First, suppose that the time rate of change of the magnetic flux is negligible, so that the electric field is essentially free of circulation. This means that no matter what closed contour C is chosen, the line integral of E must vanish.

We will find that this condition prevails in electroquasistatic systems and that all of the fields in Sec. 1.3 satisfy this requirement.

Illustration. A Field Having No Circulation

A static field between plane parallel sheets of uniform charge density has no circulation. Such a field, E = E_o i_x, exists in the region 0 < y < s between the sheets of surface charge density shown in Fig. 1.6.2. The most convenient contour for testing this claim is denoted C₁ in Fig. 1.6.2.

Figure 1.6.2. Uniform electric field intensity E_o, between plane parallel uniform distributions of surface charge density, has no circulation about contours C₁ and C₂.

Along path 1, E ds = E_o dy, and integration from y = 0 to y = s gives sE_o for the EMF of point (a) relative to point (b). Note that the EMF between the plane parallel surfaces in Fig. 1.6.2 is the same regardless of where the points (a) and (b) are located in the respective surfaces.

On segments 2 and 4, E is orthogonal to ds, so there is no contribution to the line integral on these two sections. Because ds has a direction opposite to E on segment 3, the line integral is the integral from y = 0 to y = s of E ds = -E_o dy. The result of this integration is -sE_o, so the contributions from segments 1 and 3 cancel, and the circulation around the closed contour is indeed zero.

In this planar geometry, a field that has only a y component cannot be a function of x without incurring a circulation. This is evident from carrying out this integration for such a field on the rectangular contour C₁. Contributions to paths 1 and 3 cancel only if E is independent of x.

Example 1.6.1. Contour Integration

To gain some appreciation for what it means to require of E that it have no circulation, no matter what contour is chosen, consider the somewhat more complicated contour C₂ in the uniform field region of Fig. 1.6.2. Here, C₂ is composed of the semicircle (5) and the straight segment (6). On the latter, E is perpendicular to ds and so there is no contribution there to the circulation.

On segment 5, the vector differential ds is first written in terms of the unit vector i, and that vector is in turn written (with the help of the vector decomposition shown in the figure) in terms of the Cartesian unit vectors.

It follows that on the segment 5 of contour C₂

and integration gives

So for contour C₂, the circulation of E is also zero.

When the electromotive force between two points is path independent, we call it the voltage between the two points. For a field having no circulation, the EMF must be independent of path. This we will recognize formally in Chap. 4.

The second limiting situation, typical of the magnetoquasistatic systems to be considered, is primarily concerned with the circulation of E, and hence with the part of the electric field generated by the time-varying magnetic flux density. The remarkable fact is that Faraday's law holds for any contour, whether in free space or in a material. Often, however, the contour of interest coincides with a conducting wire, which comprises a coil that links a magnetic flux density.

Illustration. Terminal EMF of a Coil

A coil with one turn is shown in Fig. 1.6.3. Contour (1) is inside the wire, while (2) joins the terminals along a defined path. With these contours constituting C, Faraday's integral law as given by (1) determines the terminal electromotive force. If the electrical resistance of the wire can be regarded as zero, in the sense that the electric field intensity inside the wire is negligible, the contour integral reduces to an integration from (b) to (a).

In view of the definition of the EMF, (2), this integration gives the negative of the EMF. Thus, Faraday's law gives the terminal EMF as

Figure 1.6.3. Line segment (1) through a perfectly conducting wire and (2) joining the terminals (a) and (b) form closed contour.

where _f, the total flux of magnetic field linking the coil, is defined as the flux linkage. Note that Faraday's law makes it possible to measure _oH electrically (as now demonstrated).

It follows from Faraday's integral law that the tangential electric field is continuous across a surface of discontinuity, provided that the magnetic field intensity is finite in the neighborhood of the surface of discontinuity. This can be shown by applying the integral law to the incremental surface shown in Fig. 1.4.7, much as was done in Sec. 1.4 for Ampère's law. With J set equal to zero, there is a formal analogy between Ampère's integral law, (1.4.1), and Faraday's integral law, (1). The former becomes the latter if H E , J 0, and _o E -_o H. Thus, Ampère's continuity condition (1.4.16) becomes the continuity condition associated with Faraday's law.

At a surface having the unit normal n, the tangential electric field intensity is continuous.