Question

Find the electric flux through one face of a cube if a charge \(q\) is placed at its center.

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

Hint:

Think about the solid angle a face subtends at the charge, rather than dividing the total flux directly. Since the charge is at the centre, all six faces of the cube are equally placed around it, so the total solid angle around the charge splits equally among them.

Answer 1

The Correct Option is C

Concept: According to Gauss's Law, the total electric flux through a closed surface is given by: \[ \Phi_{\text{total}} = \frac{q}{\epsilon_0} \] If a charge is placed at the center of a cube, the flux distributes equally among its six identical faces.

Step 1: Write the total flux through the cube. \[ \Phi_{\text{total}} = \frac{q}{\epsilon_0} \]

Step 2: Divide the flux equally among the six faces. \[ \Phi_{\text{one face}} = \frac{1}{6} \times \frac{q}{\epsilon_0} \] \[ \Phi_{\text{one face}} = \frac{q}{6\epsilon_0} \] Thus, the electric flux through one face is: \[ \boxed{\frac{q}{6\epsilon_0}} \]

Answer 2

Concept:

• Electric flux through any surface can be expressed using the solid angle it subtends at the charge: $\Phi = \dfrac{q}{4\pi\epsilon_0} \times \Omega$, where $\Omega$ is the solid angle in steradians.
• A charge at the centre of a cube sees all of space around it, a total solid angle of $4\pi$ steradians, and by the symmetry of the cube this total is shared equally among its six identical faces.

Step 1: Write the total solid angle around the charge.
A point charge is surrounded on all sides, so the total solid angle is $4\pi$ steradians.

Step 2: Divide the solid angle equally among the six faces of the cube.
Since the charge sits exactly at the centre, each of the 6 faces subtends the same solid angle:
$\Omega_{one\ face} = \dfrac{4\pi}{6}$

Step 3: Substitute into the flux-solid angle relation.
$\Phi_{one\ face} = \dfrac{q}{4\pi\epsilon_0} \times \dfrac{4\pi}{6} = \dfrac{q}{6\epsilon_0}$

Final Answer: $\Phi_{one\ face} = \dfrac{q}{6\epsilon_0}$