An electron and a proton are in uniform electic field. The ratio of their acceleration will be

Options

A. Zero
B. One
C. Ratio of mass of proton to that of electron
D. Ratio of mass of electron to that of proton

Answer

To find the ratio of the accelerations of an electron and a proton in a uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces on the Particles**:
- In a uniform electric field \( E \), a charged particle experiences a force given by \( F = qE \), where \( q \) is the charge of the particle.
- For an electron (charge \( -e \)), the force is \( F_e = -eE \).
- For a proton (charge \( +e \)), the force is \( F_p = eE \).

2. **Applying Newton's Second Law**:
- According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration: \( F = ma \).
- For the electron, we have:
\[ F_e = m_e a_e \quad \Rightarrow \quad -eE = m_e a_e \]
- For the proton, we have:
\[ F_p = m_p a_p \quad \Rightarrow \quad eE = m_p a_p \]

3. **Finding the Accelerations**:
- Rearranging the equations for acceleration, we get:
\[ a_e = \frac{-eE}{m_e} \quad \text{(for the electron)} \]
\[ a_p = \frac{eE}{m_p} \quad \text{(for the proton)} \]

4. **Calculating the Ratio of Accelerations**:
- We want to find the ratio of the accelerations \( \frac{a_e}{a_p} \):
\[ \frac{a_e}{a_p} = \frac{\frac{-eE}{m_e}}{\frac{eE}{m_p}} = \frac{-eE \cdot m_p}{eE \cdot m_e} \]
- The \( eE \) terms cancel out:
\[ \frac{a_e}{a_p} = \frac{-m_p}{m_e} \]

5. **Considering Magnitudes**:
- Since we are interested in the ratio of magnitudes of acceleration, we can ignore the negative sign:
\[ \frac{|a_e|}{|a_p|} = \frac{m_p}{m_e} \]

6. **Final Result**:
- The ratio of the accelerations of the electron and the proton is:
\[ \frac{a_e}{a_p} = \frac{m_p}{m_e} \]