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# 2 Calculating the electric force between three or more charges

So far, this course has only considered Coulomb’s law for a pair of point charges. The extension of this law to a collection of many particles requires vector addition. If is the total electrostatic force on a charge , this is calculated from the vector sum of the electrostatic forces that it experiences due to each of the other charges. Mathematically, this is written as

where is the electrostatic force on particle due to particle and the sum runs over all the particles that exert an appreciable electrostatic force on particle . Since the electrostatic force between each pair of charges obeys Coulomb’s law, the total electrostatic force on charge is written as follows.

## Coulomb’s law for multiple charges

Equation label:(11)

- Why are terms with excluded from Equation 11?

- Because a point-like charged particle cannot exert a force on itself.

Now consider a small number of static point charges that are not arranged in a straight line. To work out the force on a given charge, you can begin by representing all the vectors in component form, as explained in the following box. Then you can use the rules of vector algebra to combine them according to the recipe given in Equation 11.

## Cartesian components of a vector

It is often helpful to describe a vector in terms of its components along three standard directions. To do this, you can use Cartesian coordinates. This is a set of three mutually perpendicular axes (, and ) pointing outwards from an origin. The unit vectors pointing in the directions of these axes are denoted by , and .

It is conventional to use a right-handed coordinate system, as described by the right-hand rule. Start by pointing the fingers of your right hand in the direction of the -axis (indicated by the black dashed line in Figure 4). Then bend your fingers round to point in the direction of the -axis, so that your hand is in the position shown in the figure. You might need to rotate your wrist to do this. Now your outstretched thumb points along the -axis.

The crucial idea is that any vector can be split into a sum of three vectors that are aligned with each axis, as shown in Figure 4. It follows that any vector (in this case, a force ) can be expressed as

Equation label:(12)

The scalar quantities , and are the Cartesian components of the vector but they are usually just called its components. The vector components , and are all positive in Figure 4 but, in general, vector components may be positive, negative or zero.

Figure 4Splitting the vector into the sum of three vectors: , , and . You can work out the relative orientations of the Cartesian axes using the right-hand rule, as explained in the text.

If you know the magnitude and direction of a vector, you can use trigonometry to find its components. For example, Figure 4 shows

where is the magnitude of the force and is the angle between and the -axis. Similarly, if you know a vector’s components then you can use Pythagoras’ theorem to find its magnitude:

Equation label:(13)

The vector operations introduced earlier in this chapter all have simple interpretations in terms of components. For example, if the position vectors of points 1 and 2 are

then the displacement vector of point 1 from point 2 is

Equation label:(14)

Vector equations have the great advantage of brevity, but numerical calculations are usually carried out using components.

Now complete Exercise 1 where you will use the vector form of Coulomb’s law to calculate the vector components of the electrostatic force on a charge due to two nearby charges.

## Exercise 1

Two charges, and , where is positive, are stationary at points and , as shown in Figure 5.

Figure 5The positions of three stationary charges in the -plane.

Find the electrostatic force on a charge placed at the origin . Evaluate the magnitude of this force and specify its direction as a unit vector in Cartesian coordinates.

### Discussion

All the charges lie in the -plane, so you can ignore the -coordinates.

The electrostatic force on charge at the origin is given by the vector sum

This force has magnitude

and is in the direction of the unit vector

As a quick check, this is consistent with the charge being attracted towards the charge on the -axis and repelled from the charge on the -axis.

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OpenLearn - Electromagnetism: testing Coulomb’s law
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