A conservative field is a vector field in which the work done by the field on a particle moving along a closed path is zero. In other words, the work done by the field depends only on the endpoints of the path and not on the actual path taken.
The reason why a conservative field is independent of the path followed can be understood by considering the concept of a potential function. In a conservative field, there exists a scalar function called the potential function (or simply the potential) such that the field can be obtained by taking the gradient of the potential. Mathematically, if F is a vector field and φ is its potential, then F = ∇φ, where ∇ represents the gradient operator.
When a particle moves in a conservative field, the work done by the field on the particle can be expressed in terms of the potential function. The work done is given by the difference in the values of the potential function at the endpoints of the path. Since the potential function is a scalar, its value is independent of the path taken, and therefore, the work done is also independent of the path.
This independence of the path is a consequence of the fact that the gradient of a scalar field is path-independent. The gradient of a scalar function only depends on the values of the function at a given point and not on the path taken to reach that point. Therefore, the field obtained from the gradient, i.e., the conservative field, is also independent of the path.
This property of conservative fields is crucial in many areas of physics, such as in the study of conservative forces like gravitational and electrostatic forces, where the work done and potential energy are path-independent. It allows for simplifications in calculations and provides a deeper understanding of the underlying physical phenomena.