Figure 29.6 is a qualitative picture of the wire’s magnetic field. Our first task is to turn that picture into a quantitative description. Because current in a wire generates a magnetic field, and a current is a collection of moving charges, our starting point is the idea that moving charges are the source of the magnetic field. FIGURE 29.7 shows a charged particle q moving with velocity $\vec{v}$. The magnetic field of this moving charge is found to be

where r is the distance from the charge and $\theta$ is the angle between $\vec{v}$ and $\vec{r}$.

FIGURE 29.7 The magnetic field of a moving point charge.

Equation 29.1 is called the Biot-Savart law for a point charge (rhymes with Leo and bazaar), named for two French scientists whose investigations were motivated by Oersted’s observations. It is analogous to Coulomb’s law for the electric field of a point charge. Notice that the Biot-Savart law, like Coulomb’s law, is an inverse-square law. However, the Biot-Savart law is somewhat more complex than Coulomb’s law because the magnetic field depends on the angle $\theta$ between the charge’s velocity and the line to the point where the field is evaluated.

The SI unit of magnetic field strength is the tesla, abbreviated as T. The tesla is defined as

You will see later in the chapter that this definition is based on the magnetic force on a current-carrying wire. One tesla is quite a large field; most magnetic fields are a small fraction of a tesla. TABLE 29.1 lists a few magnetic field strengths.

The constant ${\mu }_0$ in Equation 29.1 is called the permeability constant. Its value is

This constant plays a role in magnetism similar to that of the permittivity constant ${ϵ}_0$ in electricity.

The right-hand rule for finding the direction of $\vec{B}$ is similar to the rule used for a current-carrying wire: Point your right thumb in the direction of $\vec{v}$. The magnetic field vector $\vec{B}$ is perpendicular to the plane of $\vec{r}$ and $\vec{v}$, pointing in the direction in which your fingers curl. In other words, the $\vec{B}$ vectors are tangent to circles drawn about the charge’s line of motion. FIGURE 29.8 shows a more complete view of the magnetic field of a moving positive charge. Notice that $\vec{B}$ is zero along the line of motion, where $\theta =0\text{°}$ or $180\text{°}$, due to the $\text{sin}\theta$ term in Equation 29.1.

FIGURE 29.8 Two views of the magnetic field of a moving positive charge.

Full Alternative Text

The requirement that a charge be moving to generate a magnetic field is explicit in Equation 29.1. If the speed v of the particle is zero, the magnetic field (but not the electric field!) is zero. This helps to emphasize a fundamental distinction between electric and magnetic fields: All charges create electric fields, but only moving charges create magnetic fields.

EXAMPLE 29.1 The magnetic field of a proton

A proton moves with velocity $\vec{v}=1.0\times {10}^7î\text{m}\text{/}\text{s}$. As it passes the origin, what is the magnetic field at the $\left(x\text{,}y\text{,}z\right)$ positions (1 mm, 0 mm, 0 mm), (0 mm, 1 mm, 0 mm), and (1 mm, 1 mm, 0 mm)?

MODEL The magnetic field is that of a moving charged particle.

VISUALIZE FIGURE 29.9 shows the geometry. The first point is on the x-axis, directly in front of the proton, with ${\theta }_1=0\text{°}$. The second point is on the y-axis, with ${\theta }_2=90\text{°}$, and the third is in the xy-plane.

FIGURE 29.9 The magnetic field of Example 29.1.

FIGURE 29.9 Full Alternative Text

SOLVE Position 1, which is along the line of motion, has ${\theta }_1=0\text{°}$. Thus $\vec{B}{}_1=\vec{0}$. Position 2 (at 0 mm, 1 mm, 0 mm) is at distance $r_2=1\text{mm}=0.001\text{m}$. Equation 29.1, the Biot-Savart law, gives us the magnetic field strength at this point as

$$\begin{array}{ccc}\hfill B& \hfill =\hfill & {\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{qv\text{sin}{\theta }_2}{r_2^2}}\hfill \\ \hfill & \hfill =\hfill & {\displaystyle \frac{1.26\times {10}^{-6}\text{T}\text{m}\text{/}\text{A}}{4\pi }}{\displaystyle \frac{(1.60\times {10}^{-19}\text{C})(1.0\times {10}^7\text{m}\text{/}\text{s})\text{sin}90\text{°}}{(0.0010\text{m}{)}^2}}\hfill \\ \hfill & \hfill =\hfill & 1.60\times {10}^{-13}\text{T}\hfill \end{array}$$

According to the right-hand rule, the field points in the positive z-direction. Thus

$$\vec{B}{}_2=1.60\times {10}^{-13}\widehat{k}\text{T}$$

where $\widehat{k}$ is the unit vector in the positive z-direction. The field at position 3, at (1 mm, 1 mm, 0 mm), also points in the z-direction, but it is weaker than at position 2 both because r is larger and because $\theta$ is smaller. From geometry we know $r_3=\sqrt{2}\text{mm}=0.00141\text{m}$ and ${\theta }_3=45\text{°}$. Another calculation using Equation 29.1 gives

$$\vec{B}{}_3=0.57\times {10}^{-13}\widehat{k}\text{T}$$

REVIEW The magnetic field of a single moving charge is very small.