**Equilibrium Separation between Pith Balls**

When two pith balls carrying equal charges are suspended from a common point by strings of equal length, they will reach an equilibrium separation, denoted as "r". This means that the electrostatic forces between the charges on the pith balls and the tension in the strings are balanced, resulting in a stable configuration.

**Effect of Rigid Clamping at Half the Height**

When the strings are rigidly clamped at half the height, the equilibrium separation between the pith balls will change. Let's analyze this situation step by step:

1. **Initial Configuration:** In the initial configuration, the pith balls are suspended from a common point by strings of equal length. The electrostatic forces between the charges on the pith balls and the tension in the strings are balanced, resulting in an equilibrium separation of "r".

2. **Rigid Clamping:** When the strings are rigidly clamped at half the height, the system becomes a rigid triangle. The strings are no longer free to move, and the pith balls are constrained to stay at a fixed distance from the clamping point.

3. **Change in Equilibrium Separation:** Due to the rigid clamping, the equilibrium separation between the pith balls will change. The new equilibrium separation, denoted as "r'", can be determined by considering the geometry of the rigid triangle formed by the clamping point and the pith balls.

4. **Determination of r':** In the new configuration, the clamping point divides the string into two equal segments. Let's consider one of the pith balls. The distance between the clamping point and the pith ball is now half of the original length of the string. Therefore, the new equilibrium separation, r', is equal to r/2.

5. **Explanation:** The rigid clamping at half the height reduces the effective length of the string supporting the pith ball. This reduction in length leads to a decrease in the equilibrium separation between the pith balls. As a result, the pith balls will be closer to each other compared to the initial configuration.

In summary, when the strings suspending the pith balls are rigidly clamped at half the height, the equilibrium separation between the balls becomes half of the initial separation. Therefore, the new equilibrium separation, denoted as "r'", is equal to r/2.

R/4