Circulation of a Vector Field

We have already seen one example of the circulation^18.1 of a vector field, though we didn't label it as such. In the previous section we computed the work done on a charge by the electric field as it moves around a closed loop in the context of the electric generator and Faraday's law. The work done per unit charge, or the EMF, is an example of the circulation of a field, in this case the electric field, $\Gamma_E$. Faraday's law can be restated as

$$\Gamma_E = - \frac{d \Phi_B}{dt} ~~~~ \mbox{(Faraday's law)} .$$ (18.7)

In the simple case of a circular loop with the field directed along the loop, the circulation is just the magnitude of the field times the circumference of the loop, as illustrated in the left panel of figure 17.3. In more complicated cases in which the field points in a direction other than the direction of the loop, just the component in the direction of traversal around the loop enters the circulation. If this component varies as one progresses around the loop, the calculation must be broken into pieces. The total circulation is then obtained by adding up the contributions from segments of the loop in which the value of the field component parallel to the motion around the loop is constant. An example of this type is the calculation of the EMF around a square loop of wire in an electric generator. Another is illustrated in the right panel of figure 17.3.

Figure 17.3: Two examples of circulation paths in a vector field.