Two tiny conducting spheres are identical and carry

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 (1 or 2) in your answer.

An object has a charge of 22.0 mC. How many electrons must be removed so that the charge becomes 13.0 mC?

Four identical metallic objects carry the following charges: 11.6, 16.2, 24.8, and 29.4 mC. 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?

Four identical metal spheres have charges of qA 5 28.0 mC, qB 5 22.0 mC, qC 5 15.0 mC, and qD 5 112.0 mC. (a) Two of the spheres are brought together so they touch, and then they are separated. Which spheres are they, if the fi nal charge on each one is 15.0 mC? (b) In a similar manner, which three spheres are brought together and then separated, if the fi nal charge on each of the three is 13.0 mC? (c) The fi nal charge on each of the three separated spheres in part (b) is 13.0 mC. How many electrons would have to be added to one of these spheres to make it electrically neutral?

Consider three identical metal spheres, A, B, and C. Sphere A carries a charge of 15q. Sphere B carries a charge of 2q. 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 23.0 mC, while a rod carries a charge of 12.0 mC. How many electrons must be transferred from the plate to the rod, so that both objects have the same charge?

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 3 1023 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 q1 5 13.5 mC. 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)?

Two spherical objects are separated by a distance that is 1.80 3 1023 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 3 10221 N. How many electrons did it take to produce the charge on one of the objects?

Two tiny conducting spheres are identical and carry charges of 220.0 mC and 150.0 mC. 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 3 1013 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 fi xed on the y axis: a negative point charge q1 5 225 mC at y1 5 10.22 m and a positive point charge q2 at y2 5 10.34 m. A third point charge q5 18.4 mC is fi xed 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 1y direction. Determine the magnitude of q2.

The drawings show three charges that have the same magnitude but may have diff erent signs. In all cases the distance d between the charges is the same. The magnitude of the charges is uqu 5 8.6 mC, and the distance between them is d 5 3.8 mm. Determine the magnitude of the net force on charge 2 for each of the three drawings.

Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is 2.0 3 1026 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.

A charge 1q is located at the origin, while an identical charge is located on the x axis at x 5 10.50 m. A third charge of 12q 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 3 1022 m, are released from rest. Immediately after the release, particle 1 has an acceleration a B1 whose magnitude is 4.60 3 103 m/s2 , while particle 2 has an acceleration a B2 whose magnitude is 8.50 3 103 m/s2 . Particle 1 has a mass of 6.00 3 1026 kg. Find (a) the charge on each particle and (b) the mass of particle 2.

A charge of 23.00 mC is fi xed at the center of a compass. Two additional charges are fi xed on the circle of the compass, which has a radius of 0.100 m. The charges on the circle are 24.00 mC at the position due north and 15.00 mC 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 (12e), 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 3 10211 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 fi xed to each corner, as shown. The 4.00 mC charge experiences a net force due to the charges qA and qB. This net force points vertically downward and has a magnitude of 405 N. Determine the magnitudes and algebraic signs of the charges qA and qB.

The drawing shows three point charges fi xed in place. The charge at the coordinate origin has a value of q1 5 18.00 mC; the other two charges have identical magnitudes, but opposite signs: q2 5 25.00 mC and q3 5 15.00 mC. (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 fl ying 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 1q on the plane and a point charge of 2q 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 (13e). The radius of the orbit is 1.76 3 10211 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 mC. Two are positive and two are negative. The charges are fi xed 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 3 1022 kg and charge 10.600 mC is hung by a thread of negligible mass. A charge of 20.900 mC is held 0.150 m away from the sphere and directly to the right of it, so the thread makes an angle u with the vertical (see the drawing). Find (a) the angle u and (b) the tension in the thread.

Two objects carry initial charges that are q1 and q2, respectively, where uq2u . uq1u. 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?

At a distance r1 from a point charge, the magnitude of the electric fi eld created by the charge is 248 N/C. At a distance r2 from the charge, the fi eld has a magnitude of 132 N/C. Find the ratio r2/r1.

Suppose you want to determine the electric fi eld 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 fi eld. (a) The object has a charge of 120.0 mC and the instrument indicates that the electric force exerted on it is 40.0 mN, due east. What are the magnitude and direction of the electric fi eld? (b) What are the magnitude and direction of the electric fi eld if the object has a charge of 210.0 mC and the instrument indicates that the force is 20.0 mN, due west?

