A bit more on energy

This article is a short continuation on the previous introduction to work and energy.

To understand this article, you will need at least a basic understanding of the following concepts:

  • Energy
  • Vectors
  • Integrals and path integrals

Variable force

In the work and energy article we saw that work is defined as

W= \vec{F} \cdot \vec{s}

But this only holds if the force is constant over the distance. If the force changes (for example a rocket ship getting further from a planet and thereby the gravity decreasing or a spring extending), we need to integrate it over the distance instead:

W=\int_a^b{\vec{F} \cdot d\vec{s}}

Here a and b signify the start and end points.

Before, if we wanted to know the work done on something moving from one point to another, then to a third point, we would have had to calculate it for each individual path and add them all together. Now that we have an integral, it is doing all the adding for us and the displacement vector has become an infinitesimal. This means that this can be interpreted as a path integral! We can find the work required to move something along any curved path from one point to another with the force varying. The force could vary as a function of distance along the curve, as a function of time, or in the case of moving through a field: as a function of position. (All these cases can be converted to each other).


Think about the work done by moving along a path in a static (unchanging) field. Imagine for example the gravitation field of a planet which gets weaker as you get further away, but always points towards the planet. If you move from one point in the field to another point “higher up” (going against the field, away from the planet), there are an infinite number of different paths you could take. You could take the direct path, or do all kinds of loops and zigzags. Moving up the field takes work, but if you fall back down again, the field returns your energy and moving perpendicular to the field will not contribute to the work. A closed loop will therefore require net zero work. You can probably convince yourself then, that the work required to go between two points will be the same for all paths, and the potential energy will therefore only depend on the positions of the points.

Using this realization, we can construct something called a potential. This is a scalar field (a value assigned to each point in space) which represents the work required to get from some reference point to every point in space (also known as the potential energy between the reference point and every other point). If you have ever seen the “drawings” of the sun and earth lying in a curved “sheet” in space, this sheet is an illustration of the combined gravitational potential of the earth and sun (or a 2D slice of it), where the height of each point represents the value of the potential. Often, the reference for a potential is chosen to be at infinite distance, for example with gravitational and electric fields. The potentials then become negative, because you gain energy (do negative work) by traveling “down from infinity”. Setting the reference point to infinity might seem weird, and even impossible, shouldn’t that cause all the potentials to be negative infinity? In some cases, yes, but the strength of gravitational and electric fields drop off with one over the radius squared (the inverse square law) and become so small that they contribute basically nothing. You can think of what happens as you start with the reference close to your point of interest (a planet for example) and then move it further and further away. In the beginning, your potential is very steep, so the planet gets “pushed down” fast, it’s potential decreases quickly. But when you get far away, the potential is basically flat and moving the reference further almost doesn’t change anything. In fact, the flatness wins out over the distance, and your potential approaches a certain value. This is really what is meant by setting the reference at infinity.

Of course, if your field strength doesn’t approach a fixed value as distance goes to infinity, this won’t work.

Electric potentials in circuits

In electronics, the reference for the electric potential is called ground. This is often set to the negative terminal of the battery, but could just as well be set anywhere else, like the positive terminal or a point which is changing potential relative to the negative terminal. As long as we stick with the same reference at all times, the calculations will work out.

If you’ve ever heard about Kirchhoff’s voltage law, it comes from the fact that we discussed earlier, that the work along a closed loop is zero.