In a hypothetical system , a partical of mass `m` and charge `-3 q` is moving around a very heavy partical chaRGE `q`. Assume that Bohr's model is applicable to this system , then velocuity of mass `m` in the first orbit is

Options

A. `(3 q^(2))/(2 epsilon_(0) h)`
B. `(3 q^(2))/(4 epsilon_(0) h)`
C. `(3 q)/(2 pi epsilon_(0) h)`
D. `(3 q^(2))/(4 pi epsilon_(0) h)`

Explanation

To find the velocity of a particle of mass `m` and charge `-3q` moving around a heavy particle with charge `q` in the first orbit using Bohr's model, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces**:
The particle of mass `m` and charge `-3q` is experiencing a centripetal force due to the electrostatic attraction between it and the heavy particle of charge `q`. The centripetal force can be expressed as:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the velocity of the particle and \( r \) is the radius of the orbit.

2. **Calculate the Electrostatic Force**:
The electrostatic force \( F_e \) between the two charges is given by Coulomb's law:
\[ F_e = \frac{|(-3q) \cdot q|}{4\pi \epsilon_0 r^2} = \frac{3q^2}{4\pi \epsilon_0 r^2} \]

3. **Set the Forces Equal**:
For the particle to remain in a stable orbit, the centripetal force must equal the electrostatic force:
\[ \frac{mv^2}{r} = \frac{3q^2}{4\pi \epsilon_0 r^2} \]

4. **Rearranging the Equation**:
Rearranging the above equation to solve for \( v^2 \):
\[ mv^2 = \frac{3q^2}{4\pi \epsilon_0 r} \]
\[ v^2 = \frac{3q^2}{4\pi \epsilon_0 mr} \]

5. **Using Bohr's Quantization Condition**:
According to Bohr's model, the angular momentum \( L \) of the particle is quantized and given by:
\[ L = mvr = n\hbar \]
For the first orbit, \( n = 1 \):
\[ mvr = \hbar \]
Therefore, we can express \( r \) in terms of \( v \):
\[ r = \frac{\hbar}{mv} \]

6. **Substituting for \( r \)**:
Substitute \( r \) back into the equation for \( v^2 \):
\[ v^2 = \frac{3q^2}{4\pi \epsilon_0 m \left(\frac{\hbar}{mv}\right)} \]
Simplifying this gives:
\[ v^2 = \frac{3q^2 v}{4\pi \epsilon_0 \hbar} \]
Rearranging gives:
\[ v = \frac{3q^2}{4\pi \epsilon_0 \hbar} \]

7. **Final Expression**:
Thus, the velocity of the mass \( m \) in the first orbit is:
\[ v = \frac{3q^2}{2\pi \epsilon_0 h} \]