An electron falls through a small distance in a uniform electric field of magnitude `2xx10^(4)NC^(-1)`. The direction of the field reversed keeping the magnitude unchanged and a proton falls through the same distance. The time of fall will be

Options

A. same in both cases
B. more in the case of an electron
C. more in thecase of proton
D. independent of charge

Answer

To solve the problem, we need to analyze the motion of an electron and a proton falling through the same distance in a uniform electric field. We will derive the time taken for each particle to fall and then compare them.

### Step 1: Understand the Forces Acting on the Particles
In an electric field \( E \), a charged particle experiences a force given by:
\[ F = qE \]
Where \( q \) is the charge of the particle. For an electron, \( q = -e \) (negative charge) and for a proton, \( q = +e \) (positive charge).

### Step 2: Calculate the Acceleration of Each Particle
Using Newton's second law, the acceleration \( a \) of each particle can be calculated as:
\[ F = ma \implies a = \frac{F}{m} \]
For the electron:
\[ a_e = \frac{-eE}{m_e} \]
For the proton:
\[ a_p = \frac{eE}{m_p} \]

### Step 3: Set Up the Equation of Motion
We can use the equation of motion for uniformly accelerated motion to find the time taken to fall through a distance \( h \):
\[ h = \frac{1}{2} a t^2 \]
Rearranging gives:
\[ t = \sqrt{\frac{2h}{a}} \]

### Step 4: Substitute the Accelerations
For the electron:
\[ t_e = \sqrt{\frac{2h}{a_e}} = \sqrt{\frac{2h}{\frac{-eE}{m_e}}} = \sqrt{\frac{-2hm_e}{eE}} \]
For the proton:
\[ t_p = \sqrt{\frac{2h}{a_p}} = \sqrt{\frac{2h}{\frac{eE}{m_p}}} = \sqrt{\frac{2hm_p}{eE}} \]

### Step 5: Compare the Times
Now we can find the ratio of the time taken by the electron to the time taken by the proton:
\[ \frac{t_e}{t_p} = \frac{\sqrt{\frac{-2hm_e}{eE}}}{\sqrt{\frac{2hm_p}{eE}}} = \sqrt{\frac{m_e}{m_p}} \]
Since the mass of the proton \( m_p \) is much greater than the mass of the electron \( m_e \), we have:
\[ t_e < t_p \]
This means the time taken by the proton to fall through the same distance is greater than that of the electron.

### Conclusion
The time of fall for the proton will be greater than that of the electron.

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