Question

Electric field in a region is given by following equation
$E = \frac{K(\vec{r}-\vec{r_0})}{|\vec{r}-\vec{r_0}|^2}$ where $\vec{r} = x\hat{i} + y\hat{j}$ and $\vec{r_0} = \hat{i} + \hat{j}$ and an imaginary cube is made as shown in the diagram.

1. Find total flux through this cube due to the given electric field.

• A. 4πK
• B. 6πK
• C. 16K
• D. 8K

Answer 1

Answer: (A)

4πK

Answer 2

Key idea:

• The field can be written in the form

$$E = K\,\frac{\mathbf{R}}{R^2},\quad \mathbf{R}=\mathbf{r}-\mathbf{r_0},$$

which is the field of a point−singularity located at $\mathbf{r_0}$.

• Away from $\mathbf{r_0}$, $\nabla\!\cdot\!E=0$.
• By Gauss’s law, the net flux through a closed surface that encloses the singularity is

$$\Phi = \oint E\cdot d\mathbf{A} = 4\pi K.$$

• Although $\mathbf{r_0}$ lies on the face, one treats it as enclosed for flux count.

Therefore, the total flux through the cube is

$$\boxed{4\pi K}.$$