Note: All charges are point charges, unless specified otherwise.
Section 18.1 The Origin of Electricity, Section 18.2 Charged Objects and the Electric Force, Section 18.3 Conductors and Insulators, Section 18.4 Charging by Contact and by Induction
How many electrons must be removed from an electrically neutral silver dollar to give it a charge of +2.4 mC?
  
A plate carries a charge of –3.0 mC, while a rod carries a charge of +2.0 mC. How many electrons must be transferred from the plate to the rod, so that both objects have the same charge?
  
Object A is metallic and electrically neutral. It is charged by induction so that it acquires a charge of –3.0×10^{–6} C. Object B is identical to object A and is also electrically neutral. It is charged by induction so that it acquires a charge of +3.0×10^{–6} C. Find the difference in mass between the charged objects and state which has the greater mass.
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?
  
Water has a mass per mole of 18.0 g/mol, and each water molecule (H_{2}O) has 10 electrons. (a) How many electrons are there in one liter (1.00×10^{–3} m^{3}) of water? (b) What is the net charge of all these electrons?
Section 18.5 Coulomb’s Law
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 oneninth of its original value?
  
The nucleus of the helium atom contains two protons that are separated by about 3.0×10^{–15} m. Find the magnitude of the electrostatic force that each proton exerts on the other. (The protons remain together in the nucleus because the repulsive electrostatic force is balanced by an attractive force called the strong nuclear force.)
The force of repulsion that two like charges exert on each other is 3.5 N. What will the force be if the distance between the charges is increased to five times its original value?
  
In a vacuum, two particles have charges of q_{1} and q_{2}, where q_{1}=+3.5 mC. They are separated by a distance of 0.26 m, and particle 1 experiences an attractive force of 3.4 N. What is q_{2} (magnitude and sign)?
Interactive LearningWare 18.1 offers some perspective on this problem. Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is 2.0×10^{–6} 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.
  
Two tiny conducting spheres are identical and carry charges of –20.0 mC and +50.0 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.
A charge of –3.00 mC is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius=0.100 m). The charges on the circle are –4.00 mC at the position due north and +5.00 mC at the position due east. What is the magnitude and direction of the net electrostatic force acting on the charge at the center? Specify the direction relative to due east.
Interactive Solution 18.15 provides a model for solving this type of problem. 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 mC, and object B has a charge of –2.00 mC. How many electrons must be removed from A and put onto B to make the electrostatic force that acts on each object an attractive force whose magnitude is 68.0 N?
  
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 mC charge experiences a net force due to the charges q_{A} and q_{B}. This net force points vertically downward in the drawing and has a magnitude of 405 N. Determine the magnitudes and algebraic signs of the charges q_{A} and q_{B}.
A point charge of –0.70 mC is fixed to one corner of a square. An identical charge is fixed to the diagonally opposite corner. A point charge q is fixed to each of the remaining corners. The net force acting on either of the charges q is zero. Find the magnitude and algebraic sign of q.
  
Interactive LearningWare 18.2 provides one approach to solving problems such as this one. The drawing shows three point charges fixed in place. The charge at the coordinate origin has a value of q_{1}=+8.00 mC; the other two have identical magnitudes, but opposite signs: q_{2}=–5.00 mC and q_{3}=+5.00 mC. (a) Determine the net force (magnitude and direction) exerted on q_{1} by the other two charges. (b) If q_{1} had a mass of 1.50 g and it were free to move, what would be its acceleration?
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?
  
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.
An electrically neutral model airplane is flying in a horizontal circle on a 3.0m 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.
  
Two objects are identical and small enough that their sizes can be ignored relative to the distance between them, which is 0.200 m. In a vacuum, each object carries a different charge, and they attract each other with a force of 1.20 N. The objects are 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 that of the initial attractive force. What is the initial charge on each object? Note that there are two answers.
A small spherical insulator of mass 8.00×10^{–2} kg and charge +0.600 mC is hung by a thin wire of negligible mass. A charge of –0.900 mC is held 0.150 m away from the sphere and directly to the right of it, so the wire makes an angle q with the vertical (see the drawing). Find (a) the angle q and (b) the tension in the wire.
  
There are four charges, each with a magnitude of 2.0 mC. Two are positive and two are negative. The charges are fixed to the corners of a 0.30m 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.
Section 18.6 The Electric Field, Section 18.7 Electric Field Lines, Section 18.8 The Electric Field Inside a Conductor: Shielding
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 mC at this spot?
  
