Question

Two pith balls carrying equal charges are suspended from a common point by strings of equal length, the equilibrium separation between them is 7. Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now become

• A. $(\frac{1}{\sqrt{2}})^2$
• B. $(\frac{r}{^3\sqrt{2}})$
• C. $\frac{2r}{\sqrt3}$
• D. $\frac{2r}{3}$

Answer 1

The Correct Option is B

 To solve this problem, we need to understand the forces acting on the charged pith balls and how the equilibrium position changes when the strings' length and angle with the vertical are altered.

Initially, the pith balls are suspended in such a way that they reach an equilibrium position where the separation between them is \( r \). The key forces at play here are the gravitational force, the tension in the strings, and the electrostatic repulsive force between the charges.

1. The electrostatic force \(F_e\) between two charges \(q\) is given by Coulomb's law: \(F_e = \frac{k q^2}{r^2}\) where \(k\) is Coulomb's constant and \(r\) is the separation between the charges.
2. The gravitational force \(F_g\) acts vertically downward on each ball and is given by: \(F_g = m g\) where \(m\) is the mass of the pith ball and \(g\) is the acceleration due to gravity.
3. The tension in the string \(T\) has two components: a vertical component that balances the gravitational force, and a horizontal component that balances the electrostatic repulsive force.
4. When the strings are clamped at half the height, the vertical component of the tension remains the same as it still has to balance the weight of the pith ball. However, the horizontal separation changes. Considering the geometry, if the original height to length of string ratio was halved, the horizontal separation changes accordingly.
5. In this new setup, the new horizontal separation \(r'\) can be derived from the principle of similar triangles: \(r' = \frac{r}{\sqrt[3]{2}}\) This is because the angle between the string and the vertical will effectively remain the same due to the symmetry and nature of force balance as the vertical component equals the gravitational force, which is unchanged. The horizontal separation is determined by the aspect ratio defined by the geometry of the setup, affected by cubed root because the system experiences a three-dimensional spatial change due to the new angle of strung length.

Therefore, the equilibrium separation between the pith balls with the strings clamped at half their original height becomes:

\(r' = \frac{r}{\sqrt[3]{2}}\)

This corresponds to the correct answer:

$(\frac{r}{^3\sqrt{2}})$

.