Question

A long string with a charge of \(\lambda\) per unit length passes through an imaginary cube of edge \(\ell\). The maximum possible flux of the electric field through the cube will be ....... (A) \(\sqrt{3}\left(\lambda \ell / \in_{0}\right)\) (B) \(\left(\lambda \ell / \in_{0}\right)\) (C) \(\sqrt{2}\left(\lambda \ell / \in_{0}\right)\) (D) \(\left[\left(6 \lambda \ell^{2}\right) / \epsilon_{0}\right]\)

Short answer

The short answer is: \(\boxed{\text{(A)}\ \sqrt{3}\left(\frac{\lambda \ell}{\epsilon_{0}}\right)}\).

Step-by-step solution

Apply Gauss's law and note the electric field symmetry

Since we are dealing with electric flux through a closed surface (cube), we can use Gauss's law for electrostatics: \[\oint \vec{E} \cdot d\vec{A} = \frac{Q_{encl}}{\epsilon_0}\] Where: - \(\vec{E}\) is the electric field vector. - \(d\vec{A}\) is the differential area vector enclosing the closed surface. - \(Q_{encl}\) is the total charge enclosed within the closed surface (cube). - \(\epsilon_0\) is the permittivity of free space (electric constant). In our case, the electric field \(\vec{E}\) is due to the charged string, and the surface is the cube of edge length ℓ. As we are interested in the maximum possible electric flux, we should place the string along a space diagonal of the cube.

Determine the total enclosed charge

To find the total charge enclosed within the cube, we multiply the charge per unit length λ by the length of the string inside the cube (along the space diagonal): \[Q_{encl} = \lambda d\] Where \(d = \sqrt{3}\ell\) is the length of the space diagonal of the cube. Therefore, the enclosed charge is: \[Q_{encl} = \lambda \sqrt{3}\ell\]

Calculate the maximum electric flux

Now we can calculate the maximum electric flux (Φ) using Gauss's law: \[\Phi = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{encl}}{\epsilon_0}\] Substituting the value for \(Q_{encl}\): \[\Phi = \frac{\lambda \sqrt{3}\ell}{\epsilon_0}\] Comparing this result with the given options, we find that the maximum possible electric flux through the cube is: \(\boxed{\text{(A)}\ \sqrt{3}\left(\frac{\lambda \ell}{\epsilon_{0}}\right)}\)