An electron of mass `m` and charge `e` is accelerated from rest through a potential difference `V` in vacuum. The final speed of the electron will be

Options

A. `V sqrt(e//m)`
B. `sqrt(eV//m)`
C. `sqrt(2eV//m)`
D. `2eV//m`

Explanation

To find the final speed of an electron that has been accelerated from rest through a potential difference \( V \), we can use the concepts of energy conservation and kinetic energy. Here’s a step-by-step solution:

### Step 1: Understand the energy conversion
When the electron is accelerated through a potential difference \( V \), the electrical potential energy is converted into kinetic energy. The work done on the electron by the electric field is equal to the change in kinetic energy.

### Step 2: Write the expression for electrical potential energy
The work done \( W \) on the electron when it is accelerated through a potential difference \( V \) is given by:
\[ W = eV \]
where \( e \) is the charge of the electron.

### Step 3: Write the expression for kinetic energy
The kinetic energy \( KE \) of the electron when it reaches its final speed \( v \) is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron.

### Step 4: Set the work done equal to the kinetic energy
Since all the work done on the electron is converted into kinetic energy, we can set the two expressions equal to each other:
\[ eV = \frac{1}{2} mv^2 \]

### Step 5: Solve for the final speed \( v \)
Rearranging the equation to solve for \( v \), we get:
\[ v^2 = \frac{2eV}{m} \]
Taking the square root of both sides gives:
\[ v = \sqrt{\frac{2eV}{m}} \]

### Final Result
Thus, the final speed of the electron after being accelerated through a potential difference \( V \) is:
\[ v = \sqrt{\frac{2eV}{m}} \]