Jennifer is holding on to an electrically charged sphere that reaches an electric potential of about 100 000 V. The device that generates this high electric potential is called a Van de Graaff generator. What causes Jennifer’s hair to stand on end like the needles of a porcupine? Why is she safe in this situation in view of the fact that 110 V from a wall outlet can kill you?(Henry Leap and Jim Lehman)
c h a p t e r
25.1 Potential Difference and Electric
25.5 Electric Potential Due to
Continuous Charge Distributions
25.2 Potential Differences in a
Uniform Electric Field
25.6 Electric Potential Due to a
25.3 Electric Potential and Potential
Energy Due to Point Charges
25.7 (Optional)The Millikan Oil-Drop
25.4 Obtaining the Value of the
Electric Field from the Electric Potential
25.8 (Optional) Applications of
25.1 Potential Difference and Electric Potential
he concept of potential energy was introduced in Chapter 8 in connection with such conservative forces as the force of gravity and the elastic force exerted by aspring. By using the law of conservation of energy, we were able to avoid working directly with forces when solving various problems in mechanics. In this chapter we see that the concept of potential energy is also of great value in the study of electricity. Because the electrostatic force given by Coulomb’s law is conservative, electrostatic phenomena can be conveniently described in terms of anelectric potential energy. This idea enables us to deﬁne a scalar quantity known as electric potential. Because the electric potential at any point in an electric ﬁeld is a scalar function, we can use it to describe electrostatic phenomena more simply than if we were to rely only on the concepts of the electric ﬁeld and electric forces. In later chapters we shall see that the concept of electricpotential is of great practical value.
POTENTIAL DIFFERENCE AND ELECTRIC POTENTIAL
When a test charge q 0 is placed in an electric ﬁeld E created by some other charged object, the electric force acting on the test charge is q 0 E. (If the ﬁeld is produced by more than one charged object, this force acting on the test charge is the vector sum of the individual forces exerted onit by the various other charged objects.) The force q 0 E is conservative because the individual forces described by Coulomb’s law are conservative. When the test charge is moved in the ﬁeld by some external agent, the work done by the ﬁeld on the charge is equal to the negative of the work done by the external agent causing the displacement. For an inﬁnitesimal displacement ds, the work done bythe electric ﬁeld on the charge is F d s q 0 E ds. As this amount of work is done by the ﬁeld, the potential energy of the charge – ﬁeld system is decreased by an amount dU q 0 E ds. For a ﬁnite displacement of the charge from a point A to a point B, the change in potential energy of the system U U B U A is
Change in potential energy
The integration isperformed along the path that q 0 follows as it moves from A to B, and the integral is called either a path integral or a line integral (the two terms are synonymous). Because the force q 0 E is conservative, this line integral does not depend on the path taken from A to B.
Quick Quiz 25.1
If the path between A and B does not make any difference in Equation 25.1, why don’t we just use theexpression U q 0 Ed, where d is the straight-line distance between A and B ?
The potential energy per unit charge U/q 0 is independent of the value of q 0 and has a unique value at every point in an electric ﬁeld. This quantity U/q 0 is called the electric potential (or simply the potential) V. Thus, the electric potential at any point in an electric ﬁeld is V U q0 (25.2)