Question

A point charge \(q\) is placed at a distance \(a/2\) directly above the center of a square of side \(a\). The electric flux through the square is:

• A. \(\frac{q}{\varepsilon_0}\)
• B. \(\frac{q}{6\varepsilon_0}\)
• C. \(\frac{q}{4\varepsilon_0}\)
• D. \(\frac{q}{2\varepsilon_0}\)

Hint:

Whenever a charge is placed at a perpendicular distance \(a/2\) from the center of a square of side \(a\), always construct a cube around it to use symmetry. The flux through the single face will be \(1/6\) of the total flux.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the electric flux through a square of side \(a\) due to a point charge \(q\) placed at a distance \(a/2\) directly above its geometric center.

Step 2: Key Formula or Approach:
We use Gauss's Law, which states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space:
\[ \Phi_{\text{total}} = \frac{q_{\text{enc}}}{\varepsilon_0} \] To apply this, we construct a symmetric Gaussian surface around the charge.

Step 3: Detailed Explanation:
Imagine the given square as one of the six identical faces of a cube of side length \(a\).
If the square lies on the \(xy\)-plane centered at the origin, the charge is located at \((0, 0, a/2)\).
This position corresponds exactly to the geometric center of a cube of side \(a\) that rests on the given square.
According to Gauss's Law, the total electric flux passing through all six faces of this entire cube is:
\[ \Phi_{\text{total}} = \frac{q}{\varepsilon_0} \] Due to spatial symmetry, the point charge is equidistant from all six faces of the cube. Therefore, the electric flux is distributed equally among all six identical square faces.
The flux through the single given square face is:
\[ \Phi_{\text{face}} = \frac{1}{6} \Phi_{\text{total}} = \frac{q}{6\varepsilon_0} \]

Step 4: Final Answer:
The correct choice is (B).