Electric fields are invisible, but they can be represented visually in a way that allows us to understand both their strength and direction. Drawings using lines to represent electric fields around charged objects are extremely useful for visualizing how charges influence the space around them.

Because the electric field has both magnitude and direction, it is a vector quantity. Like all vectors, it can be represented by an arrow whose length corresponds to its magnitude and whose direction shows the direction of the force on a positive test charge. In earlier chapters, we used arrows to represent forces; here, we extend that same idea to fields.

Figure 6.1 shows two equivalent representations of the electric field created by a positive point charge [latex]Q[/latex]. In part (a), the field is represented by arrows placed at various points in space. Each arrow shows the direction and relative strength of the force that would act on a positive test charge placed at that location. In part (b), the same information is represented using continuous lines called electric field lines.

Electric field lines provide a convenient visual map of infinitesimal force vectors. At any point in space, the electric field vector is tangent to the field line. The density of field lines (how closely spaced they are) indicates the magnitude of the field: where lines are closer together, the field is stronger.

By definition, electric field lines point in the direction that a positive test charge would move. Therefore:

• Field lines point away from positive charges.
• Field lines point toward negative charges.

This convention is important in biology and physiology, where electric field direction determines the movement of positive ions such as Na⁺, K⁺, and Ca²⁺ in tissues.

The electric field strength is proportional to the number of field lines passing through a given area. Mathematically, for a point charge:

Since the surface area of a sphere increases as [latex]r^2[/latex], the field lines spread out with distance. This geometric spreading explains why the field weakens according to the inverse-square law.

This graphical method of representing fields is used not only for electrostatics but also for gravitational and magnetic fields. In all cases, field lines never cross, and their density indicates field strength.

Figure 6.2 illustrates several important features of electric field diagrams:

• A positive charge produces field lines radiating outward.
• A negative charge produces field lines directed inward.
• A larger magnitude charge produces more field lines, indicating a stronger electric field.

Notice that in part (c), the negative charge has twice the magnitude of the positive charge in part (a). This is represented by roughly twice as many field lines. The number of lines drawn is proportional to charge magnitude.

In biological systems, differences in charge magnitude are crucial. For example, localized charge imbalances across cell membranes produce electric fields that guide ion motion and influence membrane potentials.

Electric Fields from Multiple Charges

In many realistic situations, more than one charge is present. When multiple charges exist, the total electric field at a point is found by vector addition of the individual fields produced by each charge. This is known as the principle of superposition.

Mathematically, if charges [latex]Q_1[/latex], [latex]Q_2[/latex], and [latex]Q_3[/latex] are present, the total field at a point is

Because electric fields are vectors, both magnitude and direction must be considered. In two dimensions, components are typically added separately.

This superposition principle is essential in physiology. The electric field inside and around a cell is not created by a single charge, but by the combined effect of many ions and charged macromolecules. Understanding how fields add allows us to analyze phenomena such as nerve impulse propagation and electrical signaling in cardiac tissue.

Field Patterns for Two Charges

When two charges of the same sign are placed near each other, their field lines repel one another. Field lines bend away from the region between the charges, indicating repulsion. No field lines connect the charges.

When two charges of opposite sign are placed near each other, field lines originate on the positive charge and terminate on the negative charge. The pattern between them becomes dense and directional. This configuration is called an electric dipole.

Electric dipoles are extremely important in biology. Many molecules, including water, are naturally dipolar. The electric field patterns around dipoles influence molecular interactions, protein folding, and membrane structure.

Force on a Test Charge in a Multi-Charge System

If a test charge [latex]q[/latex] is placed in a region where multiple charges create an electric field, the force on that test charge is given by

This means that once the net electric field is known, calculating the force becomes straightforward. Positive test charges experience force in the direction of the field; negative charges experience force opposite the field direction.

This directional behavior is central to understanding how negatively charged chloride ions move differently from positively charged sodium ions in electric fields across biological membranes.

Example 6.1: Adding Electric Fields

Find the magnitude and direction of the total electric field due to the two point charges, [latex]{q}_{1}[/latex] and [latex]{q}_{2}[/latex], at the origin of the coordinate system as shown in Figure 6.3.

Strategy

Since the electric field is a vector (having magnitude and direction), we add electric fields with the same vector techniques used for other types of vectors. We first must find the electric field due to each charge at the point of interest, which is the origin of the coordinate system (O) in this instance. We pretend that there is a positive test charge, [latex]q[/latex], at point O, which allows us to determine the direction of the fields [latex]{\mathbf{\text{E}}}_{1}[/latex] and [latex]{\mathbf{\text{E}}}_{2}[/latex]. Once those fields are found, the total field can be determined using vector addition.

Solution

The electric field strength at the origin due to [latex]{q}_{1}[/latex] is labeled [latex]{E}_{1}[/latex] and is calculated:

[latex]\begin{array}{}{E}_{1}=k\frac{{q}_{1}}{{r}_{1}^{2}}=\left(8\text{.}\text{99}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{N}\cdot {\text{m}}^{2}{\text{/C}}^{2}\right)\frac{\left(5\text{.}\text{00}×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{C}\right)}{{\left(2\text{.}\text{00}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}}\\ {E}_{1}=1\text{.}\text{124}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}.\end{array}[/latex]

Similarly, [latex]{E}_{2}[/latex] is

[latex]\begin{array}{}{E}_{2}=k\frac{{q}_{2}}{{r}_{2}^{2}}=\left(8\text{.}\text{99}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{N}\cdot {\text{m}}^{2}{\text{/C}}^{2}\right)\frac{\left(\text{10}\text{.}0×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{C}\right)}{{\left(4\text{.}\text{00}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}}\\ {E}_{2}=0\text{.}\text{5619}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}.\end{array}[/latex]

