Q.A point charge $+10\,\mu\text{C}$ is a distance $5\,\text{cm}$ directly above the centre of a square of side $10\,\text{cm}$, as shown in Fig. 1.31. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge $10\,\text{cm}$.)

By the hint, treat the square as one face of a cube of edge $10\,\text{cm}$. Because the charge is $5\,\text{cm}$ (half an edge) above the square's centre, it lies exactly at the cube's centre.

Gauss's law gives the total flux through the closed cube:

By symmetry the six faces share this equally, so the flux through one face is …

A single square is an open surface, so Gauss's law cannot be applied to it directly. The hint tells us to complete it into a closed surface.

Step 1 — Build the cube. The charge sits $5\,\text{cm}$ above the centre of the $10\,\text{cm}$ square. Imagine a cube of edge $10\,\text{cm}$ having this square as one face. The centre of such a cube is $5\,\text{cm}$ from each face — exactly where the charge is. So the charge is at the centre of the cube, and the given square is one of its six faces.

Step 2 — Total flux through the cube. The cube is now a closed surface enclosing $q=+10\,\mu\text{C}$. Gauss's law gives

Step 3 — Use symmetry. With the charge at the centre, the six faces are equivalent, so each receives one‑sixth of the total flux: …

Method: Gauss's Law with Symmetry (Cube Construction)

Why This Method Works

The hint suggests a powerful symmetry trick. A point charge above the centre of a square has no simple symmetry by itself — but if we imagine the square as one face of a cube with the charge at its centre, the full cube has perfect symmetry.

Steps

Step 1: Construct an imaginary cube

Place the $+10\,\mu\text{C}$ charge at the exact centre of a cube of side $10\,\text{cm}$. The given square becomes the top face of this cube.

Step 2: Apply Gauss's Law to the entire cube

Gauss's Law states:

Here, $Q_{\text{enclosed}} = +10\,\mu\text{C} = 10 \times 10^{-6}\,\text{C}$.

So:

Step 3: Use symmetry to find flux through one face

The charge is at the cube's centre. By symmetry, the total flux is divided equally among all 6 faces of the cube.

Therefore: …

Common Mistakes Students Make with This Gauss Law Problem

Mistake 1: Trying to integrate directly over the square

What students do wrong:

They attempt to compute $\Phi = \int \vec{E} \cdot d\vec{A}$ directly, setting up a double integral over the square's surface. This is messy because the electric field from a point charge varies in both magnitude and direction across the square.

Why it's wrong:

The integration is unnecessarily complex. The electric field is not uniform over the square — its magnitude changes with distance from the charge, and its direction changes relative to the surface normal. This leads to a difficult integral that most students cannot evaluate correctly.

How to avoid:

Use the hint in the problem. Place the square as one face of a cube of side $10\,\text{cm}$, with the charge at the cube's centre. By Gauss's law, the total flux through the entire cube is:

Since the charge is at the centre, the flux is equally distributed through all 6 faces. Therefore:

Mistake 2: Forgetting that the charge is not at the centre of the square

What students do wrong:

They assume the charge is at the centre of the square and use symmetry arguments incorrectly — for example, claiming the flux through the square is $\frac{q}{4\varepsilon_0}$ (as if the square were one face of a tetrahedron).

Why it's wrong:

The charge is 5 cm above the centre, not at the centre of the square itself. The square is only one face of an imaginary cube. The symmetry that works is the cubic symmetry — the charge is at the cube's centre, so all 6 faces are equivalent.

How to avoid:

Visualise the cube clearly. The square is the top face of a cube of side $10\,\text{cm}$, and the charge is at the cube's centre (5 cm below the top face). This makes all 6 faces symmetric with respect to the charge.

Mistake 3: Using the wrong value of $q$ or units

What students do wrong:

They forget to convert $10\,\mu\text{C}$ to SI units ($10 \times 10^{-6}\,\text{C}$) or use $10\,\text{cm}$ as $10\,\text{m}$ instead of $0.1\,\text{m}$.

Why it's wrong:

Gauss's law in SI form requires charge in coulombs and distances in metres. Using wrong units gives a numerically incorrect answer.

How to avoid:

Always convert to SI before plugging into formulas:

• $q = 10\,\mu\text{C} = 10 \times 10^{-6}\,\text{C} = 1.0 \times 10^{-5}\,\text{C}$
• Side of square = $10\,\text{cm} = 0.1\,\text{m}$

Mistake 4: Forgetting $\varepsilon_0$ or using the wrong value

What students do wrong:

They either omit $\varepsilon_0$ entirely or use $\varepsilon_0 = 8.85 \times 10^{-12}$ incorrectly (e.g., forgetting units).

Why it's wrong: …