Two particles are held in equilibrium by the gravitational and electrostatic forces between them. Particle-A has mass `m_(alpha)` and charge `q_(alpha) ` and particle-B has mass `m_(b)` and charge `q_(b)`. The distance between the charges may cause the chages to accelerate towards one another ?

Options

A. `m_(a)` is doubled and `m_(b)` is halved
B. `m_(a)` is doubled and `m_(b)` is doubled
C. `q_(a)` is doubled and `q_(b)` is doubled
D. `d` is doubled

Answer

To analyze the equilibrium of two charged particles held by gravitational and electrostatic forces, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Involved**:
- The two particles, A and B, experience gravitational force (\(F_G\)) and electrostatic force (\(F_E\)).
- The gravitational force is given by:
\[ F_G = \frac{G m_\alpha m_b}{d^2} \]
- The electrostatic force is given by:
\[ F_E = \frac{k |q_\alpha q_b|}{d^2} \]
- Here, \(G\) is the gravitational constant, \(k\) is Coulomb's constant, \(m_\alpha\) and \(m_b\) are the masses of particles A and B, respectively, \(q_\alpha\) and \(q_b\) are their charges, and \(d\) is the distance between them.

2. **Set the Forces Equal for Equilibrium**:
- For the particles to be in equilibrium, the magnitudes of the forces must be equal:
\[ F_G = F_E \]
- This leads to the equation:
\[ \frac{G m_\alpha m_b}{d^2} = \frac{k |q_\alpha q_b|}{d^2} \]
- The \(d^2\) cancels out, simplifying to:
\[ G m_\alpha m_b = k |q_\alpha q_b| \]

3. **Consider Changes in Mass and Charge**:
- We need to analyze how changes in mass and charge affect the forces.
- If we change the masses or charges, we can determine if the balance is disturbed.

4. **Evaluate Each Option**:
- **Option A**: If \(m_\alpha\) is doubled and \(m_b\) is halved:
- New gravitational force:
\[ F'_G = \frac{G (2m_\alpha)(\frac{1}{2}m_b)}{d^2} = \frac{G m_\alpha m_b}{d^2} \]
- Electrostatic force remains unchanged. Thus, no acceleration occurs.
- **Option B**: If both \(m_\alpha\) and \(m_b\) are doubled:
- New gravitational force:
\[ F'_G = \frac{G (2m_\alpha)(2m_b)}{d^2} = 4 \frac{G m_\alpha m_b}{d^2} \]
- This will be greater than the electrostatic force, causing acceleration towards each other.
- **Option C**: If both \(q_\alpha\) and \(q_b\) are doubled:
- New electrostatic force:
\[ F'_E = \frac{k (2q_\alpha)(2q_b)}{d^2} = 4 \frac{k |q_\alpha q_b|}{d^2} \]
- This will not cause acceleration as both forces increase equally.
- **Option D**: If \(d\) is doubled:
- Both forces decrease by a factor of 4, maintaining equilibrium.

5. **Conclusion**:
- The only option that causes the charges to accelerate towards each other is **Option B** where both masses are doubled.