Physics Bootcamp

Circulations of magnetic and electric fields play important role in understanding electric and magnetic phenomena fully. Circulation of a field combines the strength and direction properties of a field around a directed closed loop in space into one scalar quantity. If you have taken Calc III, you are already familiar with the concept of circulation of vector fields.

We calculate circulation of magnetic field \(\vec B\) around a closed path by summing the dot product \(\vec B\cdot \Delta\vec l\) for every piece \(\Delta l\text{,}\) treated as vector whose direction is the direction of traversal, and sum them all.

\begin{equation} \text{Circulation, } = \sum_\text{closed path} \vec B\cdot \Delta\vec l, \label{circulation-simple-definition}\tag{39.3.1} \end{equation}

The circulation depends on the direction you go around; if it is positive if go one way, then it will be negative if you go in the oposite direction.

Circulation of a field also depends upon the path you choose for the circulation. I will illustrate this with the field of a current in an infinitely long wire.

Circulation in circle around a straight wire.

As an example, lets work out the circulation of magnetic field of current in a straight wire. As you know from a previous section, magnetic field of a long straight wire circulates around the wire in circles with the same magnitude \(B=\mu_0 I/ 2\pi r\) at all points of one circle of radius \(r\text{.}\) Let us denote this magnetic field by \(B(r)\text{.}\) The circulation of this magnetic field around a circle is easy to work out.

In this case, magnetic field \(\vec B\) and \(\Delta \vec l\) are both tangential and are in the same direction in every part of the circular path. Therefore, we get \(B(r)\Delta l\) from the dot product, and when we sum over all parts of the circle we get

\begin{equation*} \text{Circulation } = \sum_\text{circle} \vec B\cdot \Delta\vec l = B(r) \sum_\text{circle}\Delta l = B(r) \times 2\pi r. \end{equation*}

Circulation in another path near a straight wire.

Suppose we go one third way around the circle, then go parallel to the wire, and then go one third around the circle but in the opposite direction, and then finally go parallel o the wire returning to the original place. What will be the circulation?

You can show that the circulation will be zero. This happens because the contributions on path ab and cd cancel and the dot products on paths bc and da are zero since field and path are perpendicular to each other on segments of those paths.

\begin{align*} \amp \text{Circulation around abcda } \\ \amp \ \ = B(r) \dfrac{2\pi r}{3} + 0 - B(r) \dfrac{2\pi r}{3} + 0 = 0. \end{align*}

Circulation in multiples times around a straight wire.

What will happen if you go around a long straight wire multiple times so that the path is closed in the end?

The situation for three-times around is shown in Figure 39.3.1. You can see that in subsequent circles, the magnetic field and path are in opposite directions. Hence, circulation over two adjacent circles will be zero, and we will be left with circulation over one circle only.

\begin{equation*} \text{net circulation} = B(r) \times 2 \pi r. \end{equation*}