Question

A charge of $5\mu\text{C}$ is placed at the center of a spherical shell $S_1$ of radius $10\text{ cm}$. Now this system is enclosed inside another spherical shell $S_2$ of radius $20\text{ cm}$. The ratio of the electrical flux through the surface $S_2$ to $S_1$ is:

• A. $1:1$
• B. $4:1$
• C. $2:1$
• D. $1:2$

Hint:

Electric flux measures the total number of electric field lines passing through a surface. Every single field line originating from the central charge that cuts through the inner shell $S_1$ must continue outward and cut through the outer shell $S_2$ as well. Since the line count is identical, the flux ratio must be $1:1$.

Answer 1

The Correct Option is A

Concept: According to Gauss's Law, the total net electric flux ($\Phi$) passing outwards through any closed Gaussian boundary surface depends solely on the net total enclosed electrical charge ($q_{\text{enclosed}}$): \[ \Phi = \frac{q_{\text{enclosed}}}{\varepsilon_0} \] This fundamental principle states that the geometry, radius, surface area, or shape of the enclosing boundary has no effect on the total flux passing through it.

Step 1: Analyze the charge enclosed by each individual shell boundary.

• For shell surface $S_1$: The enclosed point charge is $5\mu\text{C}$. Thus, $\Phi_1 = \frac{5\mu\text{C}}{\varepsilon_0}$.
• For shell surface $S_2$: Since $S_2$ surrounds the entire $S_1$ setup, the total charge contained inside its perimeter remains exactly the same, $5\mu\text{C}$. Thus, $\Phi_2 = \frac{5\mu\text{C}}{\varepsilon_0}$.

Step 2: Determine the flux ratio.
\[ \text{Ratio} = \frac{\Phi_2}{\Phi_1} = \frac{5\mu\text{C} / \varepsilon_0}{5\mu\text{C} / \varepsilon_0} = \frac{1}{1} \implies 1:1 \]