Four identical metal spheres have charges of qA 8.0 C, qB

How many electrons are there in one coulomb of negative charge?

An electrically neutral object acquires a net electric charge. Which one of the following statements concerning the mass of the object is true? (a) The mass does not change. (b) The mass increases if the charge is positive and decreases if it is negative. (c) The mass increases if the charge is negative and decreases if it is positive

Object A and object B are each electrically neutral. Two million electrons are removed from object A and placed on object B. Expressed in coulombs, what is the resulting charge (algebraic sign and magnitude) on object A and on object B?

Object A has a charge of 1.6 1013 C, and object B is electrically neutral. Two million electrons are removed from object A and placed on object B. Expressed in coulombs, what is the resulting charge (algebraic sign and magnitude) on object A and on object B? C

Two metal spheres are identical. They are electrically neutral and are touching. An electrically charged ebonite rod is then brought near the spheres without touching them, as the drawing shows. After a while, with the rod held in place, the spheres are separated, and the rod is then removed. The following statements refer to the masses mA and mB of the spheres after they are separated and the rod is removed. Which one or more of the statements is true? (a) mA  m B (b) mA  m B if the rod is positive (c) mA  m B if the rod is positive (d) mA  m B if the rod is negative (e) mA  m B if the rod is negative

Blow up a balloon, tie it shut, and rub it against your shirt a number of times, so that the balloon acquires a net electric charge. Now touch the balloon to the ceiling. When released, will the balloon remain stuck to the ceiling?

A rod made from insulating material carries a net charge (which may be positive or negative), whereas a copper sphere is electrically neutral. The rod is held close to the sphere but does not touch it. Which one of the following statements concerning the forces that the rod and sphere exert on each other is true? (a) The forces are always attractive. (b) The forces are always repulsive. (c) The forces are attractive when the rod is negative and repulsive when it is positive. (d) The forces are repulsive when the rod is negative and attractive when it is positive. (e) There are no forces.

Two objects, whose charges are 1.0 and 1.0 C, are separated by 1.0 km. Compared to 1.0 km, the sizes of the objects are small. Find the magnitude of the attractive force that either charge exerts on the other.

In the Bohr model of the hydrogen atom, the electron (charge  e) is in a circular orbit about the nuclear proton (charge  e) at a radius of 5.29 1011 m, as Figure 18.10 shows. The mass of the electron is 9.11 1031 kg. Determine the speed of the electron. Rea

Figure 18.12a shows three point charges that lie along the x axis in a vacuum. Determine the magnitude and direction of the net electrostatic force on q1.

Figure 18.13a shows three point charges that lie in the x, y plane in a vacuum. Find the magnitude and direction of the net electrostatic force on q1

Identical point charges are fixed to diagonally opposite corners of a square. Where does a third point charge experience the greater force? (a) At the center of the square (b) At one of the empty corners (c) The question is unanswerable because the polarities of the charges are not given.

The drawing shows three point charges arranged in three different ways. The charges are q, q, and q; each has the same magnitude, with one positive and the other two negative. In each of the arrangements the distance d is the same. Rank the arrangements in descending order (largest first) according to the magnitude of the net electrostatic force that acts on the positive charge.

A proton and an electron are held in place on the x axis. The proton is at x  d, while the electron is at x  d. They are released simultaneously, and the only force that affects their motions significantly is the electrostatic force of attraction that each applies to the other. Which particle reaches the origin first?

A particle is attached to one end of a horizontal spring, and the other end of the spring is attached to a wall. When the particle is pushed so that the spring is compressed more and more, the particle experiences a greater and greater force from the spring. Similarly, a charged particle experiences a greater and greater force when pushed closer and closer to another particle that is fixed in position and has a charge of the same polarity. Considering this similarity, will the charged particle exhibit simple harmonic motion on being released, as will the particle on the spring?

The positive test charge shown in Figure 18.15 is q0  3.0 108 C and experiences a force F  6.0 108 N in the direction shown in the drawing. (a) Find the force per coulomb that the test charge experiences. (b) Using the result of part (a), predict the force that a charge of 12 108 C would experience if it replaced q0. Reason

In Figure 18.16 the charges on the two metal spheres and the ebonite rod create an electric field at the spot indicated. This field has a magnitude of 2.0 N/C and is directed as in the drawing. Determine the force on a charge placed at that spot, if the charge has a value of (a) q0  18 108 C and (b) q0  24 108 C. Rea

Figure 18.17 shows two charged objects, A and B. Each contributes as follows to the net electric field at point P: A  3.00 N/C directed to the right, and B  2.00 N/C directed downward. Thus, A and B are perpendicular. What is the net electric field at P?

