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. What is the principle of superposition for electric forces?
The principle of superposition states that in a system of multiple point charges, the total electrostatic force on any one charge is the vector sum of all the individual forces exerted on it by all other charges. Importantly, the force between any two charges is completely unaffected by the presence of other charges.
2. How do you calculate the net force on a specific charge in a system of multiple charges?
To calculate the net force on a charge in a multi-charge system, you must follow these steps:
- First, identify the charge on which you need to find the net force (let's call it q₀).
- Next, use Coulomb's Law to calculate the force exerted by each of the other charges (q₁, q₂, q₃, etc.) on q₀ individually. Remember to determine both the magnitude and direction for each force.
- Finally, add all these individual forces together as vectors. This vector sum gives the resultant or net force acting on the charge q₀.
3. Why is it essential to use vector addition when calculating the force between multiple charges?
Force is a vector quantity, meaning it has both magnitude and direction. In a system with multiple charges, each charge exerts a force from a different position and in a different direction (either attractive or repulsive). Simply adding the magnitudes (the numerical values of the forces) would be incorrect as it ignores their directions. Vector addition correctly combines both the magnitude and direction of each individual force to determine the true net force and its precise resultant direction.
4. How does the sign of a charge (positive or negative) influence the net force in a multi-charge system?
The sign of a charge primarily determines the direction of the force. According to Coulomb's Law, like charges repel and opposite charges attract. While the magnitude of the force between any two charges is calculated using the absolute value of the charges, their signs dictate whether the resulting force vector points towards or away from the other charge. The final net force is a vector sum, so these individual directions are critical in determining the final outcome.
5. What are the key limitations of using Coulomb's Law for a system of multiple charges?
While fundamental, Coulomb's Law has limitations when applied to systems of charges:
- It is strictly accurate only for point charges that are stationary (in static equilibrium).
- For charges distributed over an object with an irregular shape, it becomes difficult to define the exact distance 'r' between them, making direct application of the law impractical. In such cases, methods involving integration are required.
- The law assumes the charges are in a vacuum. If a medium is present, its permittivity affects the force.
6. Why is the force between two specific charges unaffected by the presence of a third charge?
The electrostatic force described by Coulomb's Law is a fundamental two-body interaction. This means the force between charge q₁ and charge q₂ depends only on their magnitudes and the distance between them. A third charge, q₃, will exert its own, separate force on both q₁ and q₂, but it does not alter the original force between q₁ and q₂. The total force on q₂ is simply the vector sum of the force from q₁ and the force from q₃, which is the essence of the superposition principle.
7. How does calculating force for a system of discrete charges differ from calculating it for a continuous charge distribution?
The approach differs significantly. For a system of discrete (separate) point charges, we use the principle of superposition and algebraic summation (Σ) to find the vector sum of individual forces. For a continuous charge distribution (like charge spread over a rod or a sphere), we must use calculus. We consider the force 'dF' from an infinitesimally small charge element 'dq' and then integrate (∫) this force over the entire body to find the total force.
