Two small balls having equal positive charge Q C on each are suspended by two insulating strings of equal length L metre, from a hook fixed to a stand. The whole setup is taken into space where there is no gravity (state of weightlessness). Then the angle `theta` between the two strings is

Options

A. `0^(@)`
B. `90^(@)`
C. `180^(@)`
D. `0^(@) lt theta lt 180^(@)`

Answer

To solve the problem, we need to analyze the forces acting on the two charged balls when they are suspended in a weightless environment.

### Step-by-step Solution:

1. **Understanding the Setup**:
- We have two small balls, each with a positive charge \( Q \), suspended by insulating strings of equal length \( L \).
- In a weightless environment, the gravitational force acting on the balls is negligible.

2. **Forces Acting on the Balls**:
- In the absence of gravity, the only force acting on each ball is the electrostatic repulsion between the two positively charged balls.
- According to Coulomb's law, the force \( F \) between the two charges is given by:
\[ F = \frac{k \cdot Q^2}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance between the two charges.

3. **Geometry of the Setup**:
- Let the angle between the two strings be \( \theta \).
- The distance \( r \) between the two balls can be expressed in terms of \( L \) and \( \theta \):
\[ r = 2L \sin\left(\frac{\theta}{2}\right) \]

4. **Equilibrium Condition**:
- In the weightless condition, the electrostatic force \( F \) must balance the tension in the strings.
- The tension \( T \) in each string can be resolved into two components: one that acts vertically (which is zero in this case) and one that acts horizontally (which provides the necessary centripetal force).
- Since there is no vertical force, we can focus on the horizontal component:
\[ T \sin\left(\frac{\theta}{2}\right) = F \]

5. **Substituting for \( F \)**:
- From Coulomb's law:
\[ T \sin\left(\frac{\theta}{2}\right) = \frac{k \cdot Q^2}{(2L \sin\left(\frac{\theta}{2}\right))^2} \]

6. **Solving for \( \theta \)**:
- Since the tension \( T \) is not acting vertically, we can simplify the equation:
\[ T \sin\left(\frac{\theta}{2}\right) = \frac{k \cdot Q^2}{4L^2 \sin^2\left(\frac{\theta}{2}\right)} \]
- Rearranging gives:
\[ T = \frac{k \cdot Q^2}{4L^2 \sin\left(\frac{\theta}{2}\right)} \]
- In the absence of gravity, the strings will align themselves to minimize potential energy, leading to a configuration where the angle \( \theta \) approaches \( 180^\circ \).

7. **Conclusion**:
- Therefore, the angle \( \theta \) between the two strings in a weightless environment is:
\[ \theta = 180^\circ \]

### Final Answer:
The angle \( \theta \) between the two strings is \( 180^\circ \).