What are conservative and non-conservative forces?

The forces which do not depend on the path taken are conservative forces, while those depending upon the path taken are non-conservative forces.

An easy example is that of the Electric Force. Imagine a test charge ‘##q##’ kept at the origin of a hypothetical Cartesian coordinate system. Now, put another unit charge (##Q## = 1 Coulomb) at, say, (0,0,5). The force on the test charge ‘##q##’ is then given in magnitude by:

##F_1 = 1/(4piepsilon_0)(qQ)/r^2 units##

which for the present case becomes

## F_1 = 1/(4piepsilon_0)(q)/25 units##

Now, give the system some translation via any path, i.e., move the unit charge from its original position to a new position taking whichever path you please. You may go from (0,0,5) to (1,0,5) and then to (2,5,6) and finally to, say, (2,3,4). Or you may want to go from (0,0,5) to (1,2,3) to (3,4,5) to (5,6,7) to (-3,0,2) to (-2,-2,-2) and finally to (2,3,4). In both the cases, the end points are the same and thus, the magnitude of the force acting between the two charges will be:

##F_2 = 1/(4piepsilon_0)(q)/14 units = F_3 ##

Then the difference between the two cases ##DeltaF##, as one sees, is the same. Therefore, the electric forces depend only on the end points and not on the path chosen and they from a subset of the conservative forces of nature. Clearly, if the starting and the end points are the same in the above case, then net work done would be zero. This is a property of the conservative forces that work done, ##W = int_c vecF.dvecr = 0 ## since net change in the coordinate space is zero. Conservative forces also follow that their curl vanishes, i.e.,

## vecgrad xx vecF = 0 ##

One can always express a conservative force as the negative gradient of a scalar potential V.

## vecF = -gradV ##

A deviation from the above properties leads one to believe that a force is non-conservative and depends on the path taken to move in space. The most common example of a non-conservative force that we use in our daily life is the force of friction!

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