Question

A solid sphere of radius \(4a\) units is placed with its centre at origin. Two charges \(-2q\) at \((-5a, 0)\) and \(5q\) at \((3a, 0)\) is placed. If the flux through the sphere is \(\frac{xq}{\in_0}\) , find \(x\)
 

Answer 1

Step 1: Use Gauss's Law.

According to Gauss's law: \[ \Phi = \frac{q_{\text{enclosed}}}{\varepsilon_0} \] That means the electric flux through any closed surface depends only on the net charge enclosed by that surface.

Step 2: Identify which charges are enclosed within the sphere.

The sphere is centered at the origin with radius \( 4a \), so its surface extends from: \[ x = -4a \text{ to } x = +4a \]

Now check the given charges:

• Charge \( -2q \) is at \( x = -5a \): outside the sphere (since \( |-5a| > 4a \)).
• Charge \( 5q \) is at \( x = +3a \): inside the sphere (since \( |3a| < 4a \)).

 

Hence, the net enclosed charge inside the sphere: \[ q_{\text{enclosed}} = 5q \]

Step 3: Apply Gauss’s law.

\[ \Phi = \frac{q_{\text{enclosed}}}{\varepsilon_0} = \frac{5q}{\varepsilon_0} \] So, comparing with given expression: \[ \Phi = \frac{xq}{\varepsilon_0} \] We get: \[ x = 5 \]

Final Answer:

\[ \boxed{x = 5} \]

Answer 2

The Correct answer is : 5

From Gauss law
\(\phi=\frac{q_{enclosed}}{ε_0}=\frac{5q}{ε_0}\)