## Forces Between Multiple Charges – Principle of Superposition

### Principle of superposition

*Force on any charge due to a number of other charges is the vector
sum of all the forces on that charge due to the other charges, taken one at a
time. The individual forces are unaffected due to the presence of other charges*.

### Explanation

Note: Here we are using bold letters to represent vectors.

Consider a system of three charges q_{1}, q_{2}, and q_{3} with position vectors **r _{1}**,

**r**, and

_{2}**r**respectively as shown in Fig. 1. The forces on q

_{3}_{1}due to q

_{2}and q

_{3}are given by

**F**and

_{12}**F**respectively. The resultant force

_{13}**F**

_{1}on q

_{1}is the vector sum of

**F**and

_{12}**F**. The vector sum can be obtained by parallelogram law of vector addition.

_{13}By Coulomb’s law of electrostatic force, force on *q*_{1} due to *q*_{2 }in free space or vacuum is given by

Similarly, the force on *q*_{1} due to *q*_{3}, is given by

The resultant force **F**_{1} on q_{1} due to q_{2} and q_{3} is the vector sum of these two forces. It is given by

Like that if there are n charges and the force on charge q_{1} due to all other charges is given by

### Numerical

Q. Calculate force on any charge q, if three equal and similar charge +q is placed at the edges of an equilateral triangle.

Solution:

Force on one charge due to other two charges are given by Coulomb’s law of electrostatic force. Since charges are same which is q, and distance between them are also same as they are placed at the sides of equilateral triangle. So, forces are same in magnitude and is given by,

Direction of forces are as shown in Fig. 2.

The resultant force is the vector sum of these two forces, the magnitude of the net force is given by the equation,

We also know, if the vectors with equal magnitude makes an angle with each other, then the resultant vector will bisect the angle between them. That is the resultant vector makes an angle from both vectors.

Here angle between the vectors is 60^{0}, therefore, the resultant force makes an angle 60/2 = 30^{0} from both forces.