Question 167: Electrostatics & Capacitance

Question

Two point charges $2 q$ and $q$ are placed at vertex $A$ and centre of face $C D E F$ of the cube as shown in figure. The electric flux passing through the cube is:

Accepted Answer

According to Gauss's Law, the total electric flux $\Phi$ through a closed surface is
$$ \Phi=\frac{Q_{	ext {enclosed }}}{\epsilon_0} $$
For the flux contribution from charge at vertex $\mathrm{A}(2 \mathrm{q})$
As the charge 2 q is placed at a corner (vertex) of the cube.
A corner of a cube is a meeting point for 8 identical cubes stacked together
Since the charge is shared equally among 8 cubes, the portion of the flux entering the specific cube is $\frac{1}{8}$ of the total flux produced by the charge.
$$ \Phi_{\mathrm{A}}=\frac{1}{8} 	imes \frac{2 \mathrm{q}}{\epsilon_0}=\frac{2 \mathrm{q}}{8 \epsilon_0}=\frac{\mathrm{q}}{4 \epsilon_0} $$
For the flux contribution from charge at the centre of face CDEF (q)
The charge q is placed at the center of the face CDEF and a face of a cube is shared by exactly 2 identical cubes placed side-by-side.
Since the charge is shared equally between 2 cubes, the portion of the flux entering the specific cube is $\frac{1}{2}$ of the total flux produced by the charge.
$$ \Phi_{	ext {face }}=\frac{1}{2} 	imes \frac{q}{\epsilon_0}=\frac{q}{2 \epsilon_0} $$
The total flux passing through the cube is the sum of the contributions from both charges.
$$ \Phi_{	ext {total }}=\Phi_{\mathrm{A}}+\Phi_{	ext {face }} $$
$$\Rightarrow $$ $\Phi_{	ext {total }}=\frac{\mathrm{q}}{4 \epsilon_0}+\frac{\mathrm{q}}{2 \epsilon_0}$
$$\Rightarrow $$ $$ \Phi_{	ext {total }}=\frac{q+2 q}{4 \epsilon_0}=\frac{3 q}{4 \epsilon_0} $$
Therefore, the total electric flux passing through the cube is $\frac{3 q}{4 \epsilon_0}$.
Hence, the correct option is (B).

Answer

$\frac{3 q}{2 \epsilon_0}$

Answer

$\frac{3 q}{4 \epsilon_0}$

Answer

$\frac{q}{\epsilon_0}$

Answer

$\frac{3 q}{\epsilon_0}$

Electrostatics & Capacitance

Two point charges 2 q and q are placed at vertex A and centre of face C D E F of the cube as shown in figure. The electric flux passing through the cube is:

Choose the correct answer: