Calculate the complete force (magnitude and direction) exerted ~ above a test fee from much more than one chargeDescribe an electric field chart of a positive suggest charge; that a negative point charge through twice the magnitude of positive chargeDraw the electrical field lines in between two points of the exact same charge; in between two clues of opposite charge.

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Drawings utilizing lines to represent electric fields about charged objects are really useful in visualizing ar strength and direction. Due to the fact that the electric field has both magnitude and direction, it is a vector. Choose all vectors, the electrical field deserve to be stood for by an arrow that has length proportional come its magnitude and also that clues in the correct direction. (We have used arrows extensively to represent pressure vectors, because that example.)

Figure 1 shows two photographic representations the the same electric field developed by a positive suggest charge Q. Number 1b reflects the standard depiction using constant lines. Number 1b shows plenty of individual arrows through each arrowhead representing the pressure on a test charge q. Ar lines are basically a map that infinitesimal pressure vectors.

Figure 1. Two identical representations of the electric field as result of a positive charge Q. (a) Arrows representing the electric field’s magnitude and direction. (b) In the typical representation, the arrows are changed by continuous field lines having the very same direction in ~ any point as the electric field. The closeness that the currently is directly related to the toughness of the electrical field. A test charge placed anywhere will feeling a pressure in the direction of the ar line; this pressure will have a toughness proportional come the density of the lines (being higher near the charge, for example).

Note the the electric field is defined for a positive test fee q, so that the field lines suggest away from a optimistic charge and toward a an unfavorable charge. (See figure 2.) The electrical field toughness is specifically proportional to the number of field lines every unit area, since the magnitude of the electrical field for a allude charge is E=k\fracQr^2\\ and area is proportional come r2. This photographic representation, in which ar lines stand for the direction and also their closeness (that is, your areal thickness or the variety of lines crossing a unit area) to represent strength, is offered for every fields: electrostatic, gravitational, magnetic, and others.

Figure 2. The electric field bordering three different allude charges. (a) A optimistic charge. (b) A an adverse charge of equal magnitude. (c) A larger an unfavorable charge.

In many situations, there are multiple charges. The total electric field produced by multiple fees is the vector amount of the individual fields created by every charge. The following instance shows exactly how to include electric field vectors.

### Example 1. Adding Electric Fields

Find the magnitude and direction the the full electric field because of the two point charges, q1 and also q2, at the beginning of the coordinate device as displayed in figure 3.

Figure 3. The electric fields E1 and E2 in ~ the beginning O include to Etot.

Strategy

Since the electric field is a vector (having magnitude and direction), we include electric fields with the same vector approaches used because that other types of vectors. We an initial must uncover the electric field as result of each fee at the allude of interest, i beg your pardon is the origin of the coordinate mechanism (O) in this instance. Us pretend the there is a confident test charge, q, at suggest O, which permits us to determine the direction the the fields E1 and E2. When those fields are found, the complete field deserve to be figured out using vector addition.

Solution

The electrical field stamin at the origin as result of q1 is labeling E1 and also is calculated:

\beginarraylllE_1&=&k\fracq_1r_1^2=\left(8.99\times10^9\frac\textN\cdot\textm^2\textC^2\right)\frac\left(5.00\times10^-9\text C\right)\left(2.00\times10^-2\text m\right)^2\\E_1&=&1.124\times10^5\text N/C\endarray\\

Similarly, E2 is

\beginarraylllE_2&=&k\fracq_2r_2^2=\left(8.99\times10^9\frac\textN\cdot\textm^2\textC^2\right)\frac\left(10.0\times10^-9\text C\right)\left(4.00\times10^-2\text m\right)^2\\E_2&=&0.5619\times10^5\text N/C\endarray\\

Four digits have been kept in this equipment to illustrate that E1 is exactly twice the magnitude of E2. Now arrows are drawn to stand for the magnitudes and also directions of E1 and also E2. (See number 3.) The direction of the electrical field is that of the force on a optimistic charge therefore both arrows point directly far from the hopeful charges that develop them. The arrow for E1 is exactly twice the length of that for E2. The arrows type a appropriate triangle in this case and can be added using the Pythagorean theorem. The magnitude of the total field Etot is

\beginarraylllE_\texttot&=&\left(E^2_1+E^2_2\right)^1/2\\E_\texttot&=&\\left(1.124\times10^5\text N/C\right)^2+\left(0.5619\times10^5\text N/C\right)^2\^1/2\\E_\texttot&=&1.26\times10^5\text N/C\endarray\\

The direction is

\beginarraylll\theta&=&\tan^-1\left(\fracE_1E_2\right)\\\text &=&\tan^-1\left(\frac1.124\times10^5\text N/C0.5619\times10^5\text N/C\right)\\\text &=&63.4^\circ\endarray\\

or 63.4º above the x-axis.