Four digits have been retained in this solution to illustrate that [latex]{E}_{1}[/latex] is exactly twice the magnitude of [latex]{E}_{2}[/latex]. Now arrows are drawn to represent the magnitudes and directions of [latex]{\mathbf{\text{E}}}_{1}[/latex] and [latex]{\mathbf{\text{E}}}_{2}[/latex]. (See Figure 6.3.) The direction of the electric field is that of the force on a positive charge so both arrows point directly away from the positive charges that create them. The arrow for [latex]{\mathbf{\text{E}}}_{1}[/latex] is exactly twice the length of that for [latex]{\mathbf{\text{E}}}_{2}[/latex]. The arrows form a right triangle in this case and can be added using the Pythagorean theorem. The magnitude of the total field [latex]{E}_{\text{tot}}[/latex] is

[latex]\begin{array}{lll}{E}_{\text{tot}}& =& ({E}_{1}^{2}+{E}_{2}^{2}{)}^{\text{1/2}}\\ & =& \{{(\text{1.124}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}{)}^{2}+(\text{0.5619}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}{)}^{2}\}}^{\text{1/2}}\\ & =& \text{1.26}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C.}\end{array}[/latex]

The direction is

[latex]\begin{array}{lll}\theta & =& {\text{tan}}^{-1}\left(\frac{{E}_{1}}{{E}_{2}}\right)\\ & =& {\text{tan}}^{-1}\left(\frac{1\text{.}\text{124}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}}{0\text{.}\text{5619}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/C}}\right)\\ & =& \text{63}\text{.}4º,\end{array}[/latex]

or [latex]63.4º[/latex] above the x-axis.

Discussion

In cases where the electric field vectors to be added are not perpendicular, vector components or graphical techniques can be used. The total electric field found in this example is the total electric field at only one point in space. To find the total electric field due to these two charges over an entire region, the same technique must be repeated for each point in the region. This impossibly lengthy task (there are an infinite number of points in space) can be avoided by calculating the total field at representative points and using some of the unifying features noted next.

Figure 6.4 shows how the electric field from two point charges can be drawn by finding the total field at representative points and drawing electric field lines consistent with those points. While the electric fields from multiple charges are more complex than those of single charges, some simple features are easily noticed.

For example, the field is weaker between like charges, as shown by the lines being farther apart in that region. (This is because the fields from each charge exert opposing forces on any charge placed between them.) (See Figure 6.4 and Figure 6.5(a).) Furthermore, at a great distance from two like charges, the field becomes identical to the field from a single, larger charge.

Figure 6.5(b) shows the electric field of two unlike charges. The field is stronger between the charges. In that region, the fields from each charge are in the same direction, and so their strengths add. The field of two unlike charges is weak at large distances, because the fields of the individual charges are in opposite directions and so their strengths subtract. At very large distances, the field of two unlike charges looks like that of a smaller single charge.

We use electric field lines to visualize and analyze electric fields (the lines are a pictorial tool, not a physical entity in themselves). The properties of electric field lines for any charge distribution can be summarized as follows:

1. Field lines must begin on positive charges and terminate on negative charges, or at infinity in the hypothetical case of isolated charges.
2. The number of field lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge.
3. The strength of the field is proportional to the closeness of the field lines—more precisely, it is proportional to the number of lines per unit area perpendicular to the lines.
4. The direction of the electric field is tangent to the field line at any point in space.
5. Field lines can never cross.

The last property means that the field is unique at any point. The field line represents the direction of the field; so if they crossed, the field would have two directions at that location (an impossibility if the field is unique).

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 6.6: Charges and Fields

Section Summary

• Drawings of electric field lines are useful visual tools. The properties of electric field lines for any charge distribution are that:
• Field lines must begin on positive charges and terminate on negative charges, or at infinity in the hypothetical case of isolated charges.
• The number of field lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge.
• The strength of the field is proportional to the closeness of the field lines—more precisely, it is proportional to the number of lines per unit area perpendicular to the lines.
• The direction of the electric field is tangent to the field line at any point in space.
• Field lines can never cross.

Conceptual Questions

13. Compare and contrast the Coulomb force field and the electric field. To do this, make a list of five properties for the Coulomb force field analogous to the five properties listed for electric field lines. Compare each item in your list of Coulomb force field properties with those of the electric field—are they the same or different? (For example, electric field lines cannot cross. Is the same true for Coulomb field lines?)
14. Figure 6.7 shows an electric field extending over three regions, labeled I, II, and III. Answer the following questions. (a) Are there any isolated charges? If so, in what region and what are their signs? (b) Where is the field strongest? (c) Where is it weakest? (d) Where is the field the most uniform?

Problem Exercises

30. (a) Sketch the electric field lines near a point charge [latex]+q[/latex]. (b) Do the same for a point charge [latex]–3.00q[/latex].
31. Sketch the electric field lines a long distance from the charge distributions shown in Figure 6.5 (a) and (b)
32. Figure 6.8 shows the electric field lines near two charges [latex]{q}_{1}[/latex] and [latex]{q}_{2}[/latex]. What is the ratio of their magnitudes? (b) Sketch the electric field lines a long distance from the charges shown in the figure.
    

    

33. Sketch the electric field lines in the vicinity of two opposite charges, where the negative charge is three times greater in magnitude than the positive. (See Figure 6.8 for a similar situation).

Glossary

electric field

a three-dimensional map of the electric force extended out into space from a point charge

electric field lines

a series of lines drawn from a point charge representing the magnitude and direction of force exerted by that charge

vector

a quantity with both magnitude and direction

vector addition

mathematical combination of two or more vectors, including their magnitudes, directions, and positions