A proton and an `alpha`-particle start from rest in a uniform electric field, then the ratio of times of flight to travel same distance in the field is

Options

A. `sqrt5: sqrt2`
B. `sqrt3 : 1`
C. `2 : 1`
D. `1: sqrt2`

Answer

To solve the problem of finding the ratio of times of flight for a proton and an alpha particle traveling the same distance in a uniform electric field, we can follow these steps:

### Step 1: Understand the forces acting on the particles
Both the proton and the alpha particle will experience an electric force due to the electric field. The force \( F \) on a charged particle in an electric field \( E \) is given by:
\[ F = qE \]
where \( q \) is the charge of the particle.

### Step 2: Determine the charges and masses
- For the proton:
- Charge \( q_p = e \) (where \( e \) is the elementary charge)
- Mass \( m_p = m_p \) (mass of the proton)

- For the alpha particle (which consists of 2 protons and 2 neutrons):
- Charge \( q_\alpha = 2e \)
- Mass \( m_\alpha = 4m_p \)

### Step 3: Calculate the acceleration of each particle
Using Newton's second law, \( F = ma \), we can express the acceleration \( a \) of each particle:
- For the proton:
\[ a_p = \frac{F}{m_p} = \frac{q_p E}{m_p} = \frac{eE}{m_p} \]

- For the alpha particle:
\[ a_\alpha = \frac{F}{m_\alpha} = \frac{q_\alpha E}{m_\alpha} = \frac{2eE}{4m_p} = \frac{eE}{2m_p} \]

### Step 4: Use the kinematic equation to relate distance, acceleration, and time
The equation for displacement \( s \) when starting from rest is:
\[ s = \frac{1}{2} a t^2 \]
For the proton:
\[ s = \frac{1}{2} a_p t_p^2 = \frac{1}{2} \left(\frac{eE}{m_p}\right) t_p^2 \]

For the alpha particle:
\[ s = \frac{1}{2} a_\alpha t_\alpha^2 = \frac{1}{2} \left(\frac{eE}{2m_p}\right) t_\alpha^2 \]

### Step 5: Set the displacements equal to each other
Since both particles travel the same distance \( s \):
\[ \frac{1}{2} \left(\frac{eE}{m_p}\right) t_p^2 = \frac{1}{2} \left(\frac{eE}{2m_p}\right) t_\alpha^2 \]

### Step 6: Simplify the equation
Cancel \( \frac{1}{2} \) and \( \frac{eE}{m_p} \) from both sides:
\[ t_p^2 = \frac{1}{2} t_\alpha^2 \]

### Step 7: Find the ratio of the times
Taking the square root of both sides gives:
\[ \frac{t_p}{t_\alpha} = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the ratio of the times of flight for the proton and the alpha particle is:
\[ \frac{t_p}{t_\alpha} = \frac{1}{\sqrt{2}} \]