31.3 The Displacement Current

We introduced Ampère’s law in Chapter 29 as an alternative to the Biot-Savart law for calculating the magnetic field of a current. Whenever total current $I_{through}$ passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is

\oint \vec{B} \cdot d \vec{s} = μ_{0} I_{through}

(31.17)

FIGURE 31.13 illustrates the geometry of Ampère’s law. The sign of each current can be determined by using Tactics Box 31.1. In this case, $I_{through} = I_{1} - I_{2}$ .

The figure illustrates two current carrying wires passing through a plane surface bounded by a curve.

FIGURE 31.13 Full Alternative Text

TACTICS BOX 31.1 Determining the signs of flux and current

1 For a surface S bounded by a closed curve C, choose either the clockwise (cw) or counterclockwise (ccw) direction around C.
2 Curl the fingers of your right hand around the curve in the chosen direction, with your thumb perpendicular to the surface. Your thumb defines the positive direction.
- A flux $Φ$ through the surface is positive if the field is in the same direction as your thumb, negative if the field is in the opposite direction.
- A current through the surface in the direction of your thumb is positive, in the direction opposite your thumb is negative.

Workbook Exercises 4–6

Ampère’s law is the formal statement that currents create magnetic fields. Although Ampère’s law can be used to calculate magnetic fields in situations with a high degree of symmetry, it is more important as a statement about what types of magnetic field can and cannot exist.