In a vacuum, a proton (charge  e, mass  1.67 1027 kg) is moving parallel to a uniform electric field that is directed along the x axis (see Figure 18.18). The proton starts with a velocity of 2.5 104 m/s and accelerates in the same direction as the electric field, which has a value of 2.3 103 N/C. Find the velocity of the proton when its displacement is 2.0 mm from the starting point. Rea

There is an isolated point charge of q  15 C in a vacuum at the left in Figure 18.19a. Using a test charge of q0  0.80 C, determine the electric field at point P, which is 0.20 m away.

Two positive point charges, q1  16 C and q2  4.0 C, are separated in a vacuum by a distance of 3.0 m, as Figure 18.20 illustrates. Find the spot on the line between the charges where the net electric field is zero.

Four point charges all have the same magnitude, but they do not all have the same sign. These charges are fixed to the corners of a rectangle in two different ways, as Figure 18.21 shows. Consider the net electric field at the center C of the rectangle in each case. In which case, if either, is the net electric field greater? (a) It is greater in Figure 18.21a. (b) It is greater in Figure 18.21b. (c) The field has the same magnitude in both cases.

There is an electric field at point P. A very small positive charge is placed at this point and experiences a force. Then the positive charge is replaced by a very small negative charge that has a magnitude different from that of the positive charge. Which one of the following statements is true concerning the forces that these charges experience at P? (a) They are identical. (b) They have the same magnitude but different directions. (c) They have different magnitudes but the same direction. (d) They have different magnitudes and different directions.

Suppose that in Figure 18.21a point charges q are fixed in place at corners 1 and 3 and point charges q are fixed in place at corners 2 and 4. What then would be the net electric field at the center C of the rectangle? 1

A positive point charge q is fixed in position at the center of a square, as the drawing shows. A second point charge is fixed to corner B, C, or D. The net electric field that results at corner A is zero. (a) At which corner is the second charge located? (b) Is the second charge positive or negative? (c) Does the second charge have a greater, a smaller, or the same magnitude as the charge at the center?

A positive point charge is located to the left of a negative point charge. When both charges have the same magnitude, there is no place on the line passing through both charges where the net electric field due to the two charges is zero. Suppose, however, that the negative charge has a greater magnitude than the positive charge. On which part of the line, if any, is a place of zero net electric field now located? (a) To the left of the positive charge (b) Between the two charges (c) To the right of the negative charge (d) There is no zero place.

Three point charges are fixed to the corners of a square, one to a corner, in such a way that the net electric field at the empty corner is zero. Do these charges all have (a) the same sign and (b) the same magnitude (but, possibly, different signs)?

Consider two identical, thin, and nonconducting rods, A and B. On rod A, positive charge is spread evenly, so that there is the same amount of charge per unit length at every point. On rod B, positive charge is spread evenly over only the left half, and the same amount of negative charge is spread evenly over the right half. For each rod deduce the direction of the electric field at a point that is located directly above the midpoint of the rod.

Figure 18.28 shows four choices for the electric field lines between three negative point charges (q, q, and 2q) and one positive point charge (4q). Which of these choices is the only one of the four that could possibly show a correct representation of the field lines? (a) Figure 18.28a (b) Figure 18.28b (c) Figure 18.28c (d) Figure 18.28d Rea

Drawings A and B show two examples of electric field lines. Which (one or more) of the following statements are true, and which (one or more) are false? (a) In both A and B the electric field is the same everywhere. (b) As you move from left to right in each case, the electric field becomes stronger. (c) The electric field in A is the same everywhere, but it becomes stronger in B as you move from left to right. (d) The electric fields in both A and B could be created by negative charges located somewhere on the left and positive charges somewhere on the right. (e) Both A and B arise from a single positive point charge located somewhere on the left

A positively charged particle is moving horizontally when it enters the region between the plates of a parallel plate capacitor, as the drawing illustrates. When the particle is within the capacitor, which of the following vectors, if any, are parallel to the electric field lines inside the capacitor? (a) The particles displacement (b) Its velocity (c) Its linear momentum (d) Its acceleration

A charge q is suspended at the center of a hollow, electrically neutral, spherical, metallic conductor, as Figure 18.31 illustrates. The table shows a number of possibilities for the charges that this suspended charge induces on the interior and exterior surfaces of the conductor. Which one of the possibilities is correct?

