Calculate the full force (magnitude and also direction) exerted ~ above a test fee from an ext than one chargeDescribe an electrical field chart of a positive allude charge; the a an adverse point charge through twice the magnitude of optimistic chargeDraw the electric field lines between two points of the very same charge; in between two points of the contrary charge.

You are watching: Determine the magnitude of the electric field at point 1 in (figure 1).

Drawings utilizing lines to represent electric fields around charged objects are very useful in visualizing ar strength and also direction. Since the electrical field has actually both magnitude and also direction, the is a vector. Favor all vectors, the electrical field can be represented by an arrow that has length proportional come its magnitude and also that clues in the correct direction. (We have actually used arrows broadly to represent force vectors, for example.)

Figure 1 shows two photographic representations that the same electrical field produced by a positive point charge Q. Figure 1b reflects the standard representation using continuous lines. Figure 1b shows plenty of individual arrows with each arrow representing the pressure on a test charge q. Ar lines are essentially a map the infinitesimal force vectors.

Figure 1. Two tantamount representations of the electrical field due to a positive charge Q. (a) Arrows representing the electrical field’s magnitude and direction. (b) In the typical representation, the arrows are replaced by continuous field lines having actually the very same direction at any allude as the electric field. The closeness of the lines is straight related come the strength of the electrical field. A test charge placed everywhere will feel a pressure in the direction that the ar line; this force will have actually a toughness proportional come the thickness of the currently (being higher near the charge, for example).

Note the the electric field is characterized for a hopeful test fee q, so that the ar lines point away indigenous a confident charge and toward a an adverse charge. (See number 2.) The electrical field stamin is precisely proportional come the variety of field lines per unit area, due to the fact that the magnitude of the electric field for a point charge is E=kfracr^2\ and area is proportional to r2. This pictorial representation, in which ar lines stand for the direction and their closeness (that is, your areal thickness or the number of lines crossing a unit area) to represent strength, is supplied for all fields: electrostatic, gravitational, magnetic, and others.

Figure 2. The electric field neighboring three different suggest charges. (a) A optimistic charge. (b) A an adverse charge of same magnitude. (c) A larger negative charge.

In countless situations, there space multiple charges. The full electric field created by multiple charges is the vector sum of the individual fields developed by each charge. The following instance shows how to include electric field vectors.

### Example 1. Adding Electric Fields

Find the magnitude and also direction that the complete electric field due to the two point charges, q1 and also q2, at the origin of the coordinate mechanism as shown in number 3.

Figure 3. The electrical fields E1 and E2 at the origin O include to Etot.

Strategy

Since the electrical field is a vector (having magnitude and direction), we add electric areas with the same vector techniques used for other varieties of vectors. We first must uncover the electrical field as result of each charge at the allude of interest, which is the origin of the coordinate device (O) in this instance. Us pretend the there is a positive test charge, q, at suggest O, which enables us to recognize the direction of the fields E1 and E2. When those fields are found, the total field can be determined using vector addition.

Solution

The electric field stamin at the origin due to q1 is labeling E1 and is calculated:

eginarraylllE_1&=&kfracq_1r_1^2=left(8.99 imes10^9frac extNcdot extm^2 extC^2 ight)fracleft(5.00 imes10^-9 ext C ight)left(2.00 imes10^-2 ext m ight)^2\E_1&=&1.124 imes10^5 ext N/Cendarray\

Similarly, E2 is

eginarraylllE_2&=&kfracq_2r_2^2=left(8.99 imes10^9frac extNcdot extm^2 extC^2 ight)fracleft(10.0 imes10^-9 ext C ight)left(4.00 imes10^-2 ext m ight)^2\E_2&=&0.5619 imes10^5 ext N/Cendarray\

Four digits have been maintained in this equipment to show that E1 is exactly twice the magnitude of E2. Currently arrows are attracted to stand for the magnitudes and directions that E1 and also E2. (See number 3.) The direction that the electrical field is that of the force on a confident charge therefore both arrows point directly far from the confident charges that develop them. The arrow for E1 is exactly twice the size of the for E2. The arrows kind a best triangle in this case and also can be included using the Pythagorean theorem. The magnitude of the total field Etot is

eginarraylllE_ exttot&=&left(E^2_1+E^2_2 ight)^1/2\E_ exttot&=&\left(1.124 imes10^5 ext N/C ight)^2+left(0.5619 imes10^5 ext N/C ight)^2^1/2\E_ exttot&=&1.26 imes10^5 ext N/Cendarray\

The direction is

eginarraylll heta&=& an^-1left(fracE_1E_2 ight)\ ext &=& an^-1left(frac1.124 imes10^5 ext N/C0.5619 imes10^5 ext N/C ight)\ ext &=&63.4^circendarray\

or 63.4º above the x-axis.

