Faraday found that a current is induced when the amount of magnetic field passing through a coil or a loop of wire changes. And that’s exactly what happens as the slide wire moves down the rail in Figure 30.5! As the circuit expands, more magnetic field passes through. It’s time to define more clearly what we mean by “the amount of field passing through a loop.”

Imagine holding a rectangular loop in front of the fan shown in FIGURE 30.11. The amount of air flowing through the loop—the flux—depends on the angle of the loop. The flow is maximum if the loop is perpendicular to the flow, zero if the loop is rotated to be parallel to the flow. In general, the amount of air flowing through is proportional to the effective area of the loop (i.e., the area facing the fan):

where $A=ab$ is the area of the loop and $\theta$ is the tilt angle of the loop. A loop perpendicular to the flow, with $\theta =0\text{°}$, has $A_{\text{eff}}=A$, the full area of the loop.

FIGURE 30.11 The amount of air flowing through a loop depends on the effective area of the loop.

We can apply this idea to a magnetic field passing through a loop. FIGURE 30.12 shows a loop of area $A=ab$ in a uniform magnetic field. Think of the field vectors as if they were streamlines of air flowing into the figure. The density of streamlines (streamlines per ${\text{m}}^2$) is proportional to the strength B of the magnetic field; a stronger field would be represented by streamlines packed closer together. The number of streamlines passing through a loop of wire depends on two factors:

1. The density of streamlines, which is proportional to B, and

2. The effective area $A_{\text{eff}}=A\text{cos}\theta$ of the loop.

FIGURE 30.12 Magnetic field through a loop that is tilted at various angles.

The angle $\theta$ is the angle between the magnetic field and the axis of the loop. The maximum number of streamlines passes through the loop when it is perpendicular to the magnetic field $(\theta =0\text{°})$. No streamlines pass through the loop if it is tilted $90\text{°}$.

With this in mind, let’s define the magnetic flux ${\text{Φ}}_{\text{m}}$ as

The magnetic flux measures the amount of magnetic field passing through a loop of area A if the loop is tilted at angle $\theta$ from the field. The SI unit of magnetic flux is the weber. From Equation 30.9 you can see that

Equation 30.9 is reminiscent of the vector dot product: $\vec{A}\cdot \vec{B}=AB\text{cos}\theta$. With that in mind, let’s define an area vector $\vec{A}$ to be a vector perpendicular to the loop, with magnitude equal to the area A of the loop. Vector $\vec{A}$ has units of ${\text{m}}^2$. FIGURE 30.13a shows the area vector $\vec{A}$ for a circular loop of area A.

FIGURE 30.13 Magnetic flux can be defined in terms of an area vector $\vec{A}$.

FIGURE 30.13b shows a magnetic field passing through a loop. The angle between vectors $\vec{A}$ and $\vec{B}$ is the same angle used in Equations 30.8 and 30.9 to define the effective area and the magnetic flux. So Equation 30.9 really is a dot product, and we can define the magnetic flux more concisely as

Writing the flux as a dot product helps make clear how angle $\theta$ is defined: $\theta$ is the angle between the magnetic field and the axis of the loop.

EXAMPLE 30.4 A circular loop in a magnetic field

FIGURE 30.14 is an edge view of a 10-cm-diameter circular loop in a uniform 0.050 T magnetic field. What is the magnetic flux through the loop?

FIGURE 30.14 A circular loop in a magnetic field.

FIGURE 30.14 Full Alternative Text

SOLVE Angle $\theta$ is the angle between the loop’s area vector $\vec{A}$, which is perpendicular to the plane of the loop, and the magnetic field $\vec{B}$. In this case, $\theta =60\text{°}$, not the $30\text{°}$ angle shown in the figure. Vector $\vec{A}$ has magnitude $A=\pi r^2=7.85\times {10}^{-3}{\text{m}}^2$. Thus the magnetic flux is

$${\text{Φ}}_{\text{m}}=\vec{A}\cdot \vec{B}=AB\text{cos}\theta =2.0\times {10}^{-4}\text{Wb}$$