each of magnitude q, are located on the corners of a cube of edge s, as shown in Figure P23.69.

(a) Determine the x, y, and z components of the resultant force exerted by the other charges on the charge located at point A.

(b) What are the magnitude and direction of this resultant force?

(a) Determine the x, y, and z components of the resultant force exerted by the other charges on the charge located at point A.

(b) What are the magnitude and direction of this resultant force?

Consider the charge distribution shown in Figure P23.69.
(a) Show that the magnitude of the electric field at the center of any face of the cube has a value of 2.18keq/s2.

(b) What is the direction of the electric field at the center of the top face of the cube

(b) What is the direction of the electric field at the center of the top face of the cube

A negatively charged particle -q is placed at the center of a uniformly charged ring, where the ring has a total positive charge Q as shown in Example 23.8. The particle, confined to move along the x axis, is displaced a small distance x along the axis (where x << a) and released. Show that the particle oscillates in simple harmonic motion with a frequency given by

A line of charge with uniform density 35.0 nC/m lies along the line y = -15.0 cm, between the points with coordinates x = 0 and x = 40.0 cm. Find the electric field it creates at the origin.

An electric dipole in a uniform electric field is displaced slightly from its equilibrium position, as shown in Figure P23.73, where + is small. The separation of the charges is 2a, and the moment of inertia of the dipole is I. Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency

An electric field with a magnitude of 3.50 kN/C is applied along the x axis. Calculate the electric flux through a rectangular plane 0.350 m wide and 0.700 m long assuming that

(a) The plane is parallel to the yz plane;

(b) The plane is parallel to the xy plane;

(c) The plane contains the y axis, and its normal makes an angle of 40.0° with the x axis.

(a) The plane is parallel to the yz plane;

(b) The plane is parallel to the xy plane;

(c) The plane contains the y axis, and its normal makes an angle of 40.0° with the x axis.

A vertical electric field of magnitude 2.00 x 104 N/C exists above the Earth’s 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 flux through the bottom of the car.

A 40.0-cm-diameter loop is rotated in a uniform electric field until the position of maximum electric flux is found. The flux in this position is measured to be 5.20 x 105 Nm2/C. What is the magnitude of the electric field?

Consider a closed triangular box resting within a horizontal electric field of magnitude E = 7.80 x 104 N/C as shown in Figure P24.4. Calculate the electric flux through

(a) The vertical rectangular surface,

(b) The slanted surface, and

(c) The entire surface of the box.

(a) The vertical rectangular surface,

(b) The slanted surface, and

(c) The entire surface of the box.

A uniform electric field ai + bj intersects a surface of area A. What is the flux through this area if the surface lies?

(a) In the yz plane?

(b) In the xz plane?

(c) In the xy plane?

(a) In the yz plane?

(b) In the xz plane?

(c) In the xy plane?

A point charge q is located at the center of a uniform ring having linear charge density A and radius a, as shown in Figure P24.6. Determine the total electric flux through a sphere centered at the point charge and having radius R, where R < a.

A pyramid with horizontal square base, 6.00 m on each side, and a height of 4.00 m is placed in a vertical electric field of 52.0 N/C. Calculate the total electric flux through the pyramid’s four slanted surfaces.

A cone with base radius R and height h is located on a horizontal table. A horizontal uniform field E penetrates the cone, as shown in Figure P24.8. Determine the electric flux that enters the left-hand side of the cone.

The following charges are located inside a submarine: 5.00 μC, - 9.00 μC, 27.0 μC, and -84.0 μC.

(a) Calculate the net electric flux through the hull of the submarine.

(b) Is the number of electric field lines leaving the submarine greater than, equal to, or less than the number entering it?

(a) Calculate the net electric flux through the hull of the submarine.

(b) Is the number of electric field lines leaving the submarine greater than, equal to, or less than the number entering it?

The electric field everywhere on the surface of a thin spherical shell of radius 0.750 m is measured to be 890 N/C and points radially toward the center of the sphere.

(a) What is the net charge within the sphere’s surface?

(b) What can you conclude about the nature and distribution of the charge inside the spherical shell?

(a) What is the net charge within the sphere’s surface?

(b) What can you conclude about the nature and distribution of the charge inside the spherical shell?

Four closed surfaces, S1 through S4, together with the charges −2Q, Q, and −Q are sketched in Figure P24.11. (The colored lines are the intersections of the surfaces with the page.) Find the electric flux through each surface.

(a) A point charge q is located a distance d from an infinite plane. Determine the electric flux through the plane due to the point charge.

