### Electric Field

When Coulomb's law is used for calculating the net force produced by a distribution of charges acting on a single charge, , any variation of these charges requires an entire recalculation of the results.

 On the other hand, in the case of the gravitational attraction over masses, locating a mass near the surface of the Earth, implies that a force acts over that object. That force can only be observed when tested with a mass. In fact, the drawings on the left illustrate the process of mapping the gravitational field near the surface of the Earth. In the first drawing of the sequence, the area is shown where the field will be mapped. The second drawing represents a targeted point where, presently, no force is acting. The third drawing shows what happens when a small mass is placed at the targeted point. i.e., a force acts on the mass. This capacity for a force to appear in space, when properly tested, is called a vector field. Following this sequence, the next drawing (fourth) represents the gravitational field at that location. The vector represents the field at the point and the line represents the so called field line. If a mass is released at such a point, the mass will accelerate in the direction indicated by the direction of the force (arrow).In order to obtain an additional understanding of the field, the next set of drawings shows the testing of the field at other targeted points. In this set, it is illustrated that the length of the vector representing the field is constant and independent of the height at which the field is tested (as long as the test is near the surface of the Earth). Thus, the gravitational field near the surface of the Earth is constant. The last of the drawings shows three field lines. These lines are parallel to each other because the field lines show the paths followed by the testing object (mass) when released in the field at the different locations. Therefore, a constant field is represented by field lines that are parallel to each other. In addition, notice that for the gravitational field, the field lines never cross each other.

For the case of forces among charges, the concept of field can be equally used to achieve a visualization of the electric field. In addition,  simplifications of the calculations of  the net Coulomb's force acting on a charge and due to a charge distribution can also be obtained.

### Definition of Electric Field

In the case of electric fields, charges can be positive or negative; therefore, for testing the field, one type of charge must be selected as a test charge. By definition, a test charge is defined to be always positive. At difference with the case of the earth gravitational field, the electric interaction, even between small charges, can be very strong. Thus, a test charge is supposed to be sufficiently small such that it does not disturb the actual electric field of the distribution of charges being studied. Then, a test charge is a very small positive charge. In these notes, a test charge will be represented by .

#### Electric Field of a Point Charge

 The electric field at a point P is defined as the ratio between the Coulomb force exerted on a test charge and the magnitude of the test charge, The direction of the arrow accounts for the force on the test charge being repulsive when the charge producing the field is positive or attractive when the field is produced by a negative charge.

 The applet on the left shows the mapping of the electric field of a positive charge. In the applet a test charge is placed at different points near the charge in order to test the field. The arrow represents the force acting on the charge at the different locations. This vector also can represent the direction in which the test charge will move in case it is released at that point. The lines together with the arrows represent the field lines associated with the point charge. In this case, the field lines are not parallel to each other reflecting the variation of the field with the distance to the charge. In fact, the magnitude of the electric field of a point charge is and the field line directions are away from the positive charge.

In the case of a negative charge, the electric field is like the electric field of a positive charge but the direction of the field lines are toward the charge. Thus, electric field lines start in positive charges and end in negative charges.

 1) J Q 33
 If the electric charge q, of the Figure, is positive, the electric field line at “a” points

 N a) East b) North c) West d) South e) None of the above.

 2) J Q 34
 In reference with the Figure, the intensity of the electric field produced by the charge q at point “b” has

 a) the same strength as the electric field at point “d” and greater strength as the electric fields at points “c” and “a”. b) the same direction as the electric field at point “d” N c) the same strength as the electric field at point “d” and smaller strength as the electric fields at points “c” and “a”. d) the same strength and direction as the electric fields at points “c” and “a”. e) None of the above.

 3) J Q 35
 If the electric charge q, of the Figure, is negative, the electric field line at “a” points

 a) East b) North N c) West d) South e) None of the above.

#### Electric Field of Several Charges

The principle of superposition is also valid for electric fields. Since the electric field is obtained by dividing the Coulomb's force acting on the test charge by the value of the test charge. The same definition can be applied to the net electric force acting on the charge.

 The drawing shows a test charge at a point P in the field of six point charges. The electric field of the six charges is given by The field lines of these charges are not presented in the drawing.

The electric field of two charges is represented in the following drawing:

 Electric field of two charges, the one on the left is positive and the one on the right is negative. Electric field of two positive charges.

Looking at the electric field of two charges, one positive and the other negative, it can be seeing that a straight field lines is at the center of the configuration. Thus, parallel field lines could be obtained if two chains of such charges are properly distributed.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com