Question

In a cuboid of dimension \(2L \times 2L \times L\), a charge q is placed at the center of the surface ‘S’ having area of 4 L². The flux through the opposite surface to ‘S’ is:

• A. \( \frac{q}{2\epsilon_0} \)
• B. \( \frac{q}{6\epsilon_0} \)
• C. \( \frac{q}{12\epsilon_0} \)
• D. \( \frac{q}{3\epsilon_0} \)

Hint:

For flux calculations:

• Use Gauss’s Law \( \Phi = \frac{q}{\epsilon_0} \) for symmetric surfaces.
• Divide the flux equally among faces if the charge is symmetrically placed.

Answer 1

The Correct Option is B

To solve this problem, we need to understand the concept of electric flux and how it relates to charges enclosed by surfaces. According to Gauss's law, the total electric flux \(\Phi\) through a closed surface is given by:

\(\Phi = \frac{Q_{\text{enclosed}}}{\epsilon_0}\)

where \(Q_{\text{enclosed}}\) is the total charge enclosed by the surface, and \(\epsilon_0\) is the permittivity of free space.

In this cuboid with dimensions \((2L \times 2L \times L)\), the charge \(q\) is placed at the center of a surface \(S\) of area \(4L^2\). This surface is one of the \(2L \times 2L\) surfaces of the cuboid.

The opposite surface to ‘S’ is another \(2L \times 2L\) surface, which is parallel to ‘S’.

Since the charge is placed at the center of the surface 'S', it will influence the flux distribution through the opposite surface as follows:

• The total flux through the entire closed surface of the cuboid is \(\frac{q}{\epsilon_0}\) according to Gauss's law.
• Due to the symmetry of the charge placement, the flux will distribute evenly through each of the three pairs of opposite surfaces. Each pair of surfaces contributes equally to the total flux.
• Therefore, the flux through each pair of opposite surfaces will be one-third of the total flux.
• The flux through each single surface in a pair will be half of that pair's contribution.

So, the flux through the opposite surface to ‘S’ is:

\(\Phi_{\text{opposite to S}} = \frac{1}{3} \times \frac{q}{\epsilon_0} \div 2 = \frac{q}{6\epsilon_0}\)

Hence, the correct answer is \(\frac{q}{6\epsilon_0}\).