(b) What If? A point charge q is located a very small distance from the center of a very large square on the line perpendicular to the square and going through its center. Determine the approximate electric flux through the square due to the point charge.

(c) Explain why the answers to parts (a) and (b) are identical

(b) What If? A point charge q is located a very small distance from the center of a very large square on the line perpendicular to the square and going through its center. Determine the approximate electric flux through the square due to the point charge.

(c) Explain why the answers to parts (a) and (b) are identical

Calculate the total electric flux through the paraboloidal surface due to a uniform electric field of magnitude E 0 in the direction shown in Figure P24.13

A point charge of 12.0 %C is placed at the center of a spherical shell of radius 22.0 cm. What is the total electric flux through?

(a) The surface of the shell and

(b) Any hemispherical surface of the shell?

(c) Do the results depend on the radius? Explain.

(a) The surface of the shell and

(b) Any hemispherical surface of the shell?

(c) Do the results depend on the radius? Explain.

A point charge Q is located just above the center of the flat face of a hemisphere of radius R as shown in Figure P24.15. What is the electric flux?

(a) Through the curved surface and

(b) Through the flat face?

(a) Through the curved surface and

(b) Through the flat face?

In the air over a particular region at an altitude of 500 m above the ground the electric field is 120 N/C directed downward. At 600 m above the ground the electric field is 100 N/C downward. What is the average volume charge density in the layer of air between these two elevations? Is it positive or negative?

A point charge Q = 5.00 %C is located at the center of a cube of edge L = 0.100 m. In addition, six other identical point charges having q = -1.00 %C are positioned symmetrically around Q as shown in Figure P24.17. Determine the electric flux through one face of the cube.

A positive point charge Q is located at the center of a cube of edge L. In addition, six other identical negative point charges q are positioned symmetrically around Q as shown in Figure P24.17. Determine the electric flux through one face of the cube.

An infinitely long line charge having a uniform charge per unit length A lies a distance d from point O as shown in Figure P24.19. Determine the total electric flux through the surface of a sphere of radius R centered at O resulting from this line charge. Consider both cases, where R < d and R > d.

An uncharged non-conducting hollow sphere of radius 10.0 cm surrounds a 10.0-%C charge located at the origin of a cartesian coordinate system. A drill with a radius of 1.00 mm is aligned along the z axis, and a hole is drilled in the sphere. Calculate the electric flux through the hole.

A charge of 170 %C is at the center of a cube of edge 80.0 cm.

(a) Find the total flux through each face of the cube.

(b) Find the flux through the whole surface of the cube.

(c) What If? Would your answers to parts (a) or (b) change if the charge were not at the center? Explain.

(a) Find the total flux through each face of the cube.

(b) Find the flux through the whole surface of the cube.

(c) What If? Would your answers to parts (a) or (b) change if the charge were not at the center? Explain.

The line ag in Figure P24.22 is a diagonal of a cube. A point charge q is located on the extension of line ag, very close to vertex a of the cube. Determine the electric flux through each of the sides of the cube which meet at the point a.

Determine the magnitude of the electric field at the surface of a lead-208 nucleus, which contains 82 protons and 126 neutrons. Assume the lead nucleus has a volume 208 times that of one proton, and consider a proton to be a sphere of radius 1.20 & 10'15 m.

A solid sphere of radius 40.0 cm has a total positive charge of 26.0 %C uniformly distributed throughout its volume. Calculate the magnitude of the electric field

(a) 0 cm,

(b) 10.0 cm,

(c) 40.0 cm, and

(d) 60.0 cm from the center of the sphere

(a) 0 cm,

(b) 10.0 cm,

(c) 40.0 cm, and

(d) 60.0 cm from the center of the sphere

A 10.0-g piece of Styrofoam carries a net charge of '0.700 %C and floats above the center of a large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet?

A cylindrical shell of radius 7.00 cm and length 240 cm has its charge uniformly distributed on its curved surface. The magnitude of the electric field at a point 19.0 cm radially outward from its axis (measured from the midpoint of the shell) is 36.0 kN/C find

(a) The net charge on the shell and

(b) The electric field at a point 4.00 cm from the axis, measured radially outward from the midpoint of the shell.

(a) The net charge on the shell and

(b) The electric field at a point 4.00 cm from the axis, measured radially outward from the midpoint of the shell.

A particle with a charge of '60.0 nC is placed at the center of a non-conducting spherical shell of inner radius 20.0 cm and outer radius 25.0 cm. The spherical shell carries charge with a uniform density of -1.33 %C/m3. A proton moves in a circular orbit just outside the spherical shell. Calculate the speed of the proton.

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