FIGURE 29.24 shows a wire carrying current I into the figure and the magnetic field at distance r. The magnetic field of a current-carrying wire is everywhere tangent to a circle around the wire and has the same magnitude ${\mu }_{0}I\text{/}2\pi r$ at all points on the circle. According to Tactics Box 29.3, these conditions allow us to easily evaluate the line integral of $\vec{B}\cdot d\vec{s}$ along a circular path around the wire. Suppose we were to integrate all the way around the circle. That is, the initial point i of the integration path and the final point f will be the same point. This would be a line integral around a closed curve, which is denoted

FIGURE 29.24 Integrating the magnetic field around a wire.

The little circle on the integral sign indicates that the integration is performed around a closed curve. The notation has changed, but the meaning has not.

Because $\vec{B}$ is tangent to the circle and of constant magnitude at every point on the circle, we can use Option 2 from Tactics Box 29.3 to write

where, in this case, the path length l is the circumference $2\pi r$ of the circle. The magnetic field strength of a current-carrying wire is $B={\mu }_{0}I\text{/}2\pi r$, thus

The interesting result is that the line integral of $\vec{B}\cdot d\vec{s}$ around the current-carrying wire is independent of the radius of the circle. Any circle, from one touching the wire to one far away, would give the same result. The integral depends only on the amount of current passing through the circle that we integrated around.

This is reminiscent of Gauss’s law. In our investigation of Gauss’s law, we started with the observation that the electric flux ${\text{Φ}}_{\text{e}}$ through a sphere surrounding a point charge depends only on the amount of charge inside, not on the radius of the sphere. After examining several cases, we concluded that the shape of the surface wasn’t relevant. The electric flux through any closed surface enclosing total charge $Q_{\text{in}}$ turned out to be ${\text{Φ}}_{\text{e}}=Q_{\text{in}}\text{/}{ϵ}_0$.

Although we’ll skip the details, the same type of reasoning that we used to prove Gauss’s law shows that the result of Equation 29.14

• Is independent of the shape of the curve around the current.

• Is independent of where the current passes through the curve.

• Depends only on the total amount of current through the area enclosed by the integration path.

Thus whenever total current $I_{\text{through}}$ passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is

This result for the magnetic field is known as Ampère’s law.

To make practical use of Ampère’s law, we need to determine which currents are positive and which are negative. The right-hand rule is once again the proper tool. If you curl your right fingers around the closed path in the direction in which you are going to integrate, then any current passing though the bounded area in the direction of your thumb is a positive current. Any current in the opposite direction is a negative current. In FIGURE 29.25, for example, currents $I_2$ and $I_4$ are positive, $I_3$ is negative. Thus $I_{\text{through}}=I_2-I_3+I_4$.

FIGURE 29.25 Using Ampère’s law.

In one sense, Ampère’s law doesn’t tell us anything new. After all, we derived Ampère’s law from the Biot-Savart law. But in another sense, Ampère’s law is more important than the Biot-Savart law because it states a very general property about magnetic fields. We will use Ampère’s law to find the magnetic fields of some important current distributions that have a high degree of symmetry.

EXAMPLE 29.8 The magnetic field inside a current-carrying wire

A wire of radius R carries current I. Find the magnetic field inside the wire at distance $r<R$ from the axis.

MODEL Assume the current density is uniform over the cross section of the wire.

VISUALIZE FIGURE 29.26 shows a cross section through the wire. The wire has cylindrical symmetry, with all the charges moving parallel to the wire, so the magnetic field must be tangent to circles that are concentric with the wire. We don’t know how the strength of the magnetic field depends on the distance from the center—that’s what we’re going to find—but the symmetry of the situation dictates the shape of the magnetic field.

FIGURE 29.26 Using Ampère’s law inside a current-carrying wire.

FIGURE 29.26 Full Alternative Text

SOLVE To find the field strength at radius r, we draw a circle of radius r. The amount of current passing through this circle is

$$I_{\text{through}}=J{A}_{\text{circle}}=\pi r^{2}J$$

where J is the current density. Our assumption of a uniform current density allows us to use the full current I passing through a wire of radius R to find that

$$J={\displaystyle \frac{I}{A}}={\displaystyle \frac{I}{\pi R^2}}$$

Thus the current through the circle of radius r is

$$I_{\text{through}}={\displaystyle \frac{r^2}{R^2}}I$$

Let’s integrate $\vec{B}\cdot d\vec{s}$ around the circumference of this circle. According to Ampère’s law,

$$\oint \vec{B}\cdot d\vec{s}={\mu }_0{I}_{\text{through}}={\displaystyle \frac{{\mu }_0{r}^2}{R^2}}I$$

We know from the symmetry of the wire that $\vec{B}$ is everywhere tangent to the circle and has the same magnitude at all points on the circle. Consequently, the line integral of $\vec{B}\cdot d\vec{s}$ around the circle can be evaluated using Option 2 of Tactics Box 29.3:

$$\oint \vec{B}\cdot d\vec{s}=Bl=2\pi rB$$

where $l=2\pi r$ is the path length. If we substitute this expression into Ampère’s law, we find that

$$2\pi rB={\displaystyle \frac{{\mu }_0{r}^2}{R^2}}I$$

Solving for B, we find that the magnetic field strength at radius r inside a current-carrying wire is

$$B={\displaystyle \frac{{\mu }_{0}I}{2\pi R^2}}r$$

REVIEW The magnetic field inside a wire increases linearly with distance from the center until, at the surface of the wire, $B={\mu }_{0}I\text{/}2\pi R$ matches our earlier solution for the magnetic field outside a current-carrying wire. This agreement at $r=R$ gives us confidence in our result. The magnetic field strength both inside and outside the wire is shown graphically in FIGURE 29.27.

FIGURE 29.27 Graphical representation of the magnetic field of a current-carrying wire.

FIGURE 29.27 Full Alternative Text