18.11

Concepts & Calculations

In this chapter we have studied electric forces and electric fields. We conclude now by presenting some examples that review important features of these concepts. The three-part format of the examples stresses the role of conceptual understanding in problem solving. First, the problem statement is given. Then, there is a concept question-and-answer section, followed by the solution section. The purpose of the concept question-and-answer section is to provide help in understanding the solution and to illustrate how a review of the concepts can help in anticipating some of the characteristics of the numerical answers.

| Concepts & Calculations Example 17 | | | The Vector Nature of Electric Forces

The charges on three identical metal spheres are , , and . 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 18-40a. What is the net force (magnitude and direction) exerted on the sphere at the origin? Treat the spheres as if they were particles.

Figure 18-40 (a) Three equal charges lie on the x and y axes. (b) The net force exerted on q1 by the other two charges is .

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

Answer Yes. The conservation of electric charge states that during any process the net electric charge of an isolated system remains constant (is conserved). Therefore, the net charge on the three spheres before they touch () is the same as the net charge after they touch.

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

Answer Yes. Since the spheres are identical, the net charge distributes itself equally over the three spheres. After they are separated, each has one-third of the net charge: .

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

Answer Yes. The charges q2 and q3 have equal magnitudes and are the same distance from q1. According to Coulomb’s law, then, they 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?

Answer No. Although the two forces that act on q1 have equal magnitudes, they have different directions. The forces are repulsive forces, since all of the charges in part a of the drawing are identical. Figure 18-40b shows the force exerted on q1 by q2 and the force exerted on q1 by q3. To obtain the net force , we must take these directions into account by using vector addition.

Solution The magnitude F12 of the force exerted on q1 by q2 is given by Coulomb’s law, Equation 18.1, as

Problem solving insight Often charge magnitudes are specified in microcoulombs . When using Coulomb’s law, be sure to convert microcoulombs into coulombs before substituting for the charge magnitudes and .

Note that we have used the magnitudes of q1 and q2 in Coulomb’s law. As mentioned previously, the magnitude of the force exerted on q1 by q3 has the same value as F12, and so . Since the forces and are perpendicular to each other (see Figure 18-40b), we may use the Pythagorean theorem to find the magnitude F of the net force:

The angle that the net force makes with the −x axis (see part b of the drawing) is

| Concepts & Calculations Example 18 | | | Becoming Familiar with Electric Fields

Two point charges are lying on the y axis in Figure 18-41a: and . 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 and mass m = 1.20 g is placed at P. When it is released, what is its acceleration?

Figure 18-41 (a) Two charges q1 and q2 produce an electric field at the point P. (b) The electric fields and add to give the net electric field .

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

Answer Yes. An electric field is produced by the charges q1 and q2, and it exists throughout the entire region that surrounds them. If a test charge were placed at this point, it would experience a force due to the electric field. The force would be the product of the charge and the electric field.

The charge q1 produces an electric field at the point P. What is the direction of this field?

Answer The electric field created by a charge always points away from a positive charge and toward a negative charge. Since q1 is negative, the electric field points toward it (see Figure 18-41b).

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

Answer Since q2 is positive, the electric field that it produces points away from q2, as shown in the drawing.

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?

Answer No, because the electric fields have different directions. We must add the individual fields as vectors to obtain the net electric field. Only then can we determine its magnitude.

Solution

a. | The magnitude of the electric fields that q1 and q2 produce at P are given by Equation 18.3, where the distances are specified in the drawing:

The x and y components of these fields and the total field are given in the following table:

Electric field | x component | y component
| = 0 N/C | = + 7.56 × 104 N/C

The net electric field has only a component along the + y axis. So,

b. | According to Newton’s second law, Equation 4.2, the acceleration of an object placed at this point is equal to the net force acting on it divided by its mass. The net force is the product of the charge and the net electric field, , as indicated by Equation 18.2. Thus, the acceleration is

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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