Question

An infinitely long wire has uniform linear charge density $ \lambda = 2 \, \text{nC/m} $. The net flux through a Gaussian cube of side length $ \sqrt{3} \, \text{cm} $, if the wire passes through any two corners of the cube, that are maximally displaced from each other, would be $ x \, \text{Nm}^2 \text{C}^{-1} $, where $ x $ is:

• A. \( 2.16 \pi \)
• B. \( 0.72 \pi \)
• C. \( 6.48 \pi \)
• D. \( 1.44 \pi \)

Hint:

When calculating electric flux using Gauss's law, remember that the total flux depends on the charge enclosed by the Gaussian surface and the symmetry of the situation. For a wire passing through the cube, the flux is proportional to the length of the wire inside the cube.

Answer 1

The Correct Option is A

To resolve the issue, Gauss's law will be applied. This law asserts that the electric flux \( \Phi \) passing through a closed surface is equivalent to the enclosed charge divided by the permittivity of free space \( \epsilon_0 \):

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

Given a linear charge density \( \lambda \) of \( 2 \, \text{nC/m} \) or \( 2 \times 10^{-9} \, \text{C/m} \), and a wire traversing diagonally opposite corners of a cube with a side length of \( \sqrt{3} \, \text{cm} \), the total charge enclosed by the cube can be determined. The length of the cube's diagonal, through which the wire passes, is \( \sqrt{3} \, \text{cm} \), representing the distance within the cube.

The length of the wire segment within the cube equals the cube's side length: \( \sqrt{3} \, \text{cm} \), which is \( 0.03 \, \text{m} \).

The enclosed charge \( Q_{\text{enc}} \) within the cube is computed as:

\(Q_{\text{enc}} = \lambda \times l = 2 \times 10^{-9} \, \text{C/m} \times 0.03 \, \text{m} = 6 \times 10^{-11} \, \text{C}\)

The electric flux is then calculated using Gauss's law:

\(\Phi = \frac{Q_{\text{enc}}}{\epsilon_0} = \frac{6 \times 10^{-11} \, \text{C}}{8.85 \times 10^{-12} \, \text{C}^{2}/\text{Nm}^2}\)

The calculation yields:

\(\Phi = \frac{6 \times 10^{-11}}{8.85 \times 10^{-12}} \approx 6.78 \, \text{Nm}^2/\text{C}\)

Consequently, the flux is equivalent to \( 2.16\pi \, \text{Nm}^2/\text{C} \), corresponding to one of the provided options.

Therefore, the correct answer is \( 2.16\pi \).