Figures 18.34a and b show a thin spherical shell of radius R (for clarity, only half of the shell is shown). A positive charge q is spread uniformly over the shell. Find the magnitude of the electric field at any point (a) outside the shell and (b) inside the shell.

According to Equation 18.4, the electric field inside a parallel plate capacitor, and away from the edges, is constant and has a magnitude of E  /0 , where  is the charge density (the charge per unit area) on a plate. Use Gauss law to obtain this result.

A Gaussian surface contains a single charge within it, and as a result an electric flux passes through the surface. Suppose that the charge is then moved to another spot within the Gaussian surface. Does the flux through the surface change?

The drawing shows an arrangement of three charges. In parts a and b different Gaussian surfaces (both in blue) are shown. Through which surface, if either, does the greater electric flux pass?

The drawing shows three charges, labeled q1 , q2 , and q3 . A Gaussian surface (in blue) is drawn around q1 and q2 . (a) Which charges determine the electric flux through the Gaussian surface? (b) Which charges produce the electric field that exists at the point P?

The charges on three identical metal spheres are 12 C, 4.0 C, and 2.0 C. The spheres are brought together so they simultaneously touch each other. They are then separated and placed on the x and y axes, as in Figure 18.39a. What is the net force (magnitude and direction) exerted on the sphere at the origin? Treat the spheres as if they were particles. Co

Is the net charge on the system comprising the three spheres the same before and after touching?

After the spheres touch and are separated, do they have identical charges?

Do q2 and q3 exert forces of equal magnitude on q1?

Is the magnitude of the net force exerted on q1 equal to 2F, where F is the magnitude of the force that either q2 or q3 exerts on q1?

Two point charges are lying on the y axis in Figure 18.40a: q1 4.00 C and q2 4.00 C. They are equidistant from the point P, which lies on the x axis. (a) What is the net electric field at P? (b) A small object of charge q0 8.00 C and mass m 1.20 g is placed at P. When it is released, what is its acceleration? Prob

There is no charge at P in part (a). Is there an electric field at P?

The charge q1 produces an electric field at the point P. This field has what direction?

What is the direction of the electric field produced by q2 at P?

Is the magnitude of the net electric field equal to E1 E2 , where E1 and E2 are the magnitudes of the electric fields produced by q1 and q2?

An object carries a charge of 8.0 C, while another carries a charge of 2.0 C. How many electrons must be transferred from the first object to the second so that both objects have the same charge?

Each of three objects carries a charge. As the drawing shows, objects A and B attract each other, and objects C and A also attract each other. Which one of the following statements concerning objects B and C is true? (a) They attract each other. (b) They repel each other. (c) They neither attract nor repel each other. (d) This question cannot be answered without additional information.

Each of two identical objects carries a net charge. The objects are made from conducting material. One object is attracted to a positively charged ebonite rod, and the other is repelled by the rod. After the objects are touched together, it is found that they are each repelled by the rod. What can be concluded about the initial charges on the objects? (a) Initially both objects are positive, with both charges having the same magnitude. (b) Initially both objects are negative, with both charges having the same magnitude. (c) Initially one object is positive and one is negative, with the negative charge having a greater magnitude than the positive charge. (d) Initially one object is positive and one is negative, with the positive charge having a greater magnitude than the negative charge.

Only one of three balls A, B, and C carries a net charge q. The balls are made from conducting material and are identical. One of the uncharged balls can become charged by touching it to the charged ball and then separating the two. This process of touching one ball to another and then separating the two balls can be repeated over and over again, with the result that the three balls can take on a variety of charges. Which one of the following distributions of charges could not possibly be achieved in this fashion, even if the process were repeated a large number of times? (a) (c) (b) (d)

Three point charges have equal magnitudes and are located on the same line. The separation d between A and B is the same as the separation between B and C. One of the charges is positive and two are negative, as the drawing shows. Consider the net electrostatic force that each charge experiences due to the other two charges. Rank the net forces in descending order (greatest first) according to magnitude. (a) A, B, C (b) B, C, A (c) A, C, B (d) C, A, B (e) B, A, C

Three point charges have equal magnitudes and are fixed to the corners of an equilateral triangle. Two of the charges are positive and one is negative, as the drawing shows. At which one of the corners is the net force acting on the charge directed parallel to the x axis? (a) A (b) B (c) C

A positive point charge q1 creates an electric field of magnitude E1 at a spot located at a distance r1 from the charge. The charge is replaced by another positive point charge q2, which creates a field of magnitude at a distance of How is q2 related to q1? (a) (b) (c) (d) (e)

