Question

A uniformly charged cube of side one cm having surface charge density of $8.85\ \mu C\text{ cm}^{-2}$ is placed inside a hollow metal sphere. The total flux emerging out of the sphere in $\text{Nm}^{2}\text{C}^{-1}$ is ($\epsilon_{0}=8.85\times 10^{-12}\text{C}^{2}\text{N}^{-1}\text{m}^{-2}$)

• A. $8.85\times 10^{6}$
• B. $12\times 10^{6}$
• C. $19.7\times 10^{6}$
• D. $3\times 10^{6}$
• E. $6\times 10^{6}$

Hint:

Logic Tip: Notice how the units of $\sigma$ are given in $\mu C\text{ cm}^{-2}$ and the side is in $\text{cm}$. You don't need to convert the area to square meters because the $\text{cm}^2$ units naturally cancel out when calculating total charge!

Answer 1

Answer: (E)

Concept:
According to Gauss's Law, the total electric flux ($\Phi$) emerging from any closed surface is equal to the total enclosed charge ($q_{enc}$) divided by the permittivity of free space ($\epsilon_0$): $$\Phi = \frac{q_{enc}}{\epsilon_0}$$ The shape of the enclosing surface (the sphere) does not matter; only the total charge inside matters.
Step 1: Calculate the total surface area of the charged cube.
Side of the cube, $a = 1\text{ cm}$. A cube has 6 identical square faces. $$\text{Total Area} = 6 \times a^2 = 6 \times (1\text{ cm})^2 = 6\text{ cm}^2$$
Step 2: Calculate the total enclosed charge ($q_{enc}$).
Surface charge density, $\sigma = 8.85\ \mu C\text{ cm}^{-2} = 8.85 \times 10^{-6}\text{ C cm}^{-2}$. $$q_{enc} = \sigma \times \text{Total Area}$$ $$q_{enc} = (8.85 \times 10^{-6}\text{ C cm}^{-2}) \times (6\text{ cm}^2)$$ $$q_{enc} = 6 \times 8.85 \times 10^{-6}\text{ C}$$
Step 3: Calculate the total electric flux.
Substitute $q_{enc}$ and $\epsilon_0$ into Gauss's Law formula: $$\Phi = \frac{q_{enc}}{\epsilon_0}$$ $$\Phi = \frac{6 \times 8.85 \times 10^{-6}}{8.85 \times 10^{-12}}$$
Step 4: Simplify the expression.
The $8.85$ terms in the numerator and denominator cancel out perfectly: $$\Phi = \frac{6 \times 10^{-6}}{10^{-12}}$$ $$\Phi = 6 \times 10^{-6 - (-12)}$$ $$\Phi = 6 \times 10^6\text{ Nm}^2\text{C}^{-1}$$