An electric fi eld 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 27.0 mC at this spot?

Review the important features of electric fi eld lines discussed in Conceptual Example 12. Three point charges (1q, 12q, and 23q) are at the corners of an equilateral triangle. Sketch in six electric fi eld lines between the three charges.

Four point charges have the same magnitude of 2.4 3 10212 C and are fi xed 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 fi eld 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 fi eld at the origin.

A uniform electric fi eld exists everywhere in the x, y plane. This electric fi eld has a magnitude of 4500 N/C and is directed in the positive x direction. A point charge 28.0 3 1029 C is placed at the origin. Determine the magnitude of the net electric fi eld at (a) x 5 20.15 m, (b) x 5 10.15 m, and (c) y 5 10.15 m.

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 3 1026 C/m2 . (a) What is the magnitude of the electric fi eld within the cell membrane? (b) Find the magnitude of the electric force that would be exerted on a potassium ion (K1; charge 5 1e) placed inside the membrane.

A long, thin rod (length 5 4.0 m) lies along the x axis, with its midpoint at the origin. In a vacuum, a 18.0 mC point charge is fi xed to one end of the rod, and a 28.0 mC point charge is fi xed to the other end. Everywhere in the x, y plane there is a constant external electric fi eld (magnitude 5 5.0 3 103 N/C) that is perpendicular to the rod. With respect to the z axis, fi nd the magnitude of the net torque applied to the rod.

A 3.0 mC point charge is placed in an external uniform electric fi eld that has a magnitude of 1.6 3 104 N/C. At what distance from the charge is the net electric fi eld zero?

A tiny ball (mass 5 0.012 kg) carries a charge of 218 mC. What electric fi eld (magnitude and direction) is needed to cause the ball to fl oat above the ground?

A proton and an electron are moving due east in a constant electric fi eld that also points due east. The electric fi eld has a magnitude of 8.0 3 104 N/C. Determine the magnitude of the acceleration of the proton and the electron.

Review Conceptual Example 11 before attempting to work this problem. The magnitude of each of the charges in Figure 18.20 is 8.60 3 10212 C. The lengths of the sides of the rectangles are 3.00 cm and 5.00 cm. Find the magnitude of the electric fi eld at the center of the rectangle in Figures 18.20a and b.

Two charges are placed between the plates of a parallel plate capacitor. One charge is 1q1 and the other is q2 5 15.00 mC. The charge per unit area on each of the plates has a magnitude of s 5 1.30 3 1024 C/m2 . The magnitude of the force on q1 due to q2 equals the magnitude of the force on q1 due to the electric fi eld of the parallel plate capacitor. What is the distance r between the two charges?

A small object has a mass of 3.0 3 1023 kg and a charge of 234 mC. It is placed at a certain spot where there is an electric fi eld. When released, the object experiences an acceleration of 2.5 3 103 m/s2 in the direction of the 1x axis. Determine the magnitude and direction of the electric fi eld.

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 fi eld directed vertically upward fi lls the region containing the spring. A sphere with a mass of 5.1 3 1023 kg and a net charge of 16.6 mC 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 fi eld?

Two point charges are located along the x axis: q1 5 16.0 mC at x1 5 14.0 cm, and q2 5 16.0 mC at x2 5 24.0 cm. Two other charges are located on the y axis: q3 5 13.0 mC at y3 5 15.0 cm, and q4 5 28.0 mC at y4 5 17.0 cm. Find the net electric fi eld (magnitude and direction) at the origin

The total electric fi eld E B consists of the vector sum of two parts. One part has a magnitude of E1 5 1200 N/C and points at an angle u1 5 358 above the 1x axis. The other part has a magnitude of E2 5 1700 N/C and points at an angle u2 5 558 above the 1x axis. Find the magnitude and direction of the total fi eld. Specify the directional angle relative to the 1x axis.

A particle of charge 112 mC and mass 3.8 3 1025 kg is released from rest in a region where there is a constant electric fi eld of 1480 N/C. What is the displacement of the particle after a time of 1.6 3 1022 s?