The drawing shows a positive and a negative point charge. The negative charge has the greater magnitude. Where on the line that passes through the charges is the one spot where the total electric field is zero? (a) To the right of the negative charge (b) To the left of the positive charge (c) Between the charges, to the left of the midpoint (d) Between the charges, to the right of the midpoint

The drawing shows some electric field lines. For the points indicated, rank the magnitudes of the electric field in descending order (largest first). (a) B, C, A (b) B, A, C (c) A, B, C (d) A, C, B (e) C, A, B

The drawings show (in cross section) two solid spheres and two spherical shells. Each object is made from copper and has a net charge, as the plus and minus signs indicate. Which drawing correctly shows where the charges reside when they are in equilibrium? (a) A (b) B (c) C (d) D

A cubical Gaussian surface surrounds two charges, q1  6.0  1012 C and q2  2.0  1012 C, as the drawing shows. What is the electric flux passing through the surface? +

Iron atoms have been detected in the suns outer atmosphere, some with many of their electrons stripped away. What is the net electric charge (in coulombs) of an iron atom with 26 protons and 7 electrons? Be sure to include the algebraic sign ( or ) in your answer.

An object has a charge of 2.0 C. How many electrons must be removed so that the charge becomes 3.0 C? 3.

Four identical metallic objects carry the following charges: 1.6, 6.2, 4.8, and 9.4 C. The objects are brought simultaneously into contact, so that each touches the others. Then they are separated. (a) What is the final charge on each object? (b) How many electrons (or protons) make up the final charge on each object? 4. F

Four identical metal spheres have charges of qA  8.0 C, qB  2.0 C, qC  5.0 C, and q D  12.0 C. (a) Two of the spheres are brought together so they touch, and then they are separated. Which spheres are they, if the final charge on each one is 5.0 C? (b) In a similar manner, which three spheres are brought together and then separated, if the final charge on each of the three is 3.0 C? (c) The final charge on each of the three separated spheres in part (b) is 3.0 C. How many electrons would have to be added to one of these spheres to make it electrically neutral? 5. ssm Consid

Consider three identical metal spheres, A, B, and C. Sphere A carries a charge of 5q. Sphere B carries a charge of q. Sphere C carries no net charge. Spheres A and B are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Last, sphere C is touched to sphere B and separated from it. (a) How much charge ends up on sphere C? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

A plate carries a charge of 3.0 C, while a rod carries a charge of 2.0 C. How many electrons must be transferred from the plate to the rod, so that both objects have the same charge? * 7

Water has a mass per mole of 18.0 g/mol, and each water molecule (H2O) has 10 electrons. (a) How many electrons are there in one liter (1.00  103 m3) of water? (b) What is the net charge of all these electrons?

In a vacuum, two particles have charges of q1 and q2 , where q13.5 C. They are separated by a distance of 0.26 m, and particle 1 experiences an attractive force of 3.4 N. What is q2 (magnitude and sign)? 9

Two spherical objects are separated by a distance that is 1.80  103 m. The objects are initially electrically neutral and are very small compared to the distance between them. Each object acquires the same negative charge due to the addition of electrons. As a result, each object experiences an electrostatic force that has a magnitude of 4.55 1021 N. How many electrons did it take to produce the charge on one of the objects? 1

Two tiny conducting spheres are identical and carry charges of 20.0 C and 50.0 C. They are separated by a distance of 2.50 cm. (a) What is the magnitude of the force that each sphere experiences, and is the force attractive or repulsive? (b) The spheres are brought into contact and then separated to a distance of 2.50 cm. Determine the magnitude of the force that each sphere now experiences, and state whether the force is attractive or repulsive

Two very small spheres are initially neutral and separated by a distance of 0.50 m. Suppose that 3.0 10 13 electrons are removed from one sphere and placed on the other. (a) What is the magnitude of the electrostatic force that acts on each sphere? (b) Is the force attractive or repulsive? Why?

Two charges attract each other with a force of 1.5 N. What will be the force if the distance between them is reduced to one-ninth of its original value?

Two point charges are fixed on the y axis: a negative point charge q1  25 C at y1  0.22 m and a positive point charge q2 at y2  0.34 m. A third point charge q  8.4 C is fixed at the origin. The net electrostatic force exerted on the charge q by the other two charges has a magnitude of 27 N and points in the y direction. Determine the magnitude of q2.

