What is the 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 for 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:
F = Ke q1q2/r2
Where,
‘F’ 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.
Coulombs’ Law in Vector Form
To understand the Coulombs’ Law in vector form, students should know what a vector is. Vector is a quantity in mathematics as well as physics which has both magnitude and direction. Since, force is also a vector quantity, Coulombs’ law is better understood in vector form. Let us now understand the vector form of Coulombs’ Law.
Let the charges be q1 and q2 and their position vectors be r1 and r2 respectively. Let the force on q1 due to q2 be denoted by F12 and force on q2 by q1 be denoted by F21.
The vector form of Coulomb’s Law is written as this-
\[ F_{12} = k.q_{1}.q_{2}. r_{2}. r \]
\[F _{12} = - F_{21} \] (For repulsion)
\[ F_{12} = + F_{21} \](For attraction)
Here, F12 is the force exerted by q1 on q2, and F21 is the force exerted by q2 on q1.
Features of Coulombs’ Law
Given below are some of the characteristics of Coulombs’ Law-
It is inversely proportional to the square of the distance between the charges and proportional to the product of magnitudes of the two charges.
This law only holds for point charges only.
It also follows the superposition principle.
The value of the Coulombs’ constant is 9 × 109 Nm2/ C2 when we take the S.I unit of value of ε is 8.854 × 10-12 C2 N-1 m-2.
By using the coulombs’ law, we can easily find the force acting upon two charges and also find force present on one point.
How to Solve Problems by Coulombs’ Law?
Given below are steps to solve problems based on coulombs’ law-
Firstly, students need to determine whether the force due to the charge is attractive or repulsive and then represent the same by drawing a vector pointing towards or away from the given charge, respectively.
Secondly, the students need to determine the magnitude of force using Coulomb’s Law formula (as stated above).
Students are then required to solve the forces with the given coordinate axis and express them in vector form using i, j,and k unit vector notation.
Apply the principle of superposition to find the net force on the charges.
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 plagiarized 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?
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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.
Similarly,
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
Where
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.
Conclusion
The superposition principle is a mathematical truth that allows us to calculate seemingly complex linear mathematical equations by dividing them into smaller segments. Similar to this principle, you can decipher seemingly tougher calculations and concepts through a little practice. You can also download free PDFs of all study material and access them from anywhere.
FAQs on Force Between Multiple Charges
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 are the limitations of Coulombs’ Law?
Coulomb’s law is very important in the area of physics but it can not be used everywhere as it has some limitations. The limitations of the coulombs’ law are given below-
This law only works if the point charges are static which means that for this law to work on, the point charges need to be in the position of rest.
Coulombs’ law only works for objects which have regular and smooth charges. It is difficult to implement Coulombs’ law when the shape is arbitrary or undefined. This is because in such cases it is difficult to find the accurate distance between the charges.
3. Who invented Coulombs’ Law?
In 600 BC, a Greek philosopher observed that bodies charged with static energy will either attract or repulse each other depending on the nature of their charges. After many years, Charles Augustin de Coulomb in 1785, established a mathematical relation between two charged bodies and the force which acts between them. This is why the law, Coulombs’ Law, is named after him. The law is similar to the way magnets work in the sense that in a magnet, two poles of the same charge repel each other while two poles of opposite charge attract each other.
4. Solve the following using Coulomb’s Law. If two charges 1C and -4C are kept at a distance of 5 m, then find the force of attraction between them.
q1 = 1C
q2 = – 4C
r = 5m
According to the Coulombs’ Law,
Fe = ke q1q2/r2
= 9 × 109 × 1 × (4/ 5)2
= 1.44 x 109
