Force Between Multiple Charges: Coulomb’s Law with Superposition Theorem

Physics Electric Charge and Field 2025

Syed Aquib Ur Rahman
Updated on Oct 8, 2025 20:12 IST

By Syed Aquib Ur Rahman, Assistant Manager

The atom’s nucleus has multiple positively charged protons, packed together. Our current knowledge of Coulomb’s Law Class 12 would tell us that the protons strongly repel each other, as there are electrostatic forces present. But protons also cannot escape. That is due to the strong nuclear force, which acts only over short distances. The forces in the nucleus are also much stronger than electrostatic forces. How should we calculate the force between multiple charges, then?

Until now, in Class 12 Physics Chapter 1, you may have applied Coulomb’s Law to find the electrostatic force between two point charges, which are separated by some distance. But in real systems, just like the nucleus of the atom, you have to deal with several charged particles.

You have to take into account each and every proton that feels the pull or push of all other surrounding protons at the same time. For such scenarios, you have to know how to use or extend Coulomb’s Law using vector addition rules, using the superposition theorem. That is how you can accurately analyse the combined effect in systems where charges are interacting simultaneously. 

What You'll Learn from our notes on Force Between Multiple Charges Class 12

  • Use the superposition principle and vector addition to calculate the net electrostatic force between multiple charges.
  • Get immediately started with solving exemplar-level questions with confidence. 
Force between multiple charges
 

 

Table of content
  • Force Between Multiple Charges: Combining Vectors with the Superposition Theorem
  • Practice Calculating Force Between Multiple Charges Class 12
  • Complete Class 12 Study Material
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Force Between Multiple Charges: Combining Vectors with the Superposition Theorem

The superposition principle or theorem tells us that the total electrostatic force that static charges exert on a stationary charge is the vector sum of all these forces. 

With this vector summing, we must treat every force pair as independent, because the vector sum does not depend on the number of charges present in the system. The presence of other charges does not change the force of one charge exerted on another. 

The total force of charge is the combination of all individual forces using vector addition. That is how we can predict electrostatic forces in complex systems in the real world. 

How do we show the Pairwise Force and Notate the Vector?

With Class 11th Physics, you got a fair and practicable idea of drawing Free Body Diagrams (FBDs). You also have in-depth knowledge of scalars and vectors.
In scalar addition for charges, we just add the magnitudes using simple algebraic summation, for this is one of the basic properties of electric charge

In vector addition for force between multiple charges, we have to include the direction of force, F_12, where the force of q_2 is exerted on q_1 with the electrostatic force formula, which is

F = k |q_1q_2|/r^2

But, the electrostatic force on q 1 due to  q 2

   F 12    =    k q 1 q 2 r 12 2 r ^ 12 As force is a vector, first try to identify its exact direction, whether it’s attraction or repulsion, which shows the sign of each charge.  The vector form uses position vectors, r_21 and unit vectors, r̂_21 to show direction. You can easily draw a free-body diagram first for conceptual clarity, before calculating the magnitudes. 

The general formula for n charges in a system, while we use the additivity property of charge, we get the total force of F_1 on charge q_1 from all the other n number of forces in the system.

F 1 = F 12 + F 13 ...until it goes up to n

For simplicity, take three charges q1, q2, and q3. They are placed at a distance represented by vectors r 12 \vec{r}_{12}

The force on the charge q1 from q2 is: 

F 12 = K q 1 q 2 ( r 12 ) 2 r ^ 12

The force on q1 due to q3 is:

F 13 = K q 1 q 3 ( r 13 ) 2 r ^ 13

The total force on q1 is:

F 1 = K q 1 q 2 ( r 12 ) 2 r ^ 12 + K q 1 q 3 ( r 13 ) 2 r ^ 13

 

Just remember to 

  • Calculate each of the individual forces with Coulomb's Law
  • Calculate the direction of each force
  • Combine the forces using vector addition.

 

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Practice Calculating Force Between Multiple Charges Class 12

Practice this type of question for finding the force between multiple charges. This would be relevant to NCERT Exemplar Solutions for Chapter 1 Physics.

Q. Three charges q 1 = + 2 μ C , q 2 = - 3 μ C and q 3 = + 4 μ C are at ( 0,0 ) , ( 0.1,0 ) , and ( 0,0.1 ) meters. Find the net force on q 1 .

F 12 = 9 × 10 9 N m 2 C - 2 × 2 × 10 - 6 3 × 10 - 6 ( 0.1 ) 2 = 0.54 N (   attractive, along   + x ) F 13 = 9 × 10 9 N m 2 C - 2 × 2 × 10 - 6 4 × 10 - 6 ( 0.1 ) 2 = 0.72 N (   repulsive, along   - y )

Net force: F 1 = ( 0.54 , - 0.72 ) N magnitude ( 0.54 ) 2 + ( 0.72 ) 2 0.9 N .

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