The drawing shows a positive point charge 1q1, 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 fi eld at P is twice what it is when q1 is present alone. Given that q1 5 10.50 mC, determine q2 when it is (a) positive and (b) negative.

An electron is released from rest at the negative plate of a parallel plate capacitor. The charge per unit area on each plate is s 5 1.8 3 1027 C/m2 , and the plate separation is 1.5 3 1022 m. How fast is the electron moving just before it reaches the positive plate?

Two particles are in a uniform electric fi eld that points in the 1x direction and has a magnitude of 2500 N/C. The mass and charge of particle 1 are m1 5 1.4 3 1025 kg and q1 5 27.0 mC, while the corresponding values for particle 2 are m2 5 2.6 3 1025 kg and q2 5 118 mC. Initially the particles are at rest. The particles are both located on the same electric fi eld 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.

Two point charges of the same magnitude but opposite signs are fi xed to either end of the base of an isosceles triangle, as the drawing shows. The electric fi eld at the midpoint M between the charges has a magnitude EM. The fi eld directly above the midpoint at point P has a magnitude EP. The ratio of these two fi eld magnitudes is EM/EP 5 9.0. Find the angle a 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 3 106 m/s. The capacitor is 2.00 cm long, and * * * * * * ** ** ** its plates are separated by 0.150 cm. Assume that the electric fi eld between the plates is uniform everywhere and fi nd its magnitude.

A small plastic ball with a mass of 6.50 3 1023 kg and with a charge of 10.150 mC 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 30.08 with respect to the vertical. The area of each plate is 0.0150 m2 . What is the magnitude of the charge on each plate?

A spherical surface completely surrounds a collection of charges. Find the electric fl ux through the surface if the collection consists of (a) a single 13.5 3 1026 C charge, (b) a single 22.3 3 1026 C charge, and (c) both of the charges in (a) and (b).

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 fi eld E B in the drawing is uniform and has a magnitude of 250 N/C. Find the magnitude of the electric fl ux through (a) surface 1 and (b) surface 2.

A surface completely surrounds a 12.0 3 1026 C charge. Find the electric fl ux 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.

A circular surface with a radius of 0.057 m is exposed to a uniform external electric fi eld of magnitude 1.44 3 104 N/C. The magnitude of the electric fl ux through the surface is 78 N ? m2 /C. What is the angle (less than 908) between the direction of the electric fi eld and the normal to the surface?

A charge Q is located inside a rectangular box. The electric fl ux through each of the six surfaces of the box is: F1 5 11500 N ? m2 /C, F2 5 12200 N ? m2 /C, F3 5 14600 N ? m2 /C, F4 5 21800 N ? m2 /C, F5 5 23500 N ? m2 /C, and F6 5 25400 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 q 4 3 pR3 . Use Gauss law to show that the electric fi eld at a point within the sphere at a radius r has a magnitude of qr 4pP0R3 . (Hint: For a Gaussian surface, use a sphere of radius r centered within the solid sphere of radius. Note that the net charge within any volume is the charge density times the volume.)

Two spherical shells have a common center. A 21.6 3 1026 C charge is spread uniformly over the inner shell, which has a radius of 0.050 m. A 15.1 3 1026 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 fi eld 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 fi rst octant of the coordinate system. The edges of the cube are 0.20 m long. A uniform electric fi eld is parallel to the x, y plane and points in the direction of the 1y axis. The magnitude of the fi eld is 1500 N/C. (a) Using the outward normal for each face of the cube, fi nd the electric fl ux through each of the six faces. (b) Add the six values obtained in part (a) to show that the electric fl ux 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 fi eld created by this wire at a radial distance r has a magnitude of E 5 l/(2pP0r), where l 5 Q/L. (Hint: For a Gaussian surface, use a cylinder aligned with its axis along the wire and note that the cylinder has a fl at surface at either end, as well as a curved surface.)

Review Conceptual Example 12 as an aid in working this problem. Charges of 24q are fi xed to diagonally opposite corners of a square. A charge of 15q is fi xed to one of the remaining corners, and a charge of 13q is fi xed to the last corner. Assuming that ten electric fi eld lines emerge from the 15q charge, sketch the fi eld lines in the vicinity of the four charges.

