Imagine holding a rectangular wire loop of area A in front of a fan. As FIGURE 24.11 shows, the volume of air flowing through the loop each second depends on the angle between the loop and the direction of flow. The flow is maximum through a loop that is perpendicular to the airflow; no air goes through the same loop if it lies parallel to the flow.

FIGURE 24.11 The amount of air flowing through a loop depends on the angle between $\vec{v}$ and $\widehat{n}$.

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The flow direction is identified by the velocity vector $\vec{v}$. We can identify the loop’s orientation by defining a unit vector $\widehat{n}$ normal to the plane of the loop. Angle $\theta$ is then the angle between $\vec{v}$ and $\widehat{n}$. The loop perpendicular to the flow in Figure 24.11a has $\theta =0\text{°};$ the loop parallel to the flow in Figure 24.11b has $\theta =90\text{°}$. You can think of $\theta$ as the angle by which a loop has been tilted away from perpendicular.

You can see from Figure 24.11c that the velocity vector $\vec{v}$ can be decomposed into components $v_{\perp }=v\text{cos}\theta$ perpendicular to the loop and $v_{||}=v\text{sin}\theta$ parallel to the loop. Only the perpendicular component $v_{\perp }$ carries air through the loop. Consequently, the volume of air flowing through the loop each second is

$\theta =0\text{°}$ is the orientation for maximum flow through the loop, as expected, and no air flows through the loop if it is tilted $90\text{°}$.

An electric field doesn’t flow in a literal sense, but we can apply the same idea to an electric field passing through a surface. FIGURE 24.12 shows a surface of area A in a uniform electric field $\vec{E}$. Unit vector $\widehat{n}$ is normal to the surface, and $\theta$ is the angle between $\widehat{n}$ and $\vec{E}$. Only the component $E_{\perp }=E\text{cos}\theta$ passes through the surface.

FIGURE 24.12 An electric field passing through a surface.

With this in mind, and using Equation 24.1 as an analog, let’s define the electric flux ${\text{Φ}}_{\text{e}}$ (uppercase Greek phi) as

The electric flux measures the amount of electric field passing through a surface of area A if the normal to the surface is tilted at angle $\theta$ from the field.

Equation 24.2 looks very much like a vector dot product: $\vec{E}\cdot \vec{A}=EA\text{cos}\theta$. (You may want to review the dot product in 《 SECTION 9.3.) For this idea to work, let’s define an area vector $\vec{A}=A\widehat{n}$ to be a vector in the direction of $\widehat{n}$—that is, perpendicular to the surface—with a magnitude A equal to the area of the surface. Vector $\vec{A}$ has units of ${\text{m}}^2$. FIGURE 24.13a shows two area vectors.

FIGURE 24.13 The electric flux can be defined in terms of the area vector $\vec{A}$.

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FIGURE 24.13b shows an electric field passing through a surface of area A. The angle between vectors $\vec{A}$ and $\vec{E}$ is the same angle used in Equation 24.2 to define the electric flux, so Equation 24.2 really is a dot product. Thus we can write the electric flux as

Writing the flux as a dot product helps make clear how angle $\theta$ is defined: $\theta$ is the angle between the electric field and a line perpendicular to the plane of the surface.