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 \(x\)
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 \(x\)
Approach Solution - 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:
\[ \boxed{x = 5} \]
Approach Solution -2
The Correct answer is : 5
From Gauss law
\(\phi=\frac{q_{enclosed}}{ε_0}=\frac{5q}{ε_0}\)
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Concepts Used:
Electric Charge
What is Electric Charge
It is the property of subatomic particles that experiences a force when put in an electric and magnetic field. It is are of two types: Positive and Negative. It commonly carried by charge carriers protons and electrons.
Properties of Electric Charge
Various properties of charge include the following :-
- Additivity of Electric Charge
- Conservation of Electric Charge
- Quantization of Electric Charge
Types of electric charge
Two kinds of electric charges are there :-
Negative Charge - When an object has a negative charge it means that it has more electrons than protons.
Positive Charge - When an object has a positive charge it means that it has more protons than electrons.
When there is an identical number of positive and negative charges, the negative and positive charges would cancel out each other and the object would become neutral.