The electric flux of a surface is defined as the total electric field passing through that surface. Mathematically, it is given by the dot product of the electric field vector and the area vector of the surface, multiplied by the magnitude of the electric field. The electric flux is a measure of the number of electric field lines passing through a given area.

The correct answer to the given question is option 'd' - the electric flux is maximum when the area vector is parallel to the direction of the electric field. Let's understand why this is the case:

Area Vector and Electric Field Direction:
When the area vector is parallel to the direction of the electric field, it means that the surface is "facing" the electric field. In this configuration, the maximum number of electric field lines pass through the surface.

Explanation:
To understand this, let's consider a simple example. Imagine a flat surface with an electric field passing through it. Now, imagine tilting this surface at an angle to the electric field. As you tilt the surface, the effective area normal to the electric field decreases, resulting in a decrease in the electric flux.

On the other hand, if the surface is facing directly towards the electric field (i.e., the area vector is parallel to the electric field), the entire surface is exposed to the electric field. Consequently, the effective area normal to the electric field is at its maximum, resulting in the maximum electric flux.

In other words, when the surface is parallel to the electric field, the maximum number of electric field lines pass through the surface, leading to the maximum electric flux. This is because the dot product of the electric field vector and the area vector is maximized when the two vectors are parallel.

Therefore, the correct answer is option 'd' - the electric flux is maximum when the area vector is parallel to the direction of the electric field.