Solving the Differential Equations

Coulomb’s Law describes the force between two stationary, electrically charged particles, also known as electrostatic force. 

 Thales of Miletus was the first to observe Coulomb’s Law in 600 BC. Two bodies charged with static electricity will either repel or attract each other depending on the nature of their charge. The calculation of attraction or repulsion force between charged bodies was just an observation, not a mathematical theory.

In 1785, Charles Augustin de Coulomb, a French physicist, published the equation for the repulsion or attraction force between two electrically charged bodies after many centuries. Moreover, this was commonly known as Coulomb’s Law. Coulomb’s Law states that the intensity of the electrostatic force of attraction or repulsion between two electrically charged bodies is directly proportional to the product of their charges and inversely proportional to the square of their distances.

Principle of Coulomb’s Law

If we have two charged bodies, one positively charged, one negatively charged, and a distance between them, they will attract each other. When we increase the charge of one body while keeping the other unchanged, the force of attraction increases.

In the same way, if we increase the charge of the second body while keeping the first one unchanged, the attraction force between them increases again. Therefore, the force between the charged bodies is proportional to their charges.

Keeping charges fixed at Q1 and Q2, the force between the two points increases as the distance decreases, and the force decreases as the length between the points increases.

If the distance between the two charge bodies is d, then the force acting on them is inversely proportional to d2. F ∝ 1/d2. The force between two equally charged bodies is not the same in all mediums. Therefore εr would change for different mediums. So, the force applied can vary depending on the medium. F ∝ 1/ εr

Coulomb’s Law Equation

According to Coulomb’s Law, the energy between two charged objects is proportional to their respective charges and inversely proportional to their separation distance and can be expressed mathematically as F= k * Q1* Q2/ d2

Object ‘1’ is charged with Q1 (in Coulombs), object ‘2’ is charged with Q2, and d is the distance between the two objects (metres). The letter K represents Coulomb’s law constant k. 

The value of k depends upon the medium in which the charged objects are immersed. As for air, the value is approximately 9.0 x 109 N /m2 /c2. In the equation, k is substituted for Coulombs, removing the units of distance and charge, leaving Newtons as the unit of force.

A point charge can be described using Coulomb’s Law, which describes the force between two objects accurately, almost as if its charge were all concentrated at its centre. The charge of a conducting sphere interacts with the charge of other objects. 

The centre of charge of a sphere, regardless of how uniformly the charges are distributed, can be considered its centre.

A point charge resides at the centre of the sphere. Since Coulomb’s Law applies to point charges, the distance between the centres of charge of each object is d in the equation.

Q1 and Q2 are the charge quantities for the two interacting objects in Coulomb’s law equation. As an object can be charged positively or negatively, these quantities can be expressed as positive or negative values. In the case of a negatively charged object, the sign indicates that it has a surplus of electrons, while a positively charged object has a deficit of electrons. 

A repulsive force is derived if “+” and “-” signs are used, whereas a negative value indicates an attractive force. 

When Q1 and Q2 have the same charge, then the force value should be positive, and when Q1 and Q2 have opposite charges, the force value is negative, since one has a “+” charge, while the other has a “-” charge. Therefore, the interaction between ‘oppositely charged’ and ‘like-charged’ objects is consistent with the concept that ‘oppositely charged’ objects have an attractive interaction.

Limitations of Coulomb’s Law

In contrast to other general formulas, Coulomb’s Law is derived under definite assumptions and cannot be applied liberally. Its limits are as follows:

• Coulomb’s Law does not holds if the average number of solvent molecules between two interesting charge particles is large.
• The point charges must be at rest for Coulomb’s Law to apply.
• Coulomb’s Law is invalid if charged bodies are of limited dimension such that they cannot be considered a point charge. Thus, Coulomb’s Law does not apply to distances below 10-15 cm.
• Inverse-square Law applies to Coulomb’s Law. It is only appropriate in cases where the inverse square law applies.
• Solvent molecules between particles must be larger than both charges to be valid.
• Charges with regular and smooth shapes can be handled easily with this formula, but charges with irregular shapes become too complicated.
• A charge in an arbitrary shape makes Coulomb’s Law challenging to apply. Thus, we cannot determine the distance ‘d’ between charges when they are shaped arbitrarily. The centre of arbitrarily shaped charged bodies cannot be determined precisely, so the distance r cannot be determined correctly.

Conclusion

The electric force is inversely proportional to the square of Coulomb’s Law. Furthermore, this Law is used in deriving Gauss’ Law accurately for general cases. Charges at rest exert the following properties according to Coulomb’s Law- where like charges repel one another and unlike charges attract each other. 

The vector form of Coulomb’s Law provides the direction of electric fields caused by charges. Two negative charges repel one another, while a positive charge attracts a negative charge. In physics, charges act in accordance with their lines of attraction.