Two equal and opposite charges of masses `m_(1)` and `m_(2)` are accelerated in a uniform electric field through the same distance. What is the ratio of their accelerations, if their ratio of masses is `m_(1)/m_(2)=0.5`?

Options

A. `a_(1)/a_(2)=0.5`
B. `a_(1)/a_(2)=1`
C. `a_(1)/a_(2)=2`
D. `a_(1)/a_(2)=3`

Answer

To solve the problem, we need to find the ratio of the accelerations of two equal and opposite charges with given masses \( m_1 \) and \( m_2 \), where the ratio of their masses is \( \frac{m_1}{m_2} = 0.5 \).

### Step-by-Step Solution:

1. **Understanding the Forces**:
The force acting on a charged particle in an electric field \( E \) is given by:
\[ F = Q \cdot E \]
where \( Q \) is the charge. For the two charges, we can denote them as \( Q_1 = +Q \) and \( Q_2 = -Q \).

2. **Applying Newton's Second Law**:
According to Newton's second law, the force can also be expressed as:
\[ F = m \cdot a \]
Therefore, for the first charge (mass \( m_1 \)):
\[ F_1 = m_1 \cdot a_1 = Q \cdot E \]
And for the second charge (mass \( m_2 \)):
\[ F_2 = m_2 \cdot a_2 = -Q \cdot E \]

3. **Setting Up the Equations**:
From the equations above, we can express the accelerations \( a_1 \) and \( a_2 \):
\[ a_1 = \frac{Q \cdot E}{m_1} \]
\[ a_2 = \frac{-Q \cdot E}{m_2} \]

4. **Finding the Ratio of Accelerations**:
To find the ratio \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{\frac{Q \cdot E}{m_1}}{\frac{-Q \cdot E}{m_2}} = \frac{m_2}{m_1} \]
The negative sign indicates the direction of acceleration is opposite, but for the ratio, we are interested in the magnitudes.

5. **Using the Given Mass Ratio**:
We know from the problem that:
\[ \frac{m_1}{m_2} = 0.5 \implies m_2 = \frac{m_1}{0.5} = 2m_1 \]
Therefore:
\[ \frac{m_2}{m_1} = 2 \]

6. **Final Calculation**:
Substituting back into the ratio of accelerations:
\[ \frac{a_1}{a_2} = \frac{m_2}{m_1} = 2 \]

### Conclusion:
The ratio of the accelerations \( \frac{a_1}{a_2} \) is \( 2 \).