Question
A small plane area is rotated in an electric field. In which orientation of the area is the flux of electric field through the area maximum? In which orientation is it zero?
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Answer
The flux of an electric field $\overrightarrow{\text{E}}$ through a surface area $\Delta\overrightarrow{\text{S}}$ is given by $\Delta\phi=\overrightarrow{\text{E}}.\Delta\overrightarrow{\text{S}},$ where $\Delta\phi$ is the flux.
Therefore, $\Delta\phi={\text{E}}\Delta{\text{S}}\cos\theta.$ Here, $\theta$ is the angle between the electric field $\overrightarrow{\text{E}}$ and the normal to the surface area.
Thus, for the flux to be maximum, $\cos\theta$ should be maximum. Thus, for $\theta=0,$ the flux is maximum, i.e. the electric field lines are perpendicular to the surface area.
The flux is minimum if $\theta=90.$ Thus, $\cos\theta=0$ and, hence, flux is also 0. Thus, if the electric field lines are parallel to the surface area, the flux is minimum.
Therefore, $\Delta\phi={\text{E}}\Delta{\text{S}}\cos\theta.$ Here, $\theta$ is the angle between the electric field $\overrightarrow{\text{E}}$ and the normal to the surface area.
Thus, for the flux to be maximum, $\cos\theta$ should be maximum. Thus, for $\theta=0,$ the flux is maximum, i.e. the electric field lines are perpendicular to the surface area.
The flux is minimum if $\theta=90.$ Thus, $\cos\theta=0$ and, hence, flux is also 0. Thus, if the electric field lines are parallel to the surface area, the flux is minimum.
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