Question

consider a charge placed at a distance a/4 from the centre of a cube of side length a. Find flux trough all the face of the cube

Answer 1

\frac{q}{\epsilon_0}

Answer 2

1. Identify the Gaussian Surface and Enclosed Charge:

The given surface is a cube of side length 'a'. The charge 'q' is placed at a distance a/4 from the center of the cube. The cube extends from $-a/2$ to $+a/2$ along each axis if its center is at the origin. Since $a/4 < a/2$, the charge 'q' is located entirely inside the cube.

2. Apply Gauss's Law:

Gauss's Law states that the total electric flux ($\Phi$) through any closed surface is equal to the net electric charge ($Q_{enclosed}$) enclosed within that surface divided by the permittivity of free space ($\epsilon_0$). The formula is:

$$\Phi = \frac{Q_{enclosed}}{\epsilon_0}$$

3. Calculate the Total Flux:

In this case, the entire charge 'q' is enclosed within the cube. Therefore, $Q_{enclosed} = q$. Substituting this into Gauss's Law:

$$\Phi_{total} = \frac{q}{\epsilon_0}$$

The question asks for the "flux through all the face of the cube", which refers to the total flux passing out of the entire cube.

The position of the charge inside the cube (a/4 from the center) does not change the total flux through the entire cube, as long as it remains enclosed. It would only affect the distribution of flux among the individual faces if one were to calculate flux through each face separately (which is not asked here).

The final answer is $\frac{q}{\epsilon_0}$.