The analysis that we have done for two particles can be extended to an arbitrary number of particles; we simply repeat the analysis, two charges at a time. Specifically, we ask the question: Given *N* charges (which we refer to as source charge), what is the net electric force that they exert on some other point charge (which we call the test charge)? Note that we use these terms because we can think of the test charge being used to test the strength of the force provided by the source charges.

Like all forces that we have seen up to now, the net electric force on our test charge is simply the vector sum of each individual electric force exerted on it by each of the individual test charges. Thus, we can calculate the net force on the test charge *Q* by calculating the force on it from each source charge, taken one at a time, and then adding all those forces together (as vectors). This ability to simply add up individual forces in this way is referred to as the principle of superposition, and is one of the more important features of the electric force. In mathematical form, this becomes

### Note:

In this expression, *Q* represents the charge of the particle that is experiencing the electric force *N* source charges, and the vectors *i*th charge to the position of *Q*. Each of the *N* unit vectors points directly from its associated source charge toward the test charge. All of this is depicted in Figure 3. Please note that there is no physical difference between *Q* and *Q* being the charge we are determining the force on.

(Note that the force vector

There is a complication, however. Just as the source charges each exert a force on the test charge, so too (by Newton’s third law) does the test charge exert an equal and opposite force on each of the source charges. As a consequence, each source charge would change position. However, by Equation 8, the force on the test charge is a function of position; thus, as the positions of the source charges change, the net force on the test charge necessarily changes, which changes the force, which again changes the positions. Thus, the entire mathematical analysis quickly becomes intractable. Later, we will learn techniques for handling this situation, but for now, we make the simplifying assumption that the source charges are fixed in place somehow, so that their positions are constant in time. (The test charge is allowed to move.) With this restriction in place, the analysis of charges is known as electrostatics, where “statics” refers to the constant (that is, static) positions of the source charges and the force is referred to as an electrostatic force.

### Example 2

**The Net Force from Two Source Charges**

Three different, small charged objects are placed as shown in Figure 4. The charges

**Strategy**

We use Coulomb’s law again. The way the question is phrased indicates that

**Solution**

We have two source charges

We can’t add these forces directly because they don’t point in the same direction: *x*-direction, while *y*-direction. The net force is obtained from applying the Pythagorean theorem to its *x*- and *y*-components:

where

and

We find that

at an angle of

that is, *x*-axis, as shown in the diagram.

**Significance**

Notice that when we substituted the numerical values of the charges, we did not include the negative sign of either

It’s also worth noting that the only new concept in this example is how to calculate the electric forces; everything else (getting the net force from its components, breaking the forces into their components, finding the direction of the net force) is the same as force problems you have done earlier.

### Note:

#### Exercise 2

**Check Your Understanding** What would be different if

##### Solution

The net force would point *x*-axis.