The drawings show three charges that have the same magnitude but may have different signs. In all cases the distance d between the charges is the same. The magnitude of the charges is q  8.6 C, and the distance between them is d  3.8 mm. Determine the magnitude of the net force on charge 2 for each of the three drawings. 1

Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is 2.0 106 kg. The gravitational force that each sphere exerts on the other is balanced by the electric force. (a) What algebraic signs can the charges have? (b) Determine the charge magnitude. 1

A charge q is located at the origin, while an identical charge is located on the x axis at x  0.50 m. A third charge of 2q is located on the x axis at such a place that the net electrostatic force on the charge at the origin doubles, its direction remaining unchanged. Where should the third charge be located?

Two particles, with identical positive charges and a separation of 2.60 102 m, are released from rest. Immediately after the release, particle 1 has an acceleration 1 whose magnitude is 4.60 103 m/s2 , while particle 2 has an acceleration 2 whose magnitude is 8.50 103 m/s2 . Particle 1 has a mass of 6.00 106 kg. Find (a) the charge on each particle and (b) the mass of particle 2. 18.

A charge of 3.00 C is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass, which has a radius of 0.100 m. The charges on the circle are 4.00 C at the position due north and 5.00 C at the position due east. What are the magnitude and direction of the net electrostatic force acting on the charge at the center? Specify the direction relative to due east. *

Multiple-Concept Example 3 provides some pertinent background for this problem. Suppose a single electron orbits about a nucleus containing two protons (2e), as would be the case for a helium atom from which one of the two naturally occurring electrons is removed. The radius of the orbit is 2.65 1011 m. Determine the magnitude of the electrons centripetal acceleration. *

The drawing shows an equilateral triangle, each side of which has a length of 2.00 cm. Point charges are fixed to each corner, as shown. The 4.00 C charge experiences a net force due to the charges qA and q B. This net force points vertically downward and has a magnitude of 405 N. Determine the magnitudes and algebraic signs of the charges qA and q B.

The drawing shows three point charges fixed in place. The charge at the coordinate origin has a value of q1  8.00 C; the other two charges have identical magnitudes, but opposite signs: q2  5.00 C and q3  5.00 C. (a) Determine the net force (magnitude and direction) exerted on q1 by the other two charges. (b) If q1 had a mass of 1.50 g and it were free to move, what would be its acceleration?

An electrically neutral model airplane is flying in a horizontal circle on a 3.0-m guideline, which is nearly parallel to the ground. The line breaks when the kinetic energy of the plane is 50.0 J. Reconsider the same situation, except that now there is a point charge of q on the plane and a point charge of q at the other end of the guideline. In this case, the line breaks when the kinetic energy of the plane is 51.8 J. Find the magnitude of the charges.

Multiple-Concept Example 3 illustrates several of the concepts that come into play in this problem. A single electron orbits a lithium nucleus that contains three protons (3e). The radius of the orbit is 1.76 1011 m. Determine the kinetic energy of the electron. *

An unstrained horizontal spring has a length of 0.32 m and a spring constant of 220 N/m. Two small charged objects are attached to this spring, one at each end. The charges on the objects have equal magnitudes. Because of these charges, the spring stretches by 0.020 m relative to its unstrained length. Determine (a) the possible algebraic signs and (b) the magnitude of the charges.

In the rectangle in the drawing, a charge is to be placed at the empty corner to make the net force on the charge at corner A point along the vertical direction. What charge (magnitude and algebraic sign) must be placed at the empty corner?

There are four charges, each with a magnitude of 2.0 C. Two are positive and two are negative. The charges are fixed to the corners of a 0.30-m square, one to a corner, in such a way that the net force on any charge is directed toward the center of the square. Find the magnitude of the net electrostatic force experienced by any charge.

A small spherical insulator of mass 8.00 102 kg and charge 0.600 C is hung by a thread of negligible mass. A charge of 0.900 C is held 0.150 m away from the sphere and directly to the right of it, so the thread makes an angle  with the vertical (see the drawing). Find (a) the angle  and (b) the tension in the thread. A

Two objects carry initial charges that are q1 and q2, respectively, where q2  q1. They are located 0.200 m apart and behave like point charges. They attract each other with a force that has a magnitude of 1.20 N. The objects are then brought into contact, so the net charge is shared equally, and then they are returned to their initial positions. Now it is found that the objects repel one another with a force whose magnitude is equal to the magnitude of the initial attractive force. What are the magnitudes of the initial charges on the objects? Se

At a distance r1 from a point charge, the magnitude of the electric field created by the charge is 248 N/C. At a distance r 2 from the charge, the field has a magnitude of 132 N/C. Find the ratio r 2/r1.