The masses of the earth and moon are 5.98 3 1024 and 7.35 3 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?

Conceptual Example 13 deals with the hollow spherical conductor in Figure 18.30. The conductor is initially electrically neutral, and then a charge 1q is placed at the center of the hollow space. Suppose the conductor initially has a net charge of 12q instead of being neutral. What is the total charge on the interior and on the exterior surface when the 1q charge is placed at the center?

A small drop of water is suspended motionless in air by a uniform electric fi eld that is directed upward and has a magnitude of 8480 N/C. The mass of the water drop is 3.50 3 1029 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?

Two charges are placed on the x axis. One of the charges (q1 5 18.5 mC) is at x1 5 13.0 cm and the other (q2 5 221 mC) is at x1 5 19.0 cm. Find the net electric field (magnitude and direction) at (a) x 5 0 cm and (b) x 5 16.0 cm

When point charges q1 5 18.4 mC and q2 5 15.6 mC 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 fi xed in place and separated by 3.00 cm in a vacuum. Object A has a charge of 12.00 mC, and object B has a charge of 22.00 mC. 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 fi xed to a circle. At the center of the circle they produce a net electric fi eld that is directed upward along the vertical axis. Determine the ratio uq2u/uq1u of the charge magnitudes.

At three corners of a rectangle (length 5 2d, height 5 d), the following charges are located: 1q1 (upper left corner), 1q2 (lower right corner), and 2q (lower left corner). The net electric fi eld 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 fi xed to the corners of an equilateral triangle, as the drawing shows. The magnitude of each of the charges is 5.0 mC, and the lengths of the sides of the triangle are 3.0 cm. Calculate the magnitude of the net force that each charge experiences.

A small object, which has a charge q 5 7.5 mC and mass m 5 9.0 3 1025 kg, is placed in a constant electric fi eld. Starting from rest, the object accelerates to a speed of 2.0 3 103 m/s in a time of 0.96 s. Determine the magnitude of the electric fi eld.

Four point charges have equal magnitudes. Three are positive, and one is negative, as the drawing shows. They are fi xed 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 fi eld between the plates of the fi rst capacitor is 2.2 3 105 N/C, whereas the fi eld within the second capacitor is 3.8 3 105 N/C. Determine the ratio r2/r1 of the plate radius for the second capacitor to the plate radius for the fi rst 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 3 1024 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 368 between the threads. Determine (a) the charge on each ball and (b) the tension in the threads.

In a vacuum, a proton (charge 51e, mass 5 1.67 3 10227 kg) is moving parallel to a uniform electric fi eld that is directed along the 1x axis (see the fi gure). The proton starts with a velocity of 12.5 3 104 m/s and accelerates in the same direction as the electric fi eld, which has a value of 12.3 3 103 N/C. Find the velocity of the proton when its displacement is 12.0 mm from the starting point.

The charges on three identical metal spheres are 212 mC, 14.0 mC, and 12.0 mC. The spheres are brought together so they simultaneously touch each other. They are then separated and placed on the x and y axes, as shown in the fi gure. Treat the spheres as if they were particles. Concepts: (i) Is the net charge on the system comprising the three spheres the same before and after touching? Why? (ii) After the spheres touch and are separated, how is the charge distributed, and what is the value of the charge on each sphere? (iii) Do q2 and q3 exert forces of equal magnitude * on q1? (iv) 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? Calculations: What is the net force (magnitude and direction) exerted on the sphere at the origin?

Two point charges are lying on the y axis as in the fi gure: q1 5 24.00 mC and q2 5 14.00 mC. They are equidistant from the point P, which lies on the x axis. Concepts: (i) There is no charge at P in part a of the fi gure. Is there an electric fi eld at P? (ii) What is the direction of the electric fi eld at point P due to charge q2? (iii) Is the magnitude of the electric fi eld at P equal to E1 1 E2, where E1 and E2 are the magnitudes of the electric fi elds produced by q1 and q2? Explain. Calculations: (a) What is the net electric fi eld at P? (b) A small object of charge q0 5 18.00 mC and mass m 5 1.2 g is placed at P. When it is released, what is its acceleration?