Forces Between Multiple Charges

Coulomb’s Law, sometimes known as Coulomb’s Inverse Square Law, was discovered in 1785 by Charles-Augustin de Coulomb, a French physicist. This scientifically proven law quantifies the forces applied by a static charged particle on some other static charged particle.

Assume two particles which are charged & static with energies of ‘q1’ and ‘q2’ respectively. The force applied by one particle over the other is provided as follows, when the two particles are divided by a distance of ‘r’ across their centres.

F = Ke q1q2/r2

Electrostatic Force Between Two Point Charges

• According to this law, force between multiple charges experiment can be understood with the electrostatic force between two point charges q1 and q2 is as follows. This is also the force between multiple charges equations.

  • directly proportional to the product of their magnitude of charges

Feq1q2…………(1)

• inversely proportional to the square of the distance between two point charges

F1r2……………(2)

• By equations (1) and (2), we can write,

Fq1q2r2

F=140q1q2r2

• The force is along the line joining the point charges, where:0=permittivity of free space=8.8510-12C2N-m2

140 =9109N-m2C2

• If both point charges are surrounded by a medium of dielectric constant or relative permeability denoted by K or r, then the electrostatic force becomes:

F=140rq1q2r2

     =14mq1q2r2, where m=0r is the permittivity of the medium.

• The electrostatic force is an inverse square force; therefore, it is conservative in nature.

Coulomb’s Law in Vector Form:

The vector form of Coulomb’s law is given as:

Where the relative position of q2 with respect to q1 is r21. Therefore, Force on charge q2 by q1 is, Fq2/q1=140q1q2r212r21=140q1q2r2–r13(r2–r1)

• Put the values of charges with a sign while applying the vector form of Coulomb’s Law.

• Electrostatic force follows Newton’s Third Law,

Fq2/q1=-Fq1/q2

Forces Between Multiple Charges:

The principle of superposition is a mathematical idea that enables us to deconstruct and analyse seemingly difficult linear mathematical problems. Daniel Bernoulli developed the notion of superposition in 1753. The net reactivity of two or more stimuli is the linear summation stimulus’s reactions, per the principle of superposition. As per the principle of superposition, for a linear function F(x), the law of additivity relates that F(x1 + x2) = F(x1) + F(x2). 

According to Coulomb’s equation between several charges, the total electrostatic force exerted on a statically charged particle by two or even more statically charged particles is equivalent to the scalar summation forces exerted on that unit by those primary particles.

Using the theory of superposition, the energies between several charges are computed as follows:

F1 + F2 + F3 +… Fn = FTotal

Where, Fnet in the given system of n particles, is the overall electrostatic force on a particle, and F1, F2, F3... are the electrostatic forces applied on the particles in the corresponding system. The mutual electric force that particles exert 1, 2, 3,… n are represented by Fn.

• The principle of superposition is a powerful and helpful technique.

How to Use the Superposition Principle to Determine Force Between Multiple Charges

Let’s say we have a system of four particles, q1, q2, q3, and q4, for which we need to determine Coulomb’s law forces between numerous charges.

The force on q1 owing to q2, q3, and q4 will be calculated.

Clearly, this system is made up of numerous particles rather than just two. So, what are our options?

The principle of superposition will be used to determine the mutual electric force between numerous charges.

First, we’ll figure out the separate forces acting on q1 as a result of q2, q3, and q4.

F12 = (Ke × q1 ×q2)/  (r12 )2

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

Similarly,

F13 = (Ke × q1 × q3)/ (r13) 2

F14 = (Ke × q1×q4)/(r14)2

The mutual electric force between many charges as per the principle of superposition is now understood to be responsible for the overall force F on q1 owing to q2, q3, and q4.

F = F12 + F13 + F14 + F15 + F16 + F17 + F18 + F19 + Fn

Where,

F12 = (Ke× q1 × q1)/ (r12)2, 

F13 =(Ke× q1× q3)/(r13)2 

F14 = (Ke × q1× q4)/(r14)2

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

F = F12 + F13 + F14 +…. F1n

F12, F13, F14… F1n are the mutual electric force acting on particle q1 as a result of q2,q3,q4,… qn.

Conclusion

Every particle in nature imposes some sort of force on other particles. This is true at all levels, from the subatomic to the cosmic level. The range, size, and nature of the force generated by objects over each other varies. The nature of the forces exerted is determined by a variety of physical processes as well as the values of certain fundamental constants that characterise our universe’s current state. The mutual electric force or the electrostatic force between many charged particles has been discussed here.