19.7

Concepts & Calculations

The conservation of energy (Chapter 6) and the conservation of linear momentum (Chapter 7) are two of the most broadly applicable principles in all of science. In this chapter, we have seen that electrically charged particles obey the conservation-of-energy principle, provided that the electric potential energy is taken into account. The behavior of electrically charged particles, however, must also be consistent with the conservation-of-momentum principle, as the first example in this section emphasizes.

| Concepts & Calculations Example 13 | | | Conservation Principles

Particle 1 has a mass of , while particle 2 has a mass of . Each has the same electric charge. These particles are initially held at rest, and the two-particle system has an initial electric potential energy of 0.150 J. Suddenly, the particles are released and fly apart because of the repulsive electric force that acts on each one (see Figure 19-30). The effects of the gravitational force are negligible, and no other forces act on the particles. At one instant following the release, the speed of particle 1 is measured to be . What is the electric potential energy of the two-particle system at this instant?

Figure 19-30 (a) Two particles have different masses, but the same electric charge q. They are initially held at rest. (b) At an instant following the release of the particles, they are flying apart due to the mutual force of electric repulsion.

Figure 19-31 Example 14 determines the electric field and the electric potential at the midpoint (rA = rB) between the identical charges (qA = qB).

Concept Questions and Answers What type of energy does the two-particle system have initially?

Answer Initially, the particles are at rest, so they have no kinetic energy. However, they do have electric potential energy. They also have gravitational potential energy, but it is negligible.

What type of energy does the two-particle system have at the instant illustrated in part b of the drawing?

Answer Since each particle is moving, each has kinetic energy. The two-particle system also has electric potential energy at this instant. Gravitational potential energy remains negligible.

Does the principle of conservation of energy apply?

Answer Yes. Since the gravitational force is negligible, the only force acting here is the conservative electric force. Nonconservative forces are absent. Thus, the principle of conservation of energy applies.

Does the principle of conservation of linear momentum apply to the two particles as they fly apart?

Answer Yes. This principle states that the total linear momentum of an isolated system remains constant. An isolated system is one for which the vector sum of the external forces acting on the system is zero. There are no external forces acting here. The only appreciable force acting on each of the two particles is the force of electric repulsion, which is an internal force.

Solution The conservation-of-energy principle indicates that the total energy of the two-particle system is the same at the later instant as it was initially. Considering that only kinetic energy and electric potential energy are present, the principle can be stated as follows:

We have used the fact that KE0 is zero, since the particles are initially at rest. The final kinetic energy of the system is . In this expression, we have values for both masses and the speed vf1. We can find the speed vf2 by using the principle of conservation of momentum:

We have again used the fact that the particles are initially at rest, so that . Using this result for vf2, we can now obtain the final electric potential energy from the conservation-of-energy equation:

Chapter 18 introduces the electric field, and the present chapter introduces the electric potential. These two concepts are central to the study of electricity, and it is important to distinguish between them, as the next example emphasizes.

| Concepts & Calculations Example 14 | | | Electric Field and Electric Potential

Two identical point charges are fixed in place, separated by 0.50 m. (see Figure 19-31). Find the electric field and the electric potential at the midpoint of the line between the charges qA and qB. Concept Questions and Answers The electric field is a vector and has a direction. At the midpoint, what are the directions of the individual electric-field contributions from qA and qB?

Answer Since both charges are positive, the individual electric-field contributions from qA and qB point away from each charge. Thus, at the midpoint they point in opposite directions.

Is the magnitude of the net electric field at the midpoint greater than, less than, or equal to zero?

Answer The magnitude of the net electric field is zero. This is because the electric field from qA cancels the electric field from qB. These charges have the same magnitude q, and the midpoint is the same distance r from each of them. According to (Equation 18.3), then, each charge produces a field of the same strength. Since the individual electric-field contributions from qA and qB point in opposite directions at the midpoint, their vector sum is zero.

Is the total electric potential at the midpoint positive, negative, or zero?

Answer The total electric potential at the midpoint is the algebraic sum of the individual contributions from each charge. According to (Equation 19.6), each contribution is positive, since each charge is positive. Thus, the total electric potential is also positive.

Does the total electric potential have a direction associated with it?

Answer No. The electric potential is a scalar quantity, not a vector quantity. Therefore, it has no direction associated with it.

Solution As discussed in the second concept question, the total electric field at the midpoint is . Using (Equation 19.6), we find that the total potential at the midpoint is

Since , we have

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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