# 1.1 Coulomb’s law in vector form

Equation 4 is adequate for describing the interaction of two charges, but it cannot handle three or more charges that are not arranged in a straight line. Before considering a more general representation of Coulomb’s law, you need to be familiar with vector addition and displacement vectors.

## Adding and subtracting vectors

Suppose that a single particle simultaneously feels two different forces, and . It responds just as if a single force, , had been applied to it. This is called the vector sum of the individual forces.

The geometric rule for adding two vectors is shown in Figure 2. Arrows representing the vectors are drawn with the head of the first arrow, , meeting the tail of the second arrow, . The arrow joining the tail of to the head of then represents the vector sum . This is called the triangle rule. Any number of vectors can be added by repeating the application of this rule.

Vector subtraction is defined by multiplying by a negative scalar and using vector addition. The vector is interpreted as the sum of and .

Figure 2The triangle rule for vector addition.

An important use of vector subtraction is in describing the displacement of one point from another.

## Working with displacement vectors

Figure 3 shows two vectors and whose arrows start at the origin O and end at charges and . These vectors are called the position vectors of and . A position vector has dimensions of length, where the SI unit of length is the metre (m).

Figure 3The vectors and define the positions of the point charges and with respect to the origin O. The displacement vector points from to , and is parallel to the unit vector .

The figure also shows , which is the displacement vector of from . Using the triangle rule:

Equation label:(5)

which rearranges to

Equation label:(6)

Using the unit vector , this becomes

Equation label:(7)

This notation is convenient because the indices 1 and 2 are in the same order on both sides of the equation. However, remember that the displacement is from to . The left-hand index labels the end-point and the right-hand index labels the start-point.

Returning now to the discussion of Coulomb’s law for the force between a pair of charged particles, suppose that charges and are at positions and . The displacement vector of from makes it possible to express Coulomb’s law as:

Equation label:(8)

The left-hand side of this vector equation is the electrostatic force on charge 1 due to charge 2. The force on charge 2 due to charge 1 is written as . The order of indices matters here because these two forces point in opposite directions.

The right-hand side of the equation is the product of the scalar factor and the unit vector . The unit vector ensures that the force points in the correct direction. To see how this works, suppose that both charges in Figure 3 are positive. Since is positive, the unit vector is multiplied by a positive quantity, and the force on charge 1 points in the direction of . This corresponds to a repulsion away from charge 2, as required for charges of the same sign.

It is conventional to write the constant as

where C N m (to 3 significant figures) is called the permittivity of free space. The definition of leaves you with the standard vector form of Coulomb’s law for the electrostatic force between two charges.

## Coulomb’s law for two charges

Equation label:(9)

- Using the definition of (Equation 7), how can you write Equation 9 without a unit vector?

- Using Equation 7 and noting that , the vector form of Coulomb’s law for two charges becomes

  Equation label:(10)

  You may find that working with this form of Coulomb’s law speeds up some calculations.

OpenLearn - Electromagnetism: testing Coulomb’s law
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