Discussion

In situations where the electric field vectors to be included are not perpendicular, vector components or graphical techniques can be used. The complete electric field discovered in this example is the total electric ar at just one point in space. To discover the total electric field as result of these two charges over an entire region, the same an approach must be recurring for each allude in the region. This impossibly prolonged task (there room an infinite number of points in space) can be avoided by calculating the complete field in ~ representative points and using several of the unifying features listed next.

Figure 4. 2 positive point charges q1 and q2 develop the resultant electrical field shown. The ar is calculated in ~ representative points and also then smooth ar lines attracted following the rules outlined in the text.

Figure 4 shows exactly how the electrical field indigenous two suggest charges deserve to be drawn by finding the complete field at representative points and drawing electrical field lines consistent with those points. If the electric fields native multiple charges are more facility than those of solitary charges, some straightforward features are easily noticed.

For example, the field is weaker in between like charges, as presented by the lines gift farther personally in the region. (This is because the areas from each fee exert opposing pressures on any charge placed in between them.) (See number 4 and number 5a.) Furthermore, in ~ a good distance native two choose charges, the ar becomes similar to the ar from a single, bigger charge.Figure 5b reflects the electric field of two unlike charges. The field is stronger in between the charges. In that region, the areas from each charge space in the same direction, and so their toughness add. The field of two unlike fees is weak at big distances, due to the fact that the areas of the separation, personal, instance charges room in the contrary directions and also so their staminas subtract. In ~ very big distances, the ar of two unlike fees looks prefer that the a smaller solitary charge.

Figure 5. (a) Two an unfavorable charges produce the fields shown. The is very similar to the field produced by two hopeful charges, except that the directions space reversed. The field is clearly weaker in between the charges. The individual forces on a test fee in that an ar are in opposite directions. (b) two opposite charges develop the ar shown, which is stronger in the an ar between the charges.

We use electrical field lines come visualize and also analyze electric fields (the lines space a photographic tool, no a physical entity in themselves). The nature of electrical field present for any kind of charge distribution can be summarized as follows:

Field currently must begin on optimistic charges and also terminate on an unfavorable charges, or in ~ infinity in the hypothetical case of secluded charges.The variety of field currently leaving a confident charge or beginning a negative charge is proportional come the size of the charge.The toughness of the ar is proportional come the closeness the the field lines—more precisely, it is proportional to the number of lines every unit area perpendicular come the lines.The direction the the electric field is tangent come the ar line at any suggest in space.Field lines have the right to never cross.

The last property way that the ar is distinct at any kind of point. The ar line represents the direction the the field; for this reason if castle crossed, the field would have two directions at that location (an impossibility if the field is unique).

## PhET Explorations: Charges and Fields

Move suggest charges around on the play field and then see the electrical 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 nature of electrical field present for any kind of charge circulation are that:Field lines must begin on hopeful charges and terminate on an adverse charges, or in ~ infinity in the hypothetical situation of isolated charges.The variety of field lines leaving a positive charge or entering a negative charge is proportional come the size of the charge.The stamin of the field is proportional come the closeness of the field lines—more precisely, it is proportional to the number of lines every 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 deserve to never cross.

### Conceptual Questions

Compare and also contrast the Coulomb force field and the electric field. To perform this, make a list of five properties because that the Coulomb force field analogous to the five properties provided for electric field lines. Compare each item in your list that Coulomb force ar properties with those the the electric field—are they the same or different? (For example, electrical field lines can not cross. Is the exact same true because that Coulomb field lines?) mirrors an electrical field expanding over three regions, labeling I, II, and also III. Answer the complying with questions. (a) space there any kind of isolated charges? If so, in what an ar and what are their signs? (b) wherein is the ar strongest? (c) wherein is that weakest? (d) where is the ar the many uniform?

Figure 6.

### Problems & Exercises

(a) sketch the electric field lines close to a point charge +q. (b) do the exact same for a allude charge −3.00q.Sketch the electric field lines a lengthy distance indigenous the fee distributions displayed in number 5a and also 5b.Figure 8 shows the electric field lines near two fees q_1 and q_2 . What is the ratio of your magnitudes? (b) lay out the electrical field lines a long distance native the charges presented in the figure.

Figure 7. The electrical field close to two charges.

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Sketch the electrical field lines in the vicinity of 2 opposite charges, wherein the negative charge is 3 times greater in magnitude 보다 the positive. (See number 7 because that a similar situation).

## Glossary

electric field: a three-dimensional map of the electric force extended out into room from a allude charge

electric field lines: a collection of lines attracted from a point charge representing the magnitude and direction of force exerted by the charge

vector: a quantity with both magnitude and also direction

vector addition: mathematical mix of 2 or more vectors, consisting of their magnitudes, directions, and positions