Review Conceptual Example 12 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.
A tiny ball (mass=0.012 kg) carries a charge of –18 mC. What electric field (magnitude and direction) is needed to cause the ball to float above the ground?
  
At a distance r_{1} 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}/r_{1}.
Two charges, –16 and +4.0 mC, are fixed in place and separated by 3.0 m. (a) At what spot along a line through the charges is the net electric field zero? Locate this spot relative to the positive charge.
(b) What would be the force on a charge of +14 mC placed at this spot?
  
Background pertinent to this problem is available in Interactive LearningWare 18.3 . A 3.0mC point charge is placed in an external uniform electric field of 1.6×10^{4} N/C. At what distance from the charge is the net electric field zero?
  
A charge of q=+7.50 mC is located in an electric field. The x and y components of the electric field are E_{x}=6.00×10^{3} N/C and E_{y}=8.00×10^{3} N/C, respectively. (a) What is the magnitude of the force on the charge? (b) Determine the angle that the force makes with the +x axis.
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 3.50×10^{–9} 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?
  
Review Conceptual Example 11 before attempting this problem. The magnitude of each of the charges in Figure 18.21 is 8.60×10^{–12} 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 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 2.2×10^{5} N/C, while the field within the second capacitor is 3.8×10^{5} N/C. Determine the ratio r_{2}/r_{1} of the plate radius for the second capacitor to the plate radius for the first capacitor.
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=1.8×10^{–7} C/m^{2}, and the plates are separated by a distance of 1.5×10^{–2} m. How fast is the electron moving just before it reaches the positive plate?
A rectangle has a length of 2d and a height of d. Each of the following three charges is located at a corner of the rectangle: +q_{1} (upper left corner), +q_{2} (lower right corner), and –q (lower left corner). The net electric field at the (empty) upper right corner is zero. Find the magnitudes of q_{1} and q_{2}. Express your answers in terms of q.
  
A uniform electric field has a magnitude of 2.3×10^{3} N/C. In a vacuum, a proton begins with a speed of 2.5×10^{4} m/s and moves in the direction of this field. Find the speed of the proton after it has moved a distance of 2.0 mm.
Review Interactive Solution 18.41 for help with this problem. The drawing shows two positive charges q_{1} and q_{2} 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 q_{2}/q_{1} of the charge magnitudes.
  
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 7.00×10^{6} 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.
The magnitude of the electric field between the plates of a parallel plate capacitor is 480 N/C. A silver dollar is placed between the plates and oriented parallel to the plates. (a) Ignoring the edges of the coin, find the induced charge density s on each face of the coin. (b) Assuming the coin has a radius of 1.9 cm, find the magnitude of the total charge on each face of the coin.
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 E_{M}. The field directly above the midpoint at point P has a magnitude E_{P}. The ratio of these two field magnitudes is E_{M}/E_{P}=9.0. Find the angle a in the drawing.
  
Section 18.9 Gauss’ Law
A rectangular surface (0.16 m×0.38 m) is oriented in a uniform electric field of 580 N/C. What is the maximum possible electric flux through the surface?
The drawing shows an edgeon view of two planar surfaces that intersect and are mutually perpendicular. Surface 1 has an area of 1.7 m^{2}, while surface 2 has an area of 3.2 m^{2}. The electric field E in the drawing is uniform and has a magnitude of 250 N/C. Find the electric flux through (a) surface 1 and (b) surface 2.
  
A spherical surface completely surrounds a collection of charges. Find the electric flux through the surface if the collection consists of (a) a single +3.5×10^{–6} C charge, (b) a single –2.3×10^{–6} C charge, and (c) both of the charges in (a) and (b).
A vertical wall (5.9 m×2.5 m) in a house faces due east. A uniform electric field has a magnitude of 150 N/C. This field is parallel to the ground and points 35° north of east. What is the electric flux through the wall?
A cube is located with one corner at the origin of an x, y, z coordinate system. One of the cube’s 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) Find the electric flux through each of the six faces of the cube. (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.
  
Refer to Concept Simulation 18.3 for a perspective that is useful in solving this problem. Two spherical shells have a common center. A –1.6×10^{–6} C charge is spread uniformly over the inner shell, which has a radius of 0.050 m. A +5.1×10^{–6} 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 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=l/(2pe_{0}r), where l=Q/L.

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