We introduced the idea of a field as a way to understand the long-range electric force. Although this idea appeared rather far-fetched, it turned out to be very useful. We need a similar idea to understand the long-range force exerted by a current on a compass needle.

Let us define the magnetic field $\vec{B}$ as having the following properties:

1. A magnetic field is created at all points in space surrounding a current-carrying wire.

2. The magnetic field at each point is a vector. It has both a magnitude, which we call the magnetic field strength B, and a direction.

3. The magnetic field exerts forces on magnetic poles. The force on a north pole is parallel to $\vec{B}$; the force on a south pole is opposite $\vec{B}$.

FIGURE 29.5 shows a compass needle in a magnetic field. The field vectors are shown at several points, but keep in mind that the field is present at all points in space. A magnetic force is exerted on each of the two poles of the compass, parallel to $\vec{B}$ for the north pole and opposite $\vec{B}$ for the south pole. This pair of opposite forces exerts a torque on the needle, rotating the needle until it is parallel to the magnetic field at that point.

FIGURE 29.5 The magnetic field exerts forces on the poles of a compass.

Notice that the north pole of the compass needle, when it reaches the equilibrium position, is in the direction of the magnetic field. Thus a compass needle can be used as a probe of the magnetic field, just as a charge was a probe of the electric field. Magnetic forces cause a compass needle to become aligned parallel to a magnetic field, with the north pole of the compass showing the direction of the magnetic field at that point.

Look back at the compass alignments around the current-carrying wire in Figure 29.4. Because compass needles align with the magnetic field, the magnetic field at each point must be tangent to a circle around the wire. FIGURE 29.6a shows the magnetic field by drawing field vectors. Notice that the field is weaker (shorter vectors) at greater distances from the wire.

FIGURE 29.6 The magnetic field around a current-carrying wire.

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Another way to picture the field is with the use of magnetic field lines. These are imaginary lines drawn through a region of space so that

• A tangent to a field line is in the direction of the magnetic field, and

• The field lines are closer together where the magnetic field strength is larger.

FIGURE 29.6b shows the magnetic field lines around a current-carrying wire. Notice that magnetic field lines form loops, with no beginning or ending point. This is in contrast to electric field lines, which stop and start on charges.

TACTICS BOX 29.1 Right-hand rule for fields

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Workbook Exercises 6–8