College Physics for AP® Courses

# 19.4Equipotential Lines

College Physics for AP® Courses19.4 Equipotential Lines

### Learning Objectives

By the end of this section, you will be able to:

• Explain equipotential lines (also called isolines of electric potential) and equipotential surfaces.
• Describe the action of grounding an electrical appliance.
• Compare electric field and equipotential lines.

The information presented in this section supports the following AP® learning objectives and science practices:

• 2.E.2.1 The student is able to determine the structure of isolines of electric potential by constructing them in a given electric field. (S.P. 6.4, 7.2)
• 2.E.2.2 The student is able to predict the structure of isolines of electric potential by constructing them in a given electric field and make connections between these isolines and those found in a gravitational field. (S.P. 6.4, 7.2)
• 2.E.2.3 The student is able to qualitatively use the concept of isolines to construct isolines of electric potential in an electric field and determine the effect of that field on electrically charged objects. (S.P. 1.4)

We can represent electric potentials (voltages) pictorially, just as we drew pictures to illustrate electric fields. Of course, the two are related. Consider Figure 19.12, which shows an isolated positive point charge and its electric field lines. Electric field lines radiate out from a positive charge and terminate on negative charges. While we use blue arrows to represent the magnitude and direction of the electric field, we use green lines to represent places where the electric potential is constant. These are called equipotential lines in two dimensions, or equipotential surfaces in three dimensions. The term equipotential is also used as a noun, referring to an equipotential line or surface. The potential for a point charge is the same anywhere on an imaginary sphere of radius $rr size 12{r} {}$ surrounding the charge. This is true since the potential for a point charge is given by $V=kQ/rV=kQ/r size 12{V= ital "kQ"/r} {}$ and, thus, has the same value at any point that is a given distance $rr size 12{r} {}$ from the charge. An equipotential sphere is a circle in the two-dimensional view of Figure 19.12. Since the electric field lines point radially away from the charge, they are perpendicular to the equipotential lines.

Figure 19.12 An isolated point charge $QQ size 12{Q} {}$ with its electric field lines in blue and equipotential lines in green. The potential is the same along each equipotential line, meaning that no work is required to move a charge anywhere along one of those lines. Work is needed to move a charge from one equipotential line to another. Equipotential lines are perpendicular to electric field lines in every case.

### Applying the Science Practices: Electric Potential and Peaks

Starting with Figure 19.13 as a rough example, draw diagrams of isolines for both positive and negative isolated point charges. Be sure to take care with what happens to the spacing of the isolines as you get closer to the charge. Then copy both of these sets of lines, but relabel them as gravitational equipotential lines. Then try to draw the sort of hill or hole or other shape that would have equipotential lines of this form. Does this shape exist in nature?

Figure 19.13 An example of a topographical map.

You should notice that the lines get closer together the closer you get to the point charge. The hill (or sinkhole, for the equivalent from a negative charge) should have a 1/r sort of form, which is not a very common topographical feature.

It is important to note that equipotential lines are always perpendicular to electric field lines. No work is required to move a charge along an equipotential, since $ΔV=0ΔV=0 size 12{?V=0} {}$. Thus the work is

$W=–ΔPE=–qΔV=0.W=–ΔPE=–qΔV=0. size 12{W=-?"PE"=-q?V=0} {}$
19.43

Work is zero if force is perpendicular to motion. Force is in the same direction as $EE size 12{E} {}$, so that motion along an equipotential must be perpendicular to $EE size 12{E} {}$. More precisely, work is related to the electric field by

$W = Fd cos θ = qEd cos θ = 0 . W = Fd cos θ = qEd cos θ = 0 . size 12{W= ital "Fd""cos"?= ital "qEd""cos"?=0} {}$
19.44

Note that in the above equation, $E E size 12{E} {}$ and $F F size 12{F} {}$ symbolize the magnitudes of the electric field strength and force, respectively. Neither $qq size 12{q} {}$ nor $EE$ nor $dd$ is zero, and so $cosθcosθ$ must be 0, meaning $θθ size 12{?} {}$ must be $90º 90º$. In other words, motion along an equipotential is perpendicular to $EE size 12{E} {}$.

One of the rules for static electric fields and conductors is that the electric field must be perpendicular to the surface of any conductor. This implies that a conductor is an equipotential surface in static situations. There can be no voltage difference across the surface of a conductor, or charges will flow. One of the uses of this fact is that a conductor can be fixed at zero volts by connecting it to the earth with a good conductor—a process called grounding. Grounding can be a useful safety tool. For example, grounding the metal case of an electrical appliance ensures that it is at zero volts relative to the earth.

### Grounding

A conductor can be fixed at zero volts by connecting it to the earth with a good conductor—a process called grounding.

Because a conductor is an equipotential, it can replace any equipotential surface. For example, in Figure 19.12 a charged spherical conductor can replace the point charge, and the electric field and potential surfaces outside of it will be unchanged, confirming the contention that a spherical charge distribution is equivalent to a point charge at its center.

Figure 19.14 shows the electric field and equipotential lines for two equal and opposite charges. Given the electric field lines, the equipotential lines can be drawn simply by making them perpendicular to the electric field lines. Conversely, given the equipotential lines, as in Figure 19.15(a), the electric field lines can be drawn by making them perpendicular to the equipotentials, as in Figure 19.15(b).

Figure 19.14 The electric field lines and equipotential lines for two equal but opposite charges. The equipotential lines can be drawn by making them perpendicular to the electric field lines, if those are known. Note that the potential is greatest (most positive) near the positive charge and least (most negative) near the negative charge.
Figure 19.15 (a) These equipotential lines might be measured with a voltmeter in a laboratory experiment. (b) The corresponding electric field lines are found by drawing them perpendicular to the equipotentials. Note that these fields are consistent with two equal negative charges.

One of the most important cases is that of the familiar parallel conducting plates shown in Figure 19.16. Between the plates, the equipotentials are evenly spaced and parallel. The same field could be maintained by placing conducting plates at the equipotential lines at the potentials shown.

Figure 19.16 The electric field and equipotential lines between two metal plates.

### Making Connections: Slopes and Parallel Plates

Consider the parallel plates in Figure 19.2. These have equipotential lines that are parallel to the plates in the space between, and evenly spaced. An example of this (with sample values) is given in Figure 19.16. One could draw a similar set of equipotential isolines for gravity on the hill shown in Figure 19.2. If the hill has any extent at the same slope, the isolines along that extent would be parallel to each other. Furthermore, in regions of constant slope, the isolines would be evenly spaced.

Figure 19.17 Note that a topographical map along a ridge has roughly parallel elevation lines, similar to the equipotential lines in Figure 19.16.

An important application of electric fields and equipotential lines involves the heart. The heart relies on electrical signals to maintain its rhythm. The movement of electrical signals causes the chambers of the heart to contract and relax. When a person has a heart attack, the movement of these electrical signals may be disturbed. An artificial pacemaker and a defibrillator can be used to initiate the rhythm of electrical signals. The equipotential lines around the heart, the thoracic region, and the axis of the heart are useful ways of monitoring the structure and functions of the heart. An electrocardiogram (ECG) measures the small electric signals being generated during the activity of the heart. More about the relationship between electric fields and the heart is discussed in Energy Stored in Capacitors.

### PhET Explorations: Charges and Fields

Move point charges around on the playing field and then view the electric field, voltages, equipotential lines, and more. It's colorful, it's dynamic, it's free.

Figure 19.18 Charges and Fields