# Conservative field

1. Aug 23, 2007

### hanson

hi PF.
What is exactly a conservative field?
I know the mathematical definitions such as the existence of a scalar potential, the curl of the field equals 0 (irrotational), path independence etc.
But I still don't get a physical understanding of such a field.
What's the significance of identifying whether a field is conservative or not?
I know the gravitational field is a conservative field. But it seems to me that it is just a matter of terminology...

2. Aug 23, 2007

### learningphysics

You can't have a potential energy associated with a field, unless you have a conservative field. Do you see how path independence leads to this?

3. Aug 23, 2007

### olgranpappy

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".

4. Aug 23, 2007

### rbj

my "engineering bottom line" answer is such a force field so that when an object is moved from point A to point B, the energy that is required to move it as such is exactly the negative of the energy required to move it from point B back to point A, for any given points A and B and completely independent of whatever convoluted (or straight) path you choose.

if it costs you E units of energy to move something from point A to point B, exactly that E units of energy will be returned to you to move it from point B back to point A.

that is, in my understanding, the salient meaning of a conservative field.