Discussion

In cases where the electrical field vectors come be added are not perpendicular, vector contents or graphical techniques have the right to be used. The full electric field found in this instance is the full electric field at only one suggest in space. To uncover the total electric field as result of these 2 charges over an entire region, the same an approach must be recurring for each allude in the region. This impossibly an extensive task (there room an infinite number of points in space) can be avoided by calculating the total field at representative points and using few of the unifying features noted next.

Figure 4. 2 positive allude charges q1 and also q2 produce the resultant electric field shown. The ar is calculated in ~ representative points and also then smooth ar lines attracted following the rule outlined in the text.

Figure 4 shows exactly how the electrical field native two allude charges deserve to be drawn by recognize the complete field at representative points and also drawing electrical field lines continuous with those points. If the electric fields indigenous multiple charges room more complex than those of single charges, some straightforward features are conveniently noticed.

For example, the ar is weaker in between like charges, as shown by the lines gift farther personal in the region. (This is since the fields from each fee exert opposing pressures on any kind of charge placed between them.) (See figure 4 and figure 5a.) Furthermore, in ~ a great distance from two favor charges, the field becomes similar to the ar from a single, bigger charge.Figure 5b shows the electrical field of 2 unlike charges. The field is stronger in between the charges. In that region, the areas from every charge space in the exact same direction, and so their toughness add. The field of two unlike charges is weak at huge distances, because the fields of the separation, personal, instance charges space in the contrary directions and also so their toughness subtract. In ~ very large distances, the field of two unlike fees looks choose that of a smaller single charge.

Figure 5. (a) Two negative charges produce the fields shown. That is very similar to the field produced by two confident charges, except that the directions are reversed. The ar is clearly weaker in between the charges. The individual forces on a test fee in that region are in the opposite directions. (b) two opposite charges produce the ar shown, i beg your pardon is stronger in the region between the charges.

We use electrical field lines to visualize and analyze electric fields (the lines room a photographic tool, not a physical reality in themselves). The properties of electrical field currently for any charge circulation can be summarized together follows:

Field present must begin on hopeful charges and terminate on negative charges, or at infinity in the hypothetical situation of diverted charges.The variety of field lines leaving a confident charge or beginning a negative charge is proportional come the magnitude of the charge.The stamin of the field is proportional to the closeness of the ar lines—more precisely, it is proportional to the number of lines every unit area perpendicular come the lines.The direction of the electrical field is tangent come the field line in ~ any point in space.Field lines deserve to never cross.

The critical property means that the ar is distinct at any type of point. The field line to represent the direction of the field; therefore if they crossed, the ar would have two direction at that ar (an impossibility if the ar is unique).

## PhET Explorations: Charges and also Fields

Move point charges approximately on the play field and then see the electric field, voltages, equipotential lines, and more. It’s colorful, it’s dynamic, it’s free. Click to operation the simulation.

## Section Summary

Drawings of electrical field present are advantageous visual tools. The properties of electrical field lines for any kind of charge circulation are that:Field currently must begin on positive charges and terminate on an adverse charges, or at infinity in the hypothetical case of isolated charges.The number of field present leaving a hopeful charge or start a negative charge is proportional to the magnitude of the charge.The stamin of the ar is proportional to the closeness the the ar lines—more precisely, the is proportional come the number of lines every unit area perpendicular to the lines.The direction the the electric field is tangent to the ar line at any point in space.Field lines have the right to never cross.

### Conceptual Questions

Compare and contrast the Coulomb force field and also the electric field. To do this, make a perform of five properties because that the Coulomb force field analogous come the 5 properties noted for electric field lines. To compare each item in her list that Coulomb force ar properties v those the the electric field—are they the same or different? (For example, electrical field lines cannot cross. Is the very same true for Coulomb ar lines?) reflects an electrical field extending over three regions, labeling I, II, and also III. Price the following questions. (a) room there any isolated charges? If so, in what an ar and what room their signs? (b) whereby is the ar strongest? (c) where is the weakest? (d) whereby is the field the most uniform?

Figure 6.

### Problems & Exercises

(a) sketch the electric field lines near a suggest charge +q. (b) execute the very same for a suggest charge −3.00q.Sketch the electric field lines a lengthy distance from the charge distributions displayed in number 5a and also 5b.Figure 8 shows the electrical field lines close to two fees q_1 and also q_2 . What is the proportion of their magnitudes? (b) sketch the electrical field present a lengthy distance from the charges displayed in the figure.

Figure 7. The electric field near two charges.

See more: Give Me Food And I Will Live Riddle, Give Me Food, And I Will Live What Am I

Sketch the electric field present in the vicinity of two opposite charges, whereby the an unfavorable charge is three times better in magnitude 보다 the positive. (See number 7 for a comparable situation).

## Glossary

electric field: a three-dimensional map the the electrical force expanded out into room from a point charge

electric ar lines: a series of lines drawn from a allude 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 2 or more vectors, consisting of their magnitudes, directions, and also positions