Force Between Multiple Charges

Bookmark added to your notes.
View Notes

Force Between Multiple Charges Superposition Principle

In nature, every particle exerts some kind of force on the other particles. This is true from a subatomic to a celestial level. The forces exerted by objects on each other vary in range, magnitude, and nature. The nature of these forces depends upon various physical phenomena and the value of some universal constants that define the current state of our universe. Here, we will discuss the electrostatic force between multiple charged particles. We will discuss the nature of this force and the appropriate way of calculating it using coulomb's law force between multiple charges and the principle of superposition.

Coulomb's Law for Forces Between Multiple Charges

Before we try to calculate Coulomb's law for forces between multiple charges, we need to understand Coulomb's force between two charged particles. Coulomb’s law or Coulomb’s inverse square law was discovered in 1785 by French physicist Charles-Augustin de Coulomb. The experimentally proven law quantifies the force exerted by a static charged particle on another static charged particle.

Assume two static charged particles with a charge of ‘q1’ and ‘q2’ respectively. The force exerted by one particle on the other, if they are separated by a distance of ‘r’ between their centers is given by:

Fe = ke q1q2/r2


‘Fe’ is the force between the two particles,

And ‘Ke’ is the Coulomb’s constant, with a value of 9.987 * 109 N.m2.C2.

To calculate the coulomb’s law forces between multiple charges, we use the principle of superposition.

Note: As a general statement, coulomb's law force between multiple charges is always exerted radially over a straight line joining the centers of the charged particles.

What is the Force Between the Multiple Charges by the Principle of Superposition?

The superposition principle is a mathematical truth that allows us to calculate seemingly complex linear mathematical equations by dividing them into smaller segments. Daniel Bernoulli proposed the principle of superposition in 1753.

The superposition principle states that the net response of two or more stimuli is the linear sum of the response caused by the individual stimulus. 

For a linear function F(x), according to the principle of superposition,

F( x1 + x2 ) = F( x1 ) + F( x2), law of additivity

Translating the principle into Coulomb's law force between multiple charges, we say that the total electrostatic force applied on a static charged particle by two or more static charged particles is equal to the scalar sum of individual forces applied on that particle by those individual particles.

Mathematically, the forces between multiple charges using the principle of superposition is given by:

FTotal= F1 + F2 + F3 + … Fn

Where Fnet is the total electrostatic force on a particle in a system of n particles and F1, F2, F3 … Fn are the forces applied by particle 1, 2, 3, … n, respectively.

The superposition principle is a very powerful and useful tool.

Fun Fact

Daniel Bernoulli, who discovered the superposition principle,  was an amazing physicist revered by all his peers so much that his own father became jealous of him and tried to steal his masterpiece Hydrodynamica. ‘Hydrodynamica’ was completed in 1733. Daniel’s father plagiarised the book and called it ‘Hydraulica’ and claimed to have completed it in 1932.

Solved Example

How to Calculate Force Between Multiple Charges by Superposition Principle?

(Image to be added soon)

Let us assume that we have a system of 4 particles, q1,q2,q3, and q4 and we have to calculate Coulomb’s law forces between multiple charges for these four particles.

We shall calculate the force on q1 due to d2,d3, and d4.

Clearly, this system is not made of only two particles but multiple particles. So what do we do?

To calculate the force between multiple charges, we shall use the principle of superposition.

First, we shall calculate the individual forces on q1 due to q2,q3, and q4.

F12 = ( Ke * q1 *q2 )/ r212

Where, q1 and q2 are the charges of q1 and q2 respectively and r12 is the distance between them.


F13 = ( Ke * q1 * q3 )/ (r13) 2 ,

F14 = ( Ke * q1 * q4 )/( r14) 2 

Now we know from forces between multiple charges superposition principle, that the total force F on q1 due to q2, q3, and q4 is

F = F12 + F13 + F14


F12 = ( Ke * q1 * q1 )/ (r12)2,

F13 = ( Ke * q1 * q3 )/( r13)2,

F14 = ( Ke * q1 * q4 )/( r14)2 

In general, for a system of n particles, with q1, q2, q3, q4,..... qn charged particles, the force F on q1 due to all the other particles in the system will be:

F = F12 + F13 + F14 + …. F1n

F12, F13, F14 … F1n are the forces on particle q1 due to q2,q3,q4… qn respectively.

FAQ (Frequently Asked Questions)

1. Does the Sign of the Charge Affect the Magnitude of the Force Between Multiple Objects?

Coulomb’s law is an experimental law and it is unaffected by an external electrical field. The net force on a charged object may change due to the forces between multiple electric fields due to the system of charges. No, the sign of the charge does not affect the magnitude of the force.  The force between the multiple charges principle of superposition is calculated without the use of signs. This means that the magnitude is independent of it. It’s just that the opposite charges attract each other and similar charges repel each other. But the magnitude of the force does not change. Only the direction of the force changes.

2. What Happens to the Coulomb’s Law in an External Electrical Field?

Coulomb’s law is one of the most essential and imperative concepts in the present academic curricular. The electrostatic force due to the charged particles always remains the same. From the principle of superposition, we know that the net force on a charged particle in a system of particles is the linear sum of the individual forces from each particle in the system. If we consider the external field in the system of particles as another charged particle, then, according to the force between multiple charges principle of superposition, the net force can still be calculated without altering Coulomb’s Law.