Animated Solution for Physics - Electrostatics: Consider an electric field $\mathbf{E}=E_0\hat{\mathbf{x}}$, where $E_0$ is a constant. The flux through the shaded area (as shown in the figure) due to this field is

Look closely at the 3D geometry presented in the problem. We have a shaded rectangular surface placed within an $x,y,z$ coordinate system. The vertices of this surface are clearly marked at the origin $O(0,0,0)$, a point on the y-axis $C(0,a,0)$, a point in the x-z plane $A(a,0,a)$, and a point in 3D space $B(a,a,a)$.

To find the electric flux passing through this surface, we must rely on the fundamental definition of electric flux. It is the dot product of the electric field vector and the area vector:

First, let's determine the area vector $\mathbf{S}$. By definition, the area vector is always perpendicular to the surface. Since our surface is tilted within the x-z plane (while extending uniformly along the y-axis), its normal vector will also lie entirely within the x-z plane.

What is the magnitude of this area? The surface is a rectangle. One side lies perfectly along the y-axis with a length of $a$. The adjacent side stretches from the origin to the point $(a,0,a)$. Using the 3D distance formula, the length of this side is $\sqrt{a^2+a^2}=\sqrt{2}a$.

Therefore, the total magnitude of the area is:

Now, let's look at the electric field. The problem states that the electric field is uniform and points strictly in the positive x-direction:

To calculate the dot product, we need the angle $\theta$ between the electric field and the area vector. The electric field is perfectly horizontal along the x-axis. The area vector is perpendicular to a surface that bisects the x and z axes (passing through $x=a,z=a$). This means the surface is tilted at exactly $45^{\circ }$, and consequently, its normal vector also makes a $45^{\circ }$ angle with the x-axis.

We now have all the necessary pieces of the puzzle. Let's substitute the magnitude of the electric field, the magnitude of the area, and the cosine of the angle into our flux equation:

We know that the cosine of $45^{\circ }$ is $\frac{1}{\sqrt{2}}$.

The $\sqrt{2}$ terms beautifully cancel each other out, leaving us with our final answer:

The Vector Method

Another brilliant way to verify this result is by using pure vector components. We can find the area vector by taking the cross product of the two vectors that form the adjacent sides of the rectangle:

Now, simply take the dot product of this area vector with the electric field:

The answer is exactly the same! This demonstrates the incredible consistency and beauty of physics.