29.3.2 The Vector Cross Product

In 《 SECTION 22.5, we found that the electric field of a point charge can be written

E⃗=14πϵ0qr2 r^

where r^ is a unit vector that points from the charge to the point at which we wish to calculate the field. Unit vector r^ expresses the idea “away from q.”

The unit vector r^ also allows us to write the Biot-Savart law more concisely, but we’ll need to use the form of vector multiplication called the cross product. To remind you, FIGURE 29.10 shows two vectors, C⃗ and D⃗, with angle α between them. The cross product of C⃗ and D⃗ is defined to be the vector

C⃗×D⃗=(CD sin α, direction given by the right-hand rule) (29.3)

The symbol × between the vectors is required to indicate a cross product.

FIGURE 29.10 The cross product C⃗×D⃗ is a vector perpendicular to the plane of vectors C⃗ and D⃗.

The figure illustrates two vectors on a plane and their cross product is shown perpendicular to the plane.
FIGURE 29.10 Full Alternative Text

NOTE   The cross product of two vectors and the right-hand rule used to determine the direction of the cross product were introduced in 《 SECTION 12.10 to describe torque and angular momentum. A review is worthwhile.

The Biot-Savart law, Equation 29.1, can be written in terms of the cross product as

B⃗point charge=μ04πqv ⃗×r^r2  (magnetic field of a point charge) (29.4)

where unit vector r^, shown in FIGURE 29.11, points from charge q to the point at which we want to evaluate the field. This expression for the magnetic field B⃗ has magnitude (μ0/4π)qv sin θ/r2 (because the magnitude of r^ is 1) and points in the correct direction (given by the right-hand rule), so it agrees completely with Equation 29.1.

FIGURE 29.11 Unit vector r^ defines the direction from the moving charge to the point at which we evaluate the field.

The figure illustrates a moving point charge and its magnetic field.
FIGURE 29.11 Full Alternative Text

EXAMPLE 29.2 The magnetic field direction of a moving electron

The electron in FIGURE 29.12 is moving to the right. What is the direction of the electron’s magnetic field at the dot?

FIGURE 29.12 A moving electron.

The figure illustrates the magnetic field direction of a moving electron.

VISUALIZE Because the charge is negative, the magnetic field points opposite the direction of v ⃗×r^. Unit vector r^ points from the charge toward the dot. We can use the right-hand rule to find that v ⃗×r^ points into the figure. Thus the electron’s magnetic field at the dot points out of the figure.