Question

Why is an electric flux zero when an electric field is perpendicular at a surface?

Answer 1

The electric flux through a surface is determined by the electric field passing through it. When the electric field is perpendicular to the surface, the electric flux through that surface is zero. This phenomenon is known as Gauss's Law for Electric Fields.

To understand why this is the case, let's consider the definition of electric flux. Electric flux (${\Phi }_E$) is a measure of the total number of electric field lines passing through a given surface. Mathematically, it is calculated as the dot product of the electric field ($\mathbf{E}$) and the area vector ($\mathbf{A}$) of the surface, integrated over the surface:

${\Phi }_E=\int \mathbf{E}\cdot d\mathbf{A}$

Now, if the electric field is perpendicular ($\mathbf{E}$ ⊥ $\mathbf{A}$), the dot product of the two vectors becomes zero because the cosine of the angle between them is zero:

$\mathbf{E}\cdot \mathbf{A}=E\cdot A\cdot \cos \theta =E\cdot A\cdot \cos (90^{\circ })=E\cdot A\cdot 0=0$

Since the dot product is zero, the integral of the electric field over the surface ($\int \mathbf{E}\cdot d\mathbf{A}$) evaluates to zero. Therefore, the electric flux through a surface perpendicular to an electric field is zero.

This result can be intuitively understood by considering that when the electric field lines are perpendicular to a surface, they do not pass through it but rather "slice" across it. In this configuration, there is no net flow of electric field lines into or out of the surface, resulting in zero electric flux.