Suppose you want to determine the electric field in a certain region of space. You have a small object of known charge and an instrument that measures the magnitude and direction of the force exerted on the object by the electric field. (a) The object has a charge of 20.0 C and the instrument indicates that the electric force exerted on it is 40.0 N, due east. What are the magnitude and direction of the electric field? (b) What are the magnitude and direction of the electric field if the object has a charge of 10.0 C and the instrument indicates that the force is 20.0 N, due west?

An electric field of 260 000 N/C points due west at a certain spot. What are the magnitude and direction of the force that acts on a charge of 7.0 C at this spot?

Review the important features of electric field lines discussed in Conceptual Example 13. Three point charges (q, 2q, and 3q) are at the corners of an equilateral triangle. Sketch in six electric field lines between the three charges.

Four point charges have the same magnitude of 2.4 1012 C and are fixed to the corners of a square that is 4.0 cm on a side. Three of the charges are positive and one is negative. Determine the magnitude of the net electric field that exists at the center of the square.

The drawing shows two situations in which charges are placed on the x and y axes. They are all located at the same distance of 6.1 cm from the origin O. For each of the situations in the drawing, determine the magnitude of the net electric field at the or

A uniform electric field exists everywhere in the x, y plane. This electric field has a magnitude of 4500 N/C and is directed in the positive x direction. A point charge 8.0 109 C is placed at the origin. Determine the magnitude of the net electric field at (a) x  0.15 m, (b) x  0.15 m, and (c) y  0.15 m. +2

The membrane surrounding a living cell consists of an inner and an outer wall that are separated by a small space. Assume that the membrane acts like a parallel plate capacitor in which the effective charge density on the inner and outer walls has a magnitude of 7.1 106 C/m2 . (a) What is the magnitude of the electric field within the cell membrane? (b) Find the magnitude of the electric force that would be exerted on a potassium ion (K; charge  e) placed inside the membrane.

A 3.0 C point charge is placed in an external uniform electric field that has a magnitude of 1.6 104 N/C. At what distance from the charge is the net electric field zero?

A tiny ball (mass  0.012 kg) carries a charge of 18 C. What electric field (magnitude and direction) is needed to cause the ball to float above the ground?

A proton and an electron are moving due east in a constant electric field that also points due east. The electric field has a magnitude of 8.0 104 N/C. Determine the magnitude of the acceleration of the proton and the electron.

Review Conceptual Example 12 before attempting to work this problem. The magnitude of each of the charges in Figure 18.21 is 8.60 1012 C. The lengths of the sides of the rectangles are 3.00 cm and 5.00 cm. Find the magnitude of the electric field at the center of the rectangle in Figures 18.21a and b.

Two charges are placed between the plates of a parallel plate capacitor. One charge is q1 and the other is q2  5.00 C. The charge per unit area on each of the plates has a magnitude of   1.30 104 C/m2 . The magnitude of the force on q1 due to q2 equals the magnitude of the force on q1 due to the electric field of the parallel plate capacitor. What is the distance r between the two charges? *

A small object has a mass of 3.0 103 kg and a charge of 34 C. It is placed at a certain spot where there is an electric field. When released, the object experiences an acceleration of 2.5 103 m/s2 in the direction of the x axis. Determine the magnitude and direction of the electric field. *

A spring with an unstrained length of 0.074 m and a spring constant of 2.4 N/m hangs vertically downward from the ceiling. A uniform electric field directed vertically upward fills the region containing the spring. A sphere with a mass of 5.1 103 kg and a net charge of 6.6 C is attached to the lower end of the spring. The spring is released slowly, until it reaches equilibrium. The equilibrium length of the spring is 0.059 m. What is the magnitude of the external electric field? *

Two point charges are located along the x axis: q1  6.0 C at x1  4.0 cm, and q2  6.0 C at x 2  4.0 cm. Two other charges are located on the y axis: q3  3.0 C at y3  5.0 cm, and q4  8.0 C at y4  7.0 cm. Find the net electric field (magnitude and direction) at the origin. *

The total electric field consists of the vector sum of two parts. One part has a magnitude of E1  1200 N/C and points at an angle 1  35 above the x axis. The other part has a magnitude of E2  1700 N/C and points at an angle 2  55 above the x axis. Find the magnitude and direction of the total field. Specify the directional angle relative to the x axis.

