potentials and fields

Source (Feynman lectures).

Suppose we have two large objects, \[A\] and \[B\] and a third very small one which is attracted gravitationally by the two with some resultant force \[\mathbf{F}\]. As demonstrated in the field article, the gravitational force on a particle can be written as its mass times another vector field \[\mathbf{C}\], i.e. \[\mathbf{F}=m\mathbf{C}\]. We say that the objects \[A\] and \[B\] "generate" the vector field. With this form we can analyse gravitation by imagining that there is a certain vector at every position in space which "acts" upon a mass, if we choose to place one there. Then, when an object is put in the field, the force acting on it is equal to its mass times the value of the field vector at the point where the object is put.

We can also do the same with potential energy. Assuming we're only taking into account conservative forces, we can write potential energy as an integral, which we can easily derive from the expression \[\Delta U=-W=-\int \mathbf{F}\cdot d\mathbf{s}\]. Substituting \[\mathbf{F}=m\mathbf{C}\] we get \[\Delta U=-\int m\mathbf{C}\cdot d\mathbf{s}=-m\int \mathbf{C}\cdot d\mathbf{s}=m\Psi\], where we define a new function called the potential, \[\Psi=-\int \mathbf{C}\cdot d\mathbf{s}\]. By having this function \[\Psi(x,y,z)\], at every point in space, we can immediately calculate the potential energy of an object at any point in space, namely \[\Delta U(x,y,z)=m\Psi(x,y,z)\]. Now suppose we have many point masses and we wish to know the potential \[\Psi\] at some arbitrary point \[p\]. This is simply the sum of potentials at \[p\] due to each of the individual masses, i.e. \[\Psi(p)=\sum_{i}-\frac{Gm_{i}}{r_{ip}},i=1,2,\dots\] (see gravitational potential energy). However, do note that the superposition principle does not always work for every type of potential energy.

Suppose that we know the potential energy of an object at some position \[(x,y,z)\] and we want to find what the force on the object is. Theoretically, if we move the object just very slightly in the \[x\]-direction, the work done by the force on the object would be approximately the \[x\]-component of the force times \[\Delta x\]. To show this more concretely we start with expanding \[W=\int \mathbf{F}\cdot d\mathbf{s},d\mathbf{s}=(dx,dy,dz)\] and \[\mathbf{F}=(F_{x},F_{y},F_{z})\]. Then, \[dW=\mathbf{F}\cdot d\mathbf{s}=F_{x}\,dx+F_{y}\,dy+F_{z}\,dz\]. Since our particle is only displaced along the \[x\]-direction, \[dy=dz=0\implies dW=F_{x}\,dx\]. Finally, \[W=\int_{x}^{x+\Delta x}F_{x}\,dx\approx F_{x}\int_{x}^{x+\Delta x}\,dx=F_{x}\Delta x\]. Now we have \[W=-\Delta U=F_{x}\,\Delta x\implies F_{x}=-\frac{\Delta U}{\Delta x}\]. This equation isn't exact, as we actually want the limit of it when \[\Delta x\to0\]. It makes sense for us to write \[-\frac{dU}{dx}\] in this case. However, since \[U\] also depends on \[x,y,z\], we use another set of notations to signify that only \[x\] varies while \[y\] and \[z\] do not, i.e. \[-\frac{\partial U}{\partial x}\]. Similarly, we find the other two directions by differentiating \[U\] with respect to its respective direction while keeping the other directions constant: \[F_{y}=-\frac{\partial U}{\partial y}\] and \[F_{z}=-\frac{\partial U}{\partial z}\]. This notation is also known as partial derivatives.

Now that we know how to get the force using potential energy, we can get the field from the potential in exactly the same way: \[C_{x}=-\frac{\partial \Psi}{\partial x}\], \[C_{y}=-\frac{\partial \Psi}{\partial y}\], \[C_{z}=-\frac{\partial \Psi}{\partial z}\]. Incidentally, there is another notation that we can use to represent all these hassle, \[\nabla\], which is known as the gradient. This symbol is an operator (like plus, minus, multiply, etc.), that produces a vector from a scalar. In a three-dimensional Cartesian coordinate system, the gradient, if it exists is given by \[\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}\]. With this nifty new notation, we can rewrite our formulas as \[\mathbf{F}=-\nabla U\] and \[\mathbf{C}=-\nabla \Psi\].