Potential energy of a system of charges

Potential energy due to the system of charges

A body’s potential energy is the energy that it possesses due to its position with respect to other bodies, internal tensions, electric charge, and various other elements. Throwing a ball in the air, tightening a spring, and other similar activities are examples of mechanical potential energy. All of the work done in tossing the ball and keeping the spring tightened is stored as potential energy, transformed into kinetic energy when the ball arrives on the ground, or the spring is released.

As an analogy to the idea of mechanical potential energy discussed above, there is also the concept of electrostatic potential energy, which incorporates the Potential energy of a system of charges and an electric field. These principles have been explored in greater detail in the following sections.

Electrostatic Potential: potential energy of a system of charges

When an object or particle is placed in a given position or configuration, its potential energy is stored. The external work done on it is stored in the form of potential energy. As a result, potential energy can be considered a broad sort of stored energy. Assume the amount of effort required to modify the position or configuration of an object is greater. As seen in the graph below, the quantity of potential energy stored in the object will be bigger in that instance. The quantity of potential energy stored in an object is proportional to the amount of external work performed.

Consider the following example: A mass m object is placed on the ground. m = the object’s mass To elevate the object from the ground to a height h, an external force equal to mg must be applied to the object’s surface. The effort required to elevate the object from ground level to height h is equal to mgh, known as gravitational potential energy. As a result, the work done on the object as it moves from one location to another is equal to the difference in objective potential energy between the two sites. Let’s know more about the potential energy of a system of charges.

Charge’s Electrostatic Potential:  potential energy of a system of two charges

Consider how the phrases electrostatic potential and electric potential difference formula are applied in mathematics. The electric potential energy per unit charge in a static electric field is the electrostatic potential. A unit charge’s external work equals the change in potential of a point charge divided by the number of units charged. Assume we have an electric charge q and want to move it from point A to point B, and that the external work done in moving the charge from point A to point B is WAB. 

The potential energy of a system of charges

Think about two charges, q1 and q2, that each has its own set of position vectors, with r1 being the first and r2 being the second, with respect to some origin. Assume that the charges q1 and q2, which are initially at infinity, are transported to the given locations, and compute the amount of effort required by an external agency to transfer the charges there. Pretend you’re in the following situation: First, the charge q1 is carried from infinity to the point vector r1 by use of a charge transporter. In the absence of an external field against which to measure the amount of effort expended, the amount of labour required to transform q1 from infinite to vector r1 is equal to zero.

This charge generates a potential in space. Now let r1P is the distance between a point P in space and the position of q1. According to the concept of potential, the amount of work required to move charge q2 from infinity to the point vector r2 is equal to q2 times the potential at vector r2 due to charge q1.

Work completed on q2

The distance between points 1 and 2 is represented by the number r12. Because electrostatic force is a conservative force, the work done by the system is stored as potential energy in the system’s potential energy. As a result, the potential energy of a system containing two charges q1 and q2 is equal to

Work = kq1q2/r12 

Where k = 1/4πε

The potential energy U would be the same regardless of whether q2 was transported first to its current location or whether q1 was transported first.

Potential energy due to three charges  

Determine the potential energy of a system consisting of three charges, q1, q2, and q3, each of which is placed at a vector r1, r1, and r3 in turn. There is no labour necessary to bring q1 first from infinity to the vector1. Following that, we will bring q2 from infinity to the vector r2. As was the case previously, the work completed in this step is

The charges q1 and q2 form a potential, given by the equation at any point P.

When bringing q3 from infinite to the point vector r3, the work must be done equals q3 multiplied by V1 , at vector r3.

Work = k(q1q2/r12 + kq2q3/r23 + kq1q3/r13)

Where k = 1/4πε

The total effort to build the charges at the specified locations is derived by summing the work done in each of the steps and multiplying the amount by the number of charges assembled. While the potential energy is distinctive of the current configuration state, it is not characteristic of the state’s process.

Conclusion

The electrostatic potential at a given position is defined as the product of the effort required to convey a test charge from infinity to that position and the actual charge. Two points should be made about this definition, even if it is not incorrect. To begin, only changes in electrostatic potential are significant, which means that to define electrostatic potential, a point must be chosen as the zero of potential. Infinity denotes a place that is infinitely remote from the source of the field we are assuming and at which the field exerts no force.