Animated Solution for Physics - Electrostatics: A charged particle $q$ is placed at the centre $O$ of cube of length $L(ABCDEFGH)$. Another same charge $q$ is placed at a distance $L$ from $O$. Then, the electric flux through $ABCD$ is

Imagine you are standing in front of a transparent cube, labeled $ABCDEFGH$, with a side length of $L$. Right at the dead center of this cube, at point $O$, sits a point charge $q$.

But that's not all. The problem introduces a twist: a second, identical charge $q$ is placed outside the cube, at a distance $L$ from the center $O$.

Our mission is to find the total electric flux passing through the front face of the cube, which is the square $ABCD$.

In competitive exams like JEE, time is your most valuable resource. Before we rush into setting up complex integrals or calculating solid angles to find the exact flux, let's take a step back and look at the options provided.

Recall Gauss's Law, which states that the total electric flux $\varphi$ through any closed surface is equal to the enclosed charge divided by the permittivity of free space:

This fundamental law tells us something incredibly powerful about the dimensions of electric flux. The dimension of flux must always be exactly the dimension of charge divided by ${\epsilon }_0$. Notice what is missing? There is absolutely no length term involved in the formula for flux!

Now, let's critically examine the options given in the question: (a) $\frac{q}{4\pi {\epsilon }_{0}L}$ (b) $\frac{q}{2\pi {\epsilon }_{0}L}$ (c) $\frac{q}{2\pi {\epsilon }_{0}L}$

Do you see the pattern? Every single one of these options has the length $L$ sitting in the denominator.

If we look at the dimensions of these options, they are of the form $\left[\frac{q}{{\epsilon }_{0}L}\right]$. Since $\frac{q}{{\epsilon }_0}$ is flux (which is Electric Field $\times$ Area), dividing it by length $L$ gives us Electric Field $\times$ Length.

Wait a minute! Electric Field $\times$ Length is the dimension of Electric Potential (Volts), not Electric Flux!

Because options (a), (b), and (c) possess the wrong physical dimensions, they are physically impossible answers. They are dimensionally incorrect.

Therefore, without doing a single line of heavy calculus, we can confidently conclude that the correct choice must be (d) None of these.

For the sake of pure physics curiosity, what would the actual flux be?

The total flux through the face $ABCD$ is the sum of the flux from the center charge and the flux from the outside charge. By pure symmetry, the charge exactly at the center $O$ distributes its total flux $\frac{q}{{\epsilon }_0}$ equally among all 6 faces of the cube. So, its contribution to the front face $ABCD$ is exactly $\frac{q}{6{\epsilon }_0}$.

The second charge outside the cube also contributes some non-zero flux, because its electric field lines will inevitably pierce through the face $ABCD$. Calculating that exact numerical value requires advanced solid angle geometry. Thankfully, dimensional analysis saved us from that nightmare!