Potential Energy of the System of Two Charges

Potential energy is the energy stored when work is done against an external force. For example, if an object is moved against gravity or spring, the work done is stored in the object in the form of potential energy. This work is usually done by another force. When the influence of this force is removed, the object gains kinetic energy and loses potential energy. In other words, the body starts moving, and its stored potential energy is converted to kinetic energy. Thus, the total energy of the system is conserved. Forces of this category are called conservative forces. Examples of conservative forces include spring force and Coulomb force. The Coulomb force exists between two stationary charges. 

The potential energy of a system of two charges is negative when these two charges are opposite, that is, when one charge is negative and the other is positive; the potential energy is positive when both charges are the same. 

The potential energy of a system of two charges 

We can calculate the potential energy of a system of two charges q1 and q2, which are at points with position vectors r₁ and r₂ , respectively,  by using the following steps.

• First, calculate the amount of work done to get the charges q₁ and q₂ to the points r₁ and r₂, respectively, from infinity. 
• The work done in this step can be denoted by the product of the charges and the points. So q₁r₁ and q₂r₂ are the works done to get q₁ to r₁ and q₂ to r₂ from infinity.
• This work is done against the external field as well as the field created by the charges.
• So the work done on q₁ = q₁V(r₁), and the work done on q₂ = q₂V(r₂). This is equation 1.
• There is some work done on q₂ due to the field produced by q₁. This work done on q₂ is given by (q₁q₂/4πε_or₁₂ ). This is equation 2.
• In equation 2, r₁₂ is the displacement between q₁ and q₂.
• Now equations 1 and 2 can be added by following the rule of superposition of fields along with the work done on q₂ due to the two fields.
• So the work done to bring q₂ to the point r₂ is = q₂V(r₂ ) + (q₁q₂/4πε_or₁₂).
• Hence, the potential energy of the system is the sum of the work done in setting up the whole system of two charges. This work can be expressed by expression for the potential energy of a system of two charges in an external field: q₁V(r₁) + q₂V(r₂) + (q₁q₂/4πε_or₁₂).

The potential energy of a single charge

 When the potential energy of a single charge has to be calculated, the source of the external field is considered. Often, this source is unidentified and its location is unknown.

While calculating the potential energy of a point charge in an external field, it is assumed that the point charge does not affect the source of the external field. This can be true only for point charges because they are small or if the external field is held in place through other forces or factors.

Hence, it can be assumed that the point charge does not influence the external field produced by a powerful source at infinity and extends to a limited field in the area where the point charge is present. 

In the external field, the potential and the field may vary. The potential at a point is the amount of work done to bring a point charge from infinity up to that point. This is given by the product of the charge and the potential at that point. 

Suppose the charge is q, the related external potential is V, the point in the external field is P, and the position vector of the point is r. Then the potential energy of q brought from infinity to P is = qV(r)

In this equation V(r) denotes the external potential at point r.

Electrostatics

Electrostatics is a branch of physics that studies the interaction of stationary charges. The law at the centre of this branch of physics is Coulomb’s law. This law states that two charged particles near each other exert a certain amount of force on each other. This force is given by the following equation:

F = q1q2/4πεo(r12)2

From the above equation, it can be inferred that the electrostatic force is directly proportional to the product of the two charges and is inversely proportional to the square of the distance between the two charges in question.

Conclusion

Potential energy is the work done on any object to move it against an external force. In electrostatics, this work is usually done against an external field on charges. The potential energy of a system of two charges is negative when the charges are different and it is positive when the charge of the influential particle is the same as the charge of the influenced particle. It is useful to understand the concept of the potential energy of charges in electric fields to work effectively with the way particles behave in electric fields.