Thread: Conservative field View Single Post
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 Quote by hanson hi PF. What is exactly a conservative field?
A vector field that can be written as the gradient of a scalar field. I.e., if the field $$\vec F$$is conservative, then it can be written as:

$$\vec F = -\nabla \phi$$
where the minus sign is just conventional.

The reason for the name "conservative" is that if I can write the field as the gradient of a scalar, then (for a force field) I can calculate the work done (by me) as I move some point particle from one place to another and the work done is just the change in the value of the scalar field $$\phi$$ (and thus the work is independent of path), which can then be interpreted as a potential energy.

This interpretation is useful because the work done is also equal to the change in kinetic energy of the particle and thus the sum of the potential and kinetic energy is always constant in a conservative field... I.e., the total energy is conserved--hence the name "conservative".