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

Procedure Step 1: Invoke Gauss's Law.

Gauss's Law states: \[ \Phi = \frac{q_{\text{enclosed}}}{\varepsilon_0} \] This indicates that the electric flux through any closed surface is solely determined by the total charge contained within that surface.

Step 2: Determine which charges are located within the sphere.

The sphere is centered at the origin and has a radius of \( 4a \). Consequently, its surface spans the region: \[ x = -4a \text{ to } x = +4a \]

Evaluate the provided charges:

• Charge \( -2q \) is situated at \( x = -5a \): This charge is external to the sphere (as \( |-5a|>4a \)).
• Charge \( 5q \) is located at \( x = +3a \): This charge is internal to the sphere (as \( |3a|<4a \)).

Therefore, the net charge enclosed by the sphere is: \[ q_{\text{enclosed}} = 5q \]

Step 3: Apply Gauss’s law.

\[ \Phi = \frac{q_{\text{enclosed}}}{\varepsilon_0} = \frac{5q}{\varepsilon_0} \]

By comparison with the given expression: \[ \Phi = \frac{xq}{\varepsilon_0} \]

It follows that: \[ x = 5 \]

Final Result:

\[ \boxed{x = 5} \]