In Multiple-Concept Example 9 you can see the concepts that are important in this problem. A particle of charge 12 C and mass 3.8 105 kg is released from rest in a region where there is a constant electric field of 480 N/C. What is the displacement of the particle after a time of 1.6 102 s? * 4

The drawing shows a positive point charge q1, a second point charge q2 that may be positive or negative, and a spot labeled P, all on the same straight line. The distance d between the two charges is the same as the distance between q1 and the spot P. With q2 present, the magnitude of the net electric field at P is twice what it is when q1 is present alone. Given that q1  0.50 C, determine q2 when it is (a) positive and (b) negative. *

Multiple-Concept Example 9 illustrates the concepts in this problem. An electron is released from rest at the negative plate of a parallel plate capacitor. The charge per unit area on each plate is  1.8 107 C/m2 , and the plate separation is 1.5 102 m. How fast is the electron moving just before it reaches the positive plate? **

Two particles are in a uniform electric field that points in the x direction and has a magnitude of 2500 N/C. The mass and charge of particle 1 are m1  1.4 105 kg and q1  7.0 C, while the corresponding values for particle 2 are m2  2.6 105 kg and q2  18 C. Initially the particles are at rest. The particles are both located on the same electric field line but are separated from each other by a distance d. Particle 1 is located to the left of particle 2. When released, they accelerate but always remain at this same distance from each other. Find d. ** 5

Two point charges of the same magnitude but opposite signs are fixed to either end of the base of an isosceles triangle, as the drawing shows. The electric field at the midpoint M between the charges has a magnitude EM. The field directly above the midpoint at point P has a magnitude EP. The ratio of these two field magnitudes is EM/EP  9.0. Find the angle  in the drawing.

The drawing shows an electron entering the lower left side of a parallel plate capacitor and exiting at the upper right side. The initial speed of the electron is 7.00 106 m/s. The capacitor is 2.00 cm long, and its plates are separated by 0.150 cm. Assume that the electric field between the plates is uniform everywhere and find its magnitude.

A small plastic ball with a mass of 6.50 103 kg and with a charge of 0.150 C is suspended from an insulating thread and hangs between the plates of a capacitor (see the drawing). The ball is in equilibrium, with the thread making an angle of 30.0 with respect to the vertical. The area of each plate is 0.0150 m2 . What is the magnitude of the charge on each plate? d

A spherical surface completely surrounds a collection of charges. Find the electric flux through the surface if the collection consists of (a) a single 3.5 106 C charge, (b) a single 2.3 106 C charge, and (c) both of the charges in (a) and (b). 55

The drawing shows an edge-on view of two planar surfaces that intersect and are mutually perpendicular. Surface 1 has an area of 1.7 m2 , while surface 2 has an area of 3.2 m2 . The electric field in the drawing is uniform and has a magnitude of 250 N/C. Find the magnitude of the electric flux through (a) surface 1 and (b) surface 2.

A surface completely surrounds a 2.0 106 C charge. Find the electric flux through this surface when the surface is (a) a sphere with a radius of 0.50 m, (b) a sphere with a radius of 0.25 m, and (c) a cube with edges that are 0.25 m long. 5

A circular surface with a radius of 0.057 m is exposed to a uniform external electric field of magnitude 1.44 104 N/C. The magnitude of the electric flux through the surface is 78 N m2 /C. What is the angle (less than 90) between the direction of the electric field and the normal to the surface?

A charge Q is located inside a rectangular box. The electric flux through each of the six surfaces of the box is: 1  1500 N m2 /C, 2  2200 N m2 /C, 3  4600 N m2 /C, 4  1800 N m2 /C, 5  3500 N m2 /C, and 6  5400 N m2 /C. What is Q? *

A solid nonconducting sphere has a positive charge q spread uniformly throughout its volume. The charge density or charge per unit volume, therefore, is Use Gauss law to show that the electric field at a point within the sphere at a radius r has a magnitude of qr 40R3

Two spherical shells have a common center. A 1.6106 C charge is spread uniformly over the inner shell, which has a radius of 0.050 m. A 5.1 106 C charge is spread uniformly over the outer shell, which has a radius of 0.15 m. Find the magnitude and direction of the electric field at a distance (measured from the common center) of (a) 0.20 m, (b) 0.10 m, and (c) 0.025 m. *

A cube is located with one corner situated at the origin of an x, y, z coordinate system. One of the cubes faces lies in the x, y plane, another in the y, z plane, and another in the x, z plane. In other words, the cube is in the first octant of the coordinate system. The edges of the cube are 0.20 m long. A uniform electric field is parallel to the x, y plane and points in the direction of the y axis. The magnitude of the field is 1500 N/C. (a) Using the outward normal for each face of the cube, find the electric flux through each of the six faces. (b) Add the six values obtained in part (a) to show that the electric flux through the cubical surface is zero, as Gauss law predicts, since there is no net charge within the cube.

