Question

Two point charges \( 8 \, \mu C \) and \( -2 \, \mu C \) are located at \( x = 2 \, \text{cm} \) and \( x = 4 \, \text{cm} \), respectively on the x-axis. The ratio of electric flux due to these charges through two spheres of radii 3 cm and 5 cm with their centers at the origin is _______.

• A. \( 4 : 1 \)
• B. \( 3 : 4 \)
• C. \( 4 : 3 \)
• D. \( 4 : 5 \)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
According to Gauss's Law, the total electric flux through a closed surface is directly proportional to the net electric charge enclosed by that surface.
Charges outside the surface do not contribute to the net flux.
Step 2: Key Formula or Approach:
Gauss's Law: \(\Phi = \frac{q_{enclosed}}{\epsilon_0}\).
Identify which charges are located inside each of the given spheres.
Step 3: Detailed Explanation:
Let's analyze the first sphere:
Radius \(R_1 = 3 \text{ cm}\), centered at the origin.
The charge \(q_1 = 8\ \mu\text{C}\) is at \(x = 2 \text{ cm}\), which is inside the sphere (\(2<3\)).
The charge \(q_2 = -2\ \mu\text{C}\) is at \(x = 4 \text{ cm}\), which is outside the sphere (\(4>3\)).
So, the net charge enclosed by the first sphere is \(q_{encl,1} = 8\ \mu\text{C}\).
The flux through the first sphere is \(\Phi_1 = \frac{8\ \mu\text{C}}{\epsilon_0}\).
Now, let's analyze the second sphere:
Radius \(R_2 = 5 \text{ cm}\), centered at the origin.
Both the charge \(q_1 = 8\ \mu\text{C}\) (at \(x = 2 \text{ cm}\)) and the charge \(q_2 = -2\ \mu\text{C}\) (at \(x = 4 \text{ cm}\)) are inside this sphere (\(2<5\) and \(4<5\)).
So, the net charge enclosed by the second sphere is \(q_{encl,2} = 8\ \mu\text{C} + (-2\ \mu\text{C}) = 6\ \mu\text{C}\).
The flux through the second sphere is \(\Phi_2 = \frac{6\ \mu\text{C}}{\epsilon_0}\).
Finally, find the ratio of the two fluxes:
\[ \text{Ratio} = \frac{\Phi_1}{\Phi_2} = \frac{8\ \mu\text{C} / \epsilon_0}{6\ \mu\text{C} / \epsilon_0} = \frac{8}{6} = \frac{4}{3} \] Step 4: Final Answer:
The ratio of electric flux is \(4:3\).