If two balls of given masses and charges are released, which of the following is the correct arrangement in equilibrium?

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the correct arrangement of two charged balls in equilibrium, we can follow these steps:

### Step 1: Understand the Forces Acting on the Balls
When two balls with charges and masses are released, they will experience gravitational force (weight) acting downwards and electrostatic force due to their charges acting horizontally. The tension in the string will balance these forces.

### Step 2: Analyze the Forces on Each Ball
For each ball, we can break down the forces into components:
- For the first ball (mass = 2m, charge = q1):
- Vertical component: \( T \cos(\theta_1) = 2mg \)
- Horizontal component: \( T \sin(\theta_1) = F \) (where F is the electrostatic force)

- For the second ball (mass = m, charge = q2):
- Vertical component: \( T' \cos(\theta_2) = mg \)
- Horizontal component: \( T' \sin(\theta_2) = F \)

### Step 3: Set Up the Equations
From the vertical components, we can express the tension:
- For the first ball: \( T = \frac{2mg}{\cos(\theta_1)} \)
- For the second ball: \( T' = \frac{mg}{\cos(\theta_2)} \)

From the horizontal components, we can equate the forces:
- \( T \sin(\theta_1) = T' \sin(\theta_2) = F \)

### Step 4: Relate the Angles Using Tangent
Using the relationships derived from the tension equations:
- \( \tan(\theta_1) = \frac{F}{2mg} \)
- \( \tan(\theta_2) = \frac{F}{mg} \)

### Step 5: Compare the Angles
From the equations for tangent:
- Since \( \tan(\theta_2) = 2 \tan(\theta_1) \), it follows that:
- \( \tan(\theta_2) > \tan(\theta_1) \)
- Therefore, \( \theta_2 > \theta_1 \)

### Step 6: Evaluate the Given Options
Now we can evaluate the options provided in the question:
- If \( \theta_1 > \theta_2 \) is given, that is incorrect.
- If \( \theta_2 > \theta_1 \) is given, that is correct.
- If both angles are equal, that is incorrect.
- If both masses experience the same forces, that is correct.

### Conclusion
The correct arrangements in equilibrium are options C and D, where the angles and forces are consistent with our analysis.

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