A long, thin, straight wire of length L has a positive charge Q distributed uniformly along it. Use Gauss law to show that the electric field created by this wire at a radial distance r has a magnitude of E  /(20 r), where   Q/L. (

Review Conceptual Example 13 as an aid in working this problem. Charges of 4q are fixed to diagonally opposite corners of a square. A charge of 5q is fixed to one of the remaining corners, and a charge of 3q is fixed to the last corner. Assuming that ten electric field lines emerge from the 5q charge, sketch the field lines in the vicinity of the four charges.

The masses of the earth and moon are 5.98 1024 and 7.35 1022 kg, respectively. Identical amounts of charge are placed on each body, such that the net force (gravitational plus electrical) on each is zero. What is the magnitude of the charge placed on each body? 6

Conceptual Example 14 deals with the hollow spherical conductor in Figure 18.31. The conductor is initially electrically neutral, and then a charge q is placed at the center of the hollow space. Suppose the conductor initially has a net charge of 2q instead of being neutral. What is the total charge on the interior and on the exterior surface when the q charge is placed at the center?

A small drop of water is suspended motionless in air by a uniform electric field that is directed upward and has a magnitude of 8480 N/C. The mass of the water drop is 3.50 109 kg. (a) Is the excess charge on the water drop positive or negative? Why? (b) How many excess electrons or protons reside on the drop? 6

Two charges are placed on the x axis. One of the charges (q1  8.5 C) is at x1  3.0 cm and the other (q2  21 C) is at x1  9.0 cm. Find the net electric field (magnitude and direction) at (a) x  0 cm and (b) x  6.0 cm.

When point charges q1  8.4 C and q2  5.6 C are brought near each other, each experiences a repulsive force of magnitude 0.66 N. Determine the distance between the charges.

Two small objects, A and B, are fixed in place and separated by 3.00 cm in a vacuum. Object A has a charge of 2.00 C, and object B has a charge of 2.00 C. How many electrons must be removed from object A and put onto object B to make the electrostatic force that acts on each object an attractive force whose magnitude is 68.0 N?

The drawing shows two positive charges q1 and q2 fixed to a circle. At the center of the circle they produce a net electric field that is directed upward along the vertical axis. Determine the ratio q2/q1 of the charge magnitudes. * 7

At three corners of a rectangle (length  2d, height  d ), the following charges are located: q1 (upper left corner), q2 (lower right corner), and q (lower left corner). The net electric field at the (empty) upper right corner is zero. Find the magnitudes of q1 and q2. Express your answers in terms of q.

Three point charges have equal magnitudes, two being positive and one negative. These charges are fixed to the corners of an equilateral triangle, as the drawing shows. The magnitude of each of the charges is 5.0 C, and the lengths of the sides of the triangle are 3.0 cm. Calculate the magnitude of the net force that each charge experiences.

Multiple-Concept Example 9 illustrates the concepts employed in this problem. A small object, which has a charge q  7.5 C and mass m  9.0 105 kg, is placed in a constant electric field. Starting from rest, the object accelerates to a speed of 2.0 103 m/s in a time of 0.96 s. Determine the magnitude of the electric field. *

Four point charges have equal magnitudes. Three are positive, and one is negative, as the drawing shows. They are fixed in place on the same straight line, and adjacent charges are equally separated by a distance d. Consider the net electrostatic force acting on each charge. Calculate the ratio of the largest to the smallest net force.

Two parallel plate capacitors have circular plates. The magnitude of the charge on these plates is the same. However, the electric field between the plates of the first capacitor is 2.2 105 N/C, whereas the field within the second capacitor is 3.8 105 N/C. Determine the ratio r2 /r1 of the plate radius for the second capacitor to the plate radius for the first capacitor.

Two identical small insulating balls are suspended by separate 0.25-m threads that are attached to a common point on the ceiling. Each ball has a mass of 8.0 104 kg. Initially the balls are uncharged and hang straight down. They are then given identical positive charges and, as a result, spread apart with an angle of 36 between the threads. Determine (a) the charge on each ball and (b) the tension in the threads. A

A long, thin rod (length  4.0 m) lies along the x axis, with its midpoint at the origin. In a vacuum, a 8.0 C point charge is fixed to one end of the rod, and a 8.0 C point charge is fixed to the other end. Everywhere in the x, y plane there is a constant external electric field (magnitude  5.0 103 N/C) that is perpendicular to the rod. With respect to the z axis, find the magnitude of the net torque applied to the rod. 3