Two small balls having equal positive charge Q (coulumb) on each suspended by two insulating strings of equal length L (metre) from a hook fixed to a stand. The whole set-up is taken in a satellite into space where there is no gravity (state of weightlessness). Then tension (newtons) in each string is:

Options

A. `(Q^(2))/(4piin_(0)L^(2))`
B. `(Q^(2))/(8piin_(0) L^(2))`
C. `(Q^(2))/(12 pi in_(0)L^(2))`
D. `(Q^(2))/(16 pi in_(0)L^(2))`

Explanation

To solve the problem, we will analyze the forces acting on the two charged balls when they are in a state of weightlessness in space.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two small balls, each with an equal positive charge \( Q \), suspended by insulating strings of equal length \( L \).
- The system is taken into space where there is no gravity (weightlessness).

2. **Identifying Forces**:
- In a gravitational field, the weight of each ball would exert a downward force \( mg \) (where \( m \) is the mass of the ball and \( g \) is the acceleration due to gravity).
- However, in the absence of gravity, the weight \( mg = 0 \).

3. **Electrostatic Force**:
- The only force acting on the balls is the electrostatic force due to their charges.
- Since both balls have the same charge \( Q \), they will repel each other with a force given by Coulomb's Law:
\[ F = \frac{k \cdot Q^2}{r^2} \]
- Here, \( k \) is Coulomb's constant, \( r \) is the distance between the two charges.

4. **Distance Between Charges**:
- When the balls repel each other to the maximum extent, the distance \( r \) between them will be \( 2L \) (since each string has length \( L \)).

5. **Calculating the Electrostatic Force**:
- Substituting \( r = 2L \) into the formula for electrostatic force:
\[ F = \frac{k \cdot Q^2}{(2L)^2} = \frac{k \cdot Q^2}{4L^2} \]

6. **Tension in the Strings**:
- In the state of weightlessness, the tension \( T \) in each string will equal the electrostatic force acting between the two charges:
\[ T = \frac{k \cdot Q^2}{4L^2} \]

7. **Substituting the Value of \( k \)**:
- The value of \( k \) is given by:
\[ k = \frac{1}{4\pi \epsilon_0} \]
- Therefore, substituting this into the equation for tension:
\[ T = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q^2}{4L^2} \]
- Simplifying this gives:
\[ T = \frac{Q^2}{16\pi \epsilon_0 L^2} \]

8. **Final Result**:
- The tension in each string is:
\[ T = \frac{Q^2}{16\pi \epsilon_0 L^2} \]