There is a long straight wire, uniformly charged over length. A negatively charged point mass in ceiling around the wire under its electrostatic attraction. The linear speed of point mass is

Options

A. Inversely proportional to the radius of circle
B. Directly proportional to the radius of circle
C. Independent of the radius
D. Unpredictable due to insufficient information

Answer

To find the linear speed of a negatively charged point mass moving in a circular path around a long straight wire with uniform charge density, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a long straight wire with a uniform linear charge density \( \lambda \).
- A negatively charged point mass \( m \) with charge \( -q \) is moving in a circular path of radius \( d \) around this wire.

2. **Electric Field Due to the Wire**:
- The electric field \( E \) at a distance \( d \) from an infinitely long charged wire is given by:
\[ E = \frac{2k\lambda}{d} \]
- Here, \( k \) is Coulomb's constant.

3. **Force on the Point Mass**:
- The electrostatic force \( F_E \) acting on the negatively charged mass due to the electric field is:
\[ F_E = E \cdot |q| = \frac{2k\lambda}{d} \cdot q \]
- Since the charge is negative, the direction of the force will be opposite to the direction of the electric field.

4. **Centripetal Force Requirement**:
- For the point mass to move in a circular path, the electrostatic force must provide the necessary centripetal force:
\[ F_E = \frac{mv^2}{d} \]
- Here, \( v \) is the linear speed of the mass.

5. **Equating Forces**:
- Set the electrostatic force equal to the centripetal force:
\[ \frac{2k\lambda}{d} \cdot q = \frac{mv^2}{d} \]

6. **Cancel \( d \)**:
- Since \( d \) appears in both terms, we can cancel it out:
\[ 2k\lambda q = mv^2 \]

7. **Solve for \( v \)**:
- Rearranging the equation gives:
\[ v^2 = \frac{2k\lambda q}{m} \]
- Taking the square root yields:
\[ v = \sqrt{\frac{2k\lambda q}{m}} \]

8. **Conclusion**:
- The linear speed \( v \) of the point mass is independent of the radius \( d \) of the circular path.

### Final Expression:
\[ v = \sqrt{\frac{2k\lambda q}{m}} \]