Electric Flux

As discussed in the previous section, the intensity (magnitude) of the electric field E can be thought of as the number of lines of force crossing a unit area held perpendicular to those lines, $E=\frac{\text{number of lines of force}}{\text{area enclosing the lines}}.$ Rearranging, the total number of lines of force through a flat area equals $E\times(\text{area})$.

When the area is instead INCLINED at some angle $\theta$ to the direction of the field (Fig. 10.14), the number of lines actually passing through it -- the ELECTRIC FLUX -- must be calculated more carefully. If $\theta$ is the angle between the field $\vec{E}$ and the area's own AREA VECTOR $d\vec{S}$ (a vector of magnitude equal to the area, directed along the area's outward normal), the electric flux through the small area element $dS$ is defined as the product of the COMPONENT of $d\vec{S}$ along $\vec{E}$ and the field magnitude: $d\phi=(\text{component of }dS\text{ along }E)\times(\text{area of }dS)=E(dS\cos\theta)=E\,dS\cos\theta,$ which can be written compactly as the dot product $d\phi=\vec{E}\cdot d\vec{S}.$ Flux is therefore maximum when the area directly FACES the field ($\theta=0$, area perpendicular to E) and exactly ZERO when the area lies PARALLEL to the field ($\theta=90^\circ$, so the field lines merely graze past it without actually crossing it). …