Visualizing electric fields around charged objects using line drawings is very useful for visualizing the strength and direction of a field.A vector is a quantity that has magnitude and direction.Electric fields can be visualized as arrows, which have proportional lengths to their magnitudes and are pointed in the right direction.For example, arrows have been extensively used to represent force vectors.

.Diagram 1b shows a continuous representation of the electric field.There are numerous arrows in figure 1b, each representing a force applied to a test charge q of various sizes.In essence, field lines are maps of infinitesimal force vectors.

See figure 1.Here are two equivalent representations of the electric field of a positive charge Q.Arrows showing the magnitude and direction of the electric field.In standard representations, the arrows are replaced by continuous field lines that have the same direction as the electric field at any point.A closer connection between the lines is directly related to the strength of the electric field.In any position, the charge will feel a force in the direction of the field line; this force is proportional to the density of the lines (becoming stronger near the charge, for example).

.According to Figure 2, the magnitude of the electric field for a point charge is k rac [Q][r*2]/, while area is proportional to r2.Using this pictorial representation, where field lines stand for direction and closeness (the number of lines crossing the unit area) refers to strength, all types of fields are considered: electrostatic, gravitational, magnetic, and others.

The figure below shows figure 2.Three point charges and their electric fields.A positive charge is shown.The opposite (b) a negative charge.The negative charge of larger amounts.

Many times, multiple charges are involved.Multiplying multiple charges creates a total electric field which is the sum of their individual electric fields.In the example below, we show how to add the vectors for electric fields.

### Example 1. Adding Electric Fields

To determine the magnitude and direction of the total electric field due to the two point charges, q1 and q2, at the origin of the coordinate system, please see Figure 3.

Diagram 3.At the origin O, the electric fields E1 and E2 contribute to ET.

We add electric fields using the same vector techniques used for other types of vectors since electric fields are vectors (that have magnitude and direction).As a first step, we must assess the electric field arising from each charge at the point of interest, which is the origin of the coordinate system (O).Presuming that there is a positive charge, q, at point O, we can determine the direction of fields E1 and E2.Once those fields are located, the total field can be calculated by vector addition.

We calculate the electric field strength at the origin due to q1 as follows:

E[array][lll]E_1 and=&k rac[q]1 and=&k [R]1*2 are equal to 8.99(imes10*9 Rac[NT]cdot ext[M]*2] [ext[C]*2]

E2 is likewise

Here, four digits have been retained to illustrate that E1 is exactly twice as large as E2.Now, the magnitudes and directions of E1 and E2 are represented by arrows.Figure 3 illustrates the electric field as the force on a positive charge, so both arrows point directly away from the positive charges that generated them.Two times shorter than that for E1, E1's arrow is twice as long as E2's.Pythagorean theorem can be applied to the arrows in this case because they form a right triangle.the magnitude of Etot is

The array starts with E_[ ext[total]]&=&left(E*2_1+E*2_2)

It is headed in that direction

An*[-1]left( rac[E_1][E_2] egin[array][lll] heta&=&

Or 63.44o above the x-axis.

The use of vector components or graphics can be used when the electric field vectors to be added are not perpendicular.An electric field's total electric field can only be found at one point in space in this example.To find the total electric field resulting from these two charges over an entire region, the same procedure must be followed for every point within the region.By using some unifying features noted next, we can avoid the seemingly impossible task (there are an infinite number of points in space).

The figure shows figure 4.q1 and q2 are two positively charged points that produce the electric field in the figure.We calculate the field at representative points, and then draw smooth field lines according to the instructions in the text.

As shown in Figure 4, you can draw the electric field at two separate points by finding the total field at representative points and drawing the electric field lines consistent with those points.Although the electric field from multiple charges is more complex than one from a single charge, some simple features are obviously visible.

In that region, the lines are farther apart, as a result of a weaker field between like charges.Figure 5a illustrates the electric field created by two opposite charges (because the fields from each charge exert opposing forces on any charge placed between them.) Furthermore, at a distance from two like charges, the field is identical to the electric field from a single, larger charge.Figure 5b illustrates the electric field produced by two unlike charges.Charges have a greater effect on the field.Because the fields from each charge point in the same direction, their strengths add up in that part of the region.In many cases, the field of two unlike charges appears weak when viewed from a great distance, because the fields of the individual charges take opposite directions, so their strength is reduced.When two unlike charges are very near each other, their fields appear to be like those of smaller single charges.

Picture 5.Fields created by two negative charges.In many respects, it behaves the same as the field produced by two positive charges, with the difference that the direction of the field is reversed.It is clearly weaker between the charges.Those forces are polarized in that region.(b) A field with opposite charges is shown, which is stronger between the charges.

Electric field lines are used to visualize and analyze electric fields (the lines are pictorial, not physical objects).In summary, the following properties of electric field lines can be summarized for any distribution of charges:

It is unique at every point of time, according to the last property.A field line acts as a representation of the direction of the field; if they crossed, the field would have two directions (an impossibility if the field is unique).

## PhET Explorations: Charges and Fields

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## Section Summary

### Conceptual Questions

This is figure 6.

### Problems & Exercises

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*q*

Diagram 7.An electric field near a pair of charges.

## Glossary

An electric field line represents the magnitude and direction of the force exerted by a point charge

Vectors have both magnitude and direction

A vector addition is a mathematical combination of two or more vectors, including their magnitudes, directions, and positions