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Potential energy equations

The potential energy of an object in a force field is the work that should be done by the field to bring the object from one point to another. In other words, the field "drags" the object.

The force on a particle in a uniform field $$F$$ is equal to the negative of the gradient of the potential energy of the particle at that point: $$$\vecphy{F}=-\frac{dE_{\text{p}}}{d\vecphy{s}}=-\frac{\Delta E_{\text{p}}}{\Delta\vecphy{s}}$$$ This implies that the change in potential energy between two points in a uniform field is given by the negative of the product of the force with the change in displacement (i.e. work): $$$\Delta E_{\text{p}}=-\vecphy{F}\cdot\Delta\vecphy{s}$$$

$$E_{\text{p}}=$$potential energy; $$\vecphy{F}=$$force; $$\vecphy{s}=$$displacement.