Question

Three identical charged balls each of charge \(2\) \(C\) are suspended from a common point \(P\) by silk threads of \(2\) \(m\) each (as shown in figure). They form an equilateral triangle of side \(1\) \(m\). The ratio of net force on a charged ball to the force between any two charged balls will be:

• A. \(1:1\)
• B. \(1:4\)
• C. \(\sqrt3:2\)
• D. \(\sqrt3:1\)

Answer 1

The Correct Option is D

The problem involves three identical charged balls suspended from a common point, forming an equilateral triangle. We need to find the ratio of the net force on a charged ball to the force between any two charged balls.

Step-by-step Solution 

1. Each ball has a charge of \(2\, C\) and they form an equilateral triangle with sides of \(1\,m\). The distance between any two charged balls, therefore, is \(1\,m\).
2. The force between two charged balls can be calculated using Coulomb's Law: 

\[F = \frac{k \cdot q_1 \cdot q_2}{r^2}\]

1.  where \(q_1 = q_2 = 2\,C\) and \(r = 1\,m\).
2. Substituting the values: 

\[F = \frac{9 \times 10^9 \cdot 2 \cdot 2}{1^2} = 36 \times 10^9 \, N\]

1. Now, consider the net force on one of the balls due to the other two balls. Since the arrangement is symmetrical and the charges are identical, the net force vector-wise forms at \(60^\circ\) between pairs.
2. Each force is equal in magnitude and makes an angle of \(60^\circ\) because of the equilateral triangle. We use vector addition to find the net force: 

\[F_{\text{net}} = F \cdot \sqrt{3}\]

1. Thus, the ratio of the net force on a charged ball to the force between any two charged balls is: 

\[\frac{F_{\text{net}}}{F} = \sqrt{3} \] \] So, the ratio\]

Conclusion

Therefore, the correct option is \(\sqrt{3}:1\).