Question

What is the work done by the field of a nucleus in a complete circular orbit of the electron and what if the orbit is elliptical?

Answer 1

The concept of conservative and non-conservative forces needs to be applied. The formula for work done is considered and the quantities it is dependent upon is probed upon. The concept behind work done on an object is applied and the determination of whether the work done is conservative or a non-conservative force needs to be found out in order to find the work done on the electron.

Complete answer:
The above problem revolves around the concept of the amount of work done on the electron inside a nucleus moving in an elliptical orbit.The nucleus consists of atomic particles like protons and neutrons electrons contained inside it which make up the nucleus of an atom. While the protons and neutrons are said to be stationary the electrons revolve around the nucleus of an atom in orbits which are known to be the shells of the atom. This motion of electrons is said to be around the nucleus since these electrons contain a lot of energy in them and are free to move.

A force known as the nuclear force is said to keep the electrons moving in circles around it. The region around the nucleus is what is known as the field of the nucleus, that is, the region where the electron is said to revolve as given in the question. We are asked to find out the work done on the electron by this field of the nucleus on the orbiting electron.

When there is a field around the nucleus this field also exerts a force on the electron which keeps the electron moving around it without falling on the surface of the nucleus and since the electron is in motion there is some amount of displacement associated with it. Thus, there is some amount of work that will be done on the electron. Hence, we now define the work done on the electron which depends on these terms. The equation for work is given by the formula:
$W = Fx$
Where, $F$ is force and $x$ is displacement

We can see from the above equation that work depends on force and displacement variables. Work is said to be done when there is a force applied on the body and the body moves through some distance or suffers some displacement in the direction of the applied force. Work is said to be a conservative force which means that it is independent of the path which the body travels in.

This is similar to displacement which is also path independent and only depends on the initial and final points of the body in motion. Since work is dependent on the displacement of the particle, that is, the electron in motion it is also path independent. This means that the path in which the body is travelling is not considered.

We are asked if the work done by the field of the nucleus when an electron is in a circular path varies from the work done by the field of the nucleus when the electron is in an elliptical path. The work done in both of these paths is said to be zero since the initial and final points of the electron are said to be the same. The work done along a closed loop or a closed path wherein the work done is conservative in nature, is said to be zero.

Note: There is often a common misconception that the work done along a closed path is always zero which is incorrect. The work done by the body is zero only when the force that is applied on it is conservative in nature which is its main criteria. Non-conservative forces like friction need not produce work done along a closed loop as it is path dependent.