Our investigation of this force begins more simply, with individual moving charges. A current consists of moving charges, so Ampère’s experiment implies that a magnetic field exerts a force on a moving charge. It turns out that the magnetic force is somewhat more complex than the electric force, depending not only on the charge’s velocity but also on how the velocity vector is oriented relative to the magnetic field. Consider the following experiments:

Notice that the relationship among $\vec{v}$, $\vec{B}$, and $\vec{F}$ is exactly the same as the geometric relationship among vectors $\vec{C}$, $\vec{D}$, and $\vec{C}\times \vec{D}$. The magnetic force on a charge q as it moves through a magnetic field $\vec{B}$ with velocity $\vec{v}$ can be written

where $\alpha$ is the angle between $\vec{v}$ and $\vec{B}$.

The right-hand rule is that of the cross product, shown in FIGURE 29.34. Notice that the magnetic force on a moving charged particle is perpendicular to both $\vec{v}$ and $\vec{B}$.

FIGURE 29.34 The right-hand rule for magnetic forces.

The magnetic force has several important properties:

• Only a moving charge experiences a magnetic force. There is no magnetic force on a charge at rest $(\vec{v}=\vec{0})$ in a magnetic field.

• There is no force on a charge moving parallel $(\alpha =0\text{°})$ or antiparallel $(\alpha =180\text{°})$ to a magnetic field.

• When there is a force, the force is perpendicular to the plane containing $\vec{v}$ and $\vec{B}$.

• The force is a maximum $\left|q\right|vB$ when $\vec{v}$ is perpendicular to $\vec{B}$.

• The force on a negative charge is in the direction opposite to $\vec{v}\times \vec{B}$.

FIGURE 29.35 shows the relationship among $\vec{v}$, $\vec{B}$, and $\vec{F}$ for four moving charges. (The source of the magnetic field isn’t shown, only the field itself.) You can see the inherent three-dimensionality of magnetism, with the force perpendicular to both $\vec{v}$ and $\vec{B}$. The magnetic force is very different from the electric force, which is parallel to the electric field.

FIGURE 29.35 Magnetic forces on four moving charges.

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EXAMPLE 29.10 The magnetic force on an electron

A long wire carries a 10 A current from left to right. An electron 1.0 cm above the wire is traveling to the right at a speed of $1.0\times {10}^7\text{m}\text{/}\text{s}$. What are the magnitude and the direction of the magnetic force on the electron?

MODEL The magnetic field is that of a long, straight wire.

VISUALIZE FIGURE 29.36 shows the current and an electron moving to the right. The right-hand rule tells us that the wire’s magnetic field above the wire is out of the figure, so the electron is moving perpendicular to the field.

FIGURE 29.36 An electron moving parallel to a current-carrying wire.

FIGURE 29.36 Full Alternative Text

SOLVE The electron charge is negative, thus the direction of the force is opposite the direction of $\vec{v}\times \vec{B}$. The right-hand rule shows that $\vec{v}\times \vec{B}$ points down, toward the wire, so $\vec{F}$ points up, away from the wire. The magnitude of the force is $\left|q\right|vB=evB$. The field is that of a long, straight wire at distance $r=0.010\text{\hspace{0.17em}m}$:

$$B={\displaystyle \frac{{\mu }_{0}I}{2\pi r}}=2.0\times {10}^{-4}\text{T}$$

Thus the magnitude of the force on the electron is

$$\begin{array}{ccc}\hfill F& \hfill =\hfill & evB=(1.60\times {10}^{-19}\text{C})(1.0\times {10}^7\text{m}\text{/}\text{s})(2.0\times {10}^{-4}\text{T})\hfill \\ \hfill & \hfill =\hfill & 3.2\times {10}^{-16}\text{N}\hfill \end{array}$$

The force on the electron is $\vec{F}=(3.2\times {10}^{-16}\text{N,\hspace{0.17em}up})$.

REVIEW This force will cause the electron to curve away from the wire.

We can draw an interesting and important conclusion at this point. You have seen that the magnetic field is created by moving charges. Now you also see that magnetic forces are exerted on moving charges. Thus it appears that magnetism is an interaction between moving charges.