### Learning Objectives

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

- Describe the relationship between voltage and electric field.
- Derive an expression for the electric potential and electric field.
- Calculate electric field strength given distance and voltage.

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

**2.C.5.2**The student is able to calculate the magnitude and determine the direction of the electric field between two electrically charged parallel plates, given the charge of each plate, or the electric potential difference and plate separation.**(S.P. 2.2)****2.C.5.3**The student is able to represent the motion of an electrically charged particle in the uniform field between two oppositely charged plates and express the connection of this motion to projectile motion of an object with mass in the Earth’s gravitational field.**(S.P. 1.1, 2.2, 7.1)****2.E.3.1**The student is able to apply mathematical routines to calculate the average value of the magnitude of the electric field in a region from a description of the electric potential in that region using the displacement along the line on which the difference in potential is evaluated.**(S.P. 2.2)****2.E.3.2**The student is able to apply the concept of the isoline representation of electric potential for a given electric charge distribution to predict the average value of the electric field in the region.**(S.P. 1.4, 6.4)**

In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field $\mathbf{\text{E}}$ is produced by placing a potential difference (or voltage) $\mathrm{\Delta}V$ across two parallel metal plates, labeled A and B. (See Figure 19.7.) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist’s point of view, either $\mathrm{\Delta}V$ or $\mathbf{\text{E}}$ can be used to describe any charge distribution. $\mathrm{\Delta}V$ is most closely tied to energy, whereas $\mathbf{\text{E}}$ is most closely related to force. $\mathrm{\Delta}V$ is a scalar quantity and has no direction, while $\mathbf{\text{E}}$ is a vector quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by $E$ below.) The relationship between $\mathrm{\Delta}V$ and $\mathbf{\text{E}}$ is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in Electric Potential Energy: Potential Difference, this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.

The work done by the electric field in Figure 19.7 to move a positive charge $q$ from A, the positive plate, higher potential, to B, the negative plate, lower potential, is

The potential difference between points A and B is

Entering this into the expression for work yields

Work is $W=\text{Fd}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta $; here $\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =1$, since the path is parallel to the field, and so $W=\text{Fd}$. Since $F=\text{qE}$, we see that $W=\text{qEd}$. Substituting this expression for work into the previous equation gives

The charge cancels, and so the voltage between points A and B is seen to be

where *$d$* is the distance from A to B, or the distance between the plates in Figure 19.7. Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:

### Voltage between Points A and B

where *$d$* is the distance from A to B, or the distance between the plates.

### Example 19.4

#### What Is the Highest Voltage Possible between Two Plates?

Dry air will support a maximum electric field strength of about $3.0\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{V/m}$. Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field. What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air?

#### Strategy

We are given the maximum electric field $E$ between the plates and the distance $d$ between them. The equation ${V}_{\text{AB}}=\mathrm{Ed}$ can thus be used to calculate the maximum voltage.

#### Solution

The potential difference or voltage between the plates is

Entering the given values for $E$ and $d$ gives

or

(The answer is quoted to only two digits, since the maximum field strength is approximate.)

#### Discussion

One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5 cm (1 in.) gap, or 150 kV for a 5 cm spark. This limits the voltages that can exist between conductors, perhaps on a power transmission line. A smaller voltage will cause a spark if there are points on the surface, since points create greater fields than smooth surfaces. Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air. The largest voltages can be built up, say with static electricity, on dry days.

### Making Connections: Uniform Fields

Recall from Projectile Motion (Figure 3.35) that a massive projectile launched horizontally (for example, from a cliff) in a uniform downward gravitational field (as we find near the surface of the Earth) will follow a parabolic trajectory downward until it hits the ground, as shown in Figure 19.9(a).

An identical outcome occurs for a positively charged particle in a uniform electric field (Figure 19.9(b)); it follows the electric field “downhill” until it runs into something. The difference between the two cases is that the gravitational force is always attractive; the electric force has two kinds of charges, and therefore may be either attractive or repulsive. Therefore, a negatively charged particle launched into the same field will fall “uphill”.

### Example 19.5

#### Field and Force inside an Electron Gun

(a) An electron gun has parallel plates separated by 4.00 cm and gives electrons 25.0 keV of energy. What is the electric field strength between the plates? (b) What force would this field exert on a piece of plastic with a $\text{0.500 \mu C}$ charge that gets between the plates?

#### Strategy

Since the voltage and plate separation are given, the electric field strength can be calculated directly from the expression $E=\frac{{V}_{\text{AB}}}{d}$. Once the electric field strength is known, the force on a charge is found using $\mathbf{\text{F}}=q\phantom{\rule{0.25em}{0ex}}\mathbf{\text{E}}$. Since the electric field is in only one direction, we can write this equation in terms of the magnitudes, $F=q\phantom{\rule{0.25em}{0ex}}E$.

#### Solution for (a)

The expression for the magnitude of the electric field between two uniform metal plates is

Since the electron is a single charge and is given 25.0 keV of energy, the potential difference must be 25.0 kV. Entering this value for ${V}_{\text{AB}}$ and the plate separation of 0.0400 m, we obtain

#### Solution for (b)

The magnitude of the force on a charge in an electric field is obtained from the equation

Substituting known values gives

#### Discussion

Note that the units are newtons, since $\text{1 V/m}=\text{1 N/C}$. The force on the charge is the same no matter where the charge is located between the plates. This is because the electric field is uniform between the plates.

In more general situations, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of $\mathbf{\text{E}}$ and also in the direction of lower potential $V$. Furthermore, the magnitude of $\mathbf{\text{E}}$ equals the rate of decrease of $V$ with distance. The faster $V$ decreases over distance, the greater the electric field. In equation form, the general relationship between voltage and electric field is

where $\mathrm{\Delta}s$ is the distance over which the change in potential, $\mathrm{\Delta}V$, takes place. The minus sign tells us that $\mathbf{\text{E}}$ points in the direction of decreasing potential. The electric field is said to be the * gradient* (as in grade or slope) of the electric potential.

### Relationship between Voltage and Electric Field

In equation form, the general relationship between voltage and electric field is

where $\mathrm{\Delta}s$ is the distance over which the change in potential, $\mathrm{\Delta}V$, takes place. The minus sign tells us that $\mathbf{\text{E}}$ points in the direction of decreasing potential. The electric field is said to be the * gradient* (as in grade or slope) of the electric potential.

Note that Equation (19.36) is defining the __average__ electric field over the given region.

For continually changing potentials, $\mathrm{\Delta}V$ and $\mathrm{\Delta}s$ become infinitesimals and differential calculus must be employed to determine the electric field.

### Making Connections: Non-Parallel Conducting Plates

Consider two conducting plates, placed as shown in Figure 19.10. If the plates are held at a fixed potential difference Δ*V*, the average electric field is strongest between the near edges of the plates, and weakest between the two far edges of the plates.

Now assume that the potential difference is 60 V. If the arc length along the field line labeled by A is 10 cm, what is the electric field at point A? How about point B, if the arc length along that field line is 17 cm? How would the density of the electric potential isolines, if they were drawn on the figure, compare at these two points? Can you use this concept to estimate what the electric field strength would be at a point midway between A and B?

#### Answer

Applying Equation. 19.36, we find that the electric fields at A and B are 600 V/m and 350 V/m respectively. The isolines would be denser at A than at B, and would spread out evenly from A to B. Therefore, the electric field at a point halfway between the two would have an arc length of 13.5 cm and be approximately 440 V/m.