19.4

Equipotential Surfaces and their Relation to the Electric Field

An equipotential surface is a surface on which the electric potential is the same everywhere. The easiest equipotential surfaces to visualize are those that surround an isolated point charge. According to Equation 19.6, the potential at a distance r from a point charge q is . Thus, wherever r is the same, the potential is the same, and the equipotential surfaces are spherical surfaces centered on the charge. There are an infinite number of such surfaces, one for every value of r, and Figure 19-12 illustrates two of them. The larger the distance r, the smaller is the potential of the equipotential surface.

Figure 19-12 The equipotential surfaces that surround the point charge +q are spherical. The electric force does no work as a charge moves on a path that lies on an equipotential surface, such as the path ABC. However, work is done by the electric force when a charge moves between two equipotential surfaces, as along the path AD.

The net electric force does no work as a charge moves on an equipotential surface. This important characteristic arises because when an electric force does work WAB as a charge moves from A to B, the potential changes according to (Equation 19.4). Since the potential remains the same on an equipotential surface, , and we see that . In Figure 19-12, for instance, the electric force does no work as a test charge moves along the circular arc ABC, which lies on an equipotential surface. In contrast, the electric force does work when a charge moves between equipotential surfaces, as from A to D in the picture.
The spherical equipotential surfaces that surround an isolated point charge illustrate another characteristic of all equipotential surfaces. Figure 19-13 shows two of the surfaces around a positive point charge, along with some electric field lines. The electric field lines give the direction of the electric field, and for a positive point charge, the electric field is directed radially outward. Therefore, at each location on an equipotential sphere the electric field is perpendicular to the surface and points outward in the direction of decreasing potential, as the drawing emphasizes. This perpendicular relation is valid whether or not the equipotential surfaces result from a positive charge or have a spherical shape. Problem solving insight: the electric field created by any charge or group of charges is everywhere perpendicular to the associated equipotential surfaces and points in the direction of decreasing potential. For example, Figure 19-14 shows the electric field lines around an electric dipole, along with some equipotential surfaces (shown in cross section). Since the field lines are not simply radial, the equipotential surfaces are no longer spherical but, instead, have the shape necessary to be everywhere perpendicular to the field lines.

Figure 19-13 The radially directed electric field of a point charge is perpendicular to the spherical equipotential surfaces that surround the charge. The electric field points in the direction of decreasing potential.

Figure 19-14 A cross-sectional view of the equipotential surfaces (in blue) of an electric dipole. The surfaces are drawn so that at every point they are perpendicular to the electric field lines (in red) of the dipole.

To see why an equipotential surface must be perpendicular to the electric field, consider Figure 19-15, which shows a hypothetical situation in which the perpendicular relation does not hold. If were not perpendicular to the equipotential surface, there would be a component of parallel to the surface. This field component would exert an electric force on a test charge placed on the surface. As the charge moved along the surface, work would be done by this component of the electric force. The work, according to Equation 19.4, would cause the potential to change, and, thus, the surface could not be an equipotential surface as assumed. The only way out of the dilemma is for the electric field to be perpendicular to the surface, so there is no component of the field parallel to the surface.

Figure 19-15 In this hypothetical situation, the electric field is not perpendicular to the equipotential surface. As a result, there is a component of parallel to the surface.

We have already encountered one equipotential surface. In Section 18.8, we found that the direction of the electric field just outside an electrical conductor is perpendicular to the conductor’s surface, when the conductor is at equilibrium under electrostatic conditions. Thus, the surface of any conductor is an equipotential surface under such conditions. In fact, since the electric field is zero everywhere inside a conductor whose charges are in equilibrium, the entire conductor can be regarded as an equipotential volume.
There is a quantitative relation between the electric field and the equipotential surfaces. One example that illustrates this relation is the parallel plate capacitor in Figure 19-16. As Section 18.6 discusses, the electric field between the metal plates is perpendicular to them and is the same everywhere, ignoring fringe fields at the edges. To be perpendicular to the electric field, the equipotential surfaces must be planes that are parallel to the capacitor plates, which themselves are equipotential surfaces. The potential difference between the plates is given by Equation 19.4 as , where A is a point on the positive plate and B is a point on the negative plate. The work done by the electric force as a positive test charge q0 moves from A to B is , where F refers to the electric force and to the displacement along a line perpendicular to the plates. The force equals the product of the charge and the electric field , so the work becomes . Therefore, the potential difference between the capacitor plates can be written in terms of the electric field as , or

| (19.7a)

The quantity is referred to as the potential gradient and has units of volts per meter. In general, the relation gives only the component of the electric field along the displacement ; it does not give the perpendicular component. The next example deals further with the equipotential surfaces between the plates of a capacitor.

Figure 19-16 The metal plates of a parallel plate capacitor are equipotential surfaces. Two additional equipotential surfaces are shown between the plates. These two equipotential surfaces are parallel to the plates and are perpendicular to the electric field between the plates.

| Example 9 | | | The Electric Field and Potential are Related

The plates of the capacitor in Figure 19-16 are separated by a distance of 0.032 m, and the potential difference between them is . Between the two labeled equipotential surfaces there is a potential difference of −3.0 V. Find the spacing between the two labeled surfaces.
Reasoning The electric field is . To find the spacing between the two labeled equipotential surfaces, we solve this equation for , with and E equal to the electric field between the plates of the capacitor. A value for E can be obtained by using the values given for the distance and potential difference between the plates.
Solution The electric field between the capacitor plates is

| (19.7a)

The spacing between the labeled equipotential surfaces can now be determined:

Equation 19.7a gives the relationship between the electric field and the electric potential. It gives the component of the electric field along the displacement in a region of space where the electric potential changes from place to place and applies to a wide variety of situations. When applied strictly to a parallel plate capacitor, however, this expression is often used in a slightly different form. In Figure 19-16, the metal plates of the capacitor are marked A (higher potential) and B (lower potential). Traditionally, in discussions of such a capacitor, the potential difference between the plates is referred to by using the symbol V to denote the amount by which the higher potential exceeds the lower potential . In addition, the displacement from plate A to plate B is expressed in terms of the separation d between the plates . With this nomenclature, Equation 19.7a becomes

| (19.7b)

Check Your Understanding 3

The drawing shows a cross-sectional view of two spherical equipotential surfaces and two electric field lines that are perpendicular to these surfaces. When an electron moves from point A to point B (against the electric field), the electric force does of work. What are the electric potential differences (a) , (b) , and (c) ? (The answers are given at the end of the book.)

Background: Two ideas are important here: (1) the relationship between the work done by the electric force and the electric potential difference, and (2) equipotential surfaces.
For similar questions (including conceptual counterparts), consult Self-Assessment Test 19.2, which is described at the end of Section 19.5.

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