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Illustration 25.4: Conservative Forces
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This animation shows the work done in moving a particle in two different force fields. As you move the particle, a vector shows you the direction of the force, and the bar and table show you the total work done as you move the particle around (position is given in meters and work is given in joules). You can zero out the work at any position you want by pushing the "set Work = 0" button. Restart.
Simply from the direction of the force (if we assume a positive test charge), if both of these fields were electrostatic, where could charges be located to produce this type of force? As you move the test charge around, notice that the force is biggest at y = 0 on the right edge and points to the left. As you move the charge away from the right edge at y = 0, notice that the force decreases quickly and points radially outward from a point near x = 10 m, y = 0 m. This lets you know that a positive charge could be located near x = 10 m and y = 0 m to approximately produce these fields.
One of these fields, however, cannot be an electrostatic force field because it is not conservative. In other words, the amount of work done depends on the path taken. If you move the particle to a particular point, it matters if you go straight there or take a circuitous route. Which force field is conservative and which is not? You can mark an initial point and an ending point on the grid, if you want to help keep track of where you've moved the particle, to compare the work done along different paths between the same two points.
Drag the particle away from an initial point and then bring it back to the same spot. How much work is done in the conservative force field? How much is done in the nonconservative force field? For which one of these forces could you get a different answer by moving the particle differently? This means that it takes a different amount of energy to bring the particle to the same spot. Could you uniquely define the potential energy function?
No, you could not. Only conservative forces can have potential energy functions since we can define the potential energy uniquely. Since the electrostatic force is a conservative force, we can develop an associated electric potential that provides easier ways to solve problems, in many instances.
Illustration authored by Anne J. Cox.
Script authored by Mario Belloni and Wolfgang Christian and modified by Anne J. Cox.