Two particle of equal mass m and charge q are placed at a distance of 16 cm. They do not experience any force. The value of `(q)/(m)` is

Options

A. `l`
B. `sqrt((pi epsi_(0))/(G))`
C. `sqrt((G)/(4 pi epsi_(0)))`
D. `sqrt(4 pi epsi_(0)G)`

Answer

To solve the problem, we need to find the value of \( \frac{q}{m} \) given that two particles of equal mass \( m \) and charge \( q \) are placed at a distance of 16 cm and do not experience any force.

### Step-by-Step Solution:

1. **Understanding the Forces**:
Since the two charges do not experience any net force, the electrostatic force between them must be balanced by the gravitational force acting on them.

2. **Identifying the Forces**:
- The gravitational force \( F_g \) between the two masses is given by:
\[ F_g = \frac{G m^2}{r^2} \]
where \( G \) is the gravitational constant and \( r \) is the distance between the two masses.
- The electrostatic force \( F_e \) between the two charges is given by:
\[ F_e = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r^2} \]
where \( \epsilon_0 \) is the permittivity of free space.

3. **Setting the Forces Equal**:
Since the two forces are equal (as there is no net force), we can set them equal to each other:
\[ F_g = F_e \]
Thus,
\[ \frac{G m^2}{r^2} = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r^2} \]

4. **Simplifying the Equation**:
We can cancel \( r^2 \) from both sides (since \( r \neq 0 \)):
\[ G m^2 = \frac{1}{4 \pi \epsilon_0} q^2 \]

5. **Rearranging for \( \frac{q^2}{m^2} \)**:
Rearranging the equation gives:
\[ \frac{q^2}{m^2} = 4 \pi \epsilon_0 G \]

6. **Taking the Square Root**:
To find \( \frac{q}{m} \), we take the square root of both sides:
\[ \frac{q}{m} = \sqrt{4 \pi \epsilon_0 G} \]

### Final Answer:
Thus, the value of \( \frac{q}{m} \) is:
\[ \frac{q}{m} = \sqrt{4 \pi \epsilon_0 G} \]