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The context of work in defining potential energy is always confusing. There is, however, few distinguishing aspects that we should keep in mind to be correct. If we define potential energy in terms of conservative force, then potential energy is equal to “negative” of work by conservative force. If we define potential energy in terms of external force, then potential energy is simply equal to work by external force, which does not impart kinetic energy to the particle.
Potential energy is unique in yet another important respect. Unlike other forms of energy, potential energy is directly related to conservative force. We shall establish this relation here. We know that a change in potential energy is equal to the negative of work by gravity,
$$\Delta U=-{F}_{C}\Delta r$$
For infinitesimal change, we can write the equation as,
$$\Rightarrow dU=-Fcdr$$
$$\Rightarrow {F}_{C}=-\frac{dU}{dr}$$
Thus, if we know potential energy function, we can find corresponding conservative force at a given position. Further, we can see here that force – a vector – is related to potential energy (scalar) and position in scalar form. We need to resolve this so that evaluation of the differentiation on the right yields the desired vector force.
As a matter of fact, we handle this situation in a very unique way. Here, the differentiation in itself yields a vector. In three dimensions, we define an operator called “grad” as :
$$\mathrm{grad}=\left(\frac{\partial}{\partial x}i+\frac{\partial}{\partial y}j+\frac{\partial}{\partial z}k\right)$$
where " $$\frac{\partial}{\partial x}$$ " is partial differentiation operator. This is same like normal differentiation except that it considers other dimensions (y,z) constant. In terms of “grad”,
$$\Rightarrow F=-\mathrm{grad}\phantom{\rule{2pt}{0ex}}U$$
The example given here illustrates the operation of “grad”.
Problem 1: Gravitational potential energy in a region is given by :
$$U\left(x,y,z\right)=-\left({x}^{2}y+y{z}^{2}\right)$$
Find gravitational force function.
Solution : We can obtain gravitational force in each of three mutually perpendicular directions of a rectangular coordinate system by differentiating given potential function with respect to coordinate in that direction. While differentiating with respect to a given coordinate, we consider other coordinates as constant. This type of differentiation is known as partial differentiation.
Thus,
$${F}_{x}=-\frac{\partial}{\partial x}=-\frac{\partial}{\partial x}-\left({x}^{2}y+y{z}^{2}\right)=2xy$$
$${F}_{y}=-\frac{\partial}{\partial y}=-\frac{\partial}{\partial y}-\left({x}^{2}y+y{z}^{2}\right)={x}^{2}+{y}^{2}$$
$${F}_{z}=-\frac{\partial}{\partial z}=-\frac{\partial}{\partial z}-\left({x}^{2}y+y{z}^{2}\right)=2yz$$
Hence, required gravitational force is given as :
$$\Rightarrow F=-\mathrm{grad}\phantom{\rule{2pt}{0ex}}U$$
$$F=-\left(\frac{\partial}{\partial x}i+\frac{\partial}{\partial y}j+\frac{\partial}{\partial z}k\right)U$$
$$F==2xyi+\left({x}^{2}+{y}^{2}\right)j+2yzk$$
This example illustrates how a scalar quantity (potential energy) is related to a vector quantity (force). In order to implement partial differentiation by a single operator, we define a differential vector operator “grad” a short name for “gradient” as above. For this reason, we say that conservative force is equal to gradient of potential energy.
Evaluation of the integral of potential energy is positive or negative, depending on the nature of work by conservative force.
$$U=-{W}_{C}=-\underset{\infty}{\overset{0}{\int}}{F}_{C}dr$$
The nature of work by the conservative force, on the other hand, depends on whether force is attractive or repulsive. The work by attractive force like gravitation and electrostatic force between negative and positive charges do “positive” work. In these cases, component of force and displacement are in the same direction as the particle is brought from infinity. However, as a negative sign precedes the right hand expression, potential energy of the system operated by attractive force is ultimately negative.
It means that potential energy for these conservative forces would be always a negative value. The important thing is to realize that maximum potential energy of such system is “zero” ay infinity.
On the other hand, potential energy of a system interacted by repulsive force is positive. Its minimum value